Diffusion Processes and their Sample Paths

CLASSICS IN MATHEMATICS

Kiyosi Iti)

Henry P. McKean, Jr.

Diffusion Processes

and their Sample Paths

Classics in Mathematics Kiyosi Ito Henry P. McKean, Jr.

Diffusion Processes and their Sample Paths

Springer Berlin Heidelberg New York

Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo

Kiyosi Ito Henry P. McKean, Jr.

Diffusion Processes and their Sample Paths Reprint of the 1974 Edition

Springer

Kiyosi It6

Henry P. McKean, Jr.

Kyoto University, RIMS Sakyo-Ku 606 Kyoto

New York University Courant Institute of Mathematical Sciences New York, NY 10012 USA

GAO

Japan

Originally published as Vol. 125 of the Grundlehren der mathematischen Wissenschaften

Cataloging-in-Publication Data applied for

Die Deutsche Bibliothek - CIP-Einheitsaufnahme Ito, Kiosi:

Diffusion processes and their sample paths / Kiosi ItO ; Henry P. McKean, Jr. - Reprint of the 1974 ed. - Berlin ; Heidelberg New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore ; Tokyo Springer, 1996 (Grundlehren der mathematischen Wissenschaften ; Vol. 125) (Classics in mathematics) ISBN 3-540-60629-7

NE: MacKean, Henry P.:; 1. GT

Mathematics Subject Classification (1991): 28A65, 60-02, 60G17, 60J60, 60J65, 60J70

ISBN 3-540-60629-7 Springer-Verlag Berlin Heidelberg New York

Eat

"'C

0.O

.0$

,00

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustration, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provision of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. m Springer-Verlag Berlin Heidelberg 1996

Punted in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. SPIN 10518453

41/3144- 5 4 3 2 10 - Printed on acid-free paper

K. Ito^

H. P. McKean, Jr.

Diffusion Processes and their Sample Paths

Second Printing, Corrected

Springer-Verlag

Berlin Heidelberg New York 1974

Kiyosi Ito Cornell University, Ithaca, N. Y. 14850, USA Henry P. McKean, Jr. Courant Institute of Mathematical Sciences, New York University, New York, USA

Geschaftsfuhrende Herausgeber B. Eckmann Eidgenossische Technische Hochschule Zurich

J. K. Moser Courant Institute of Mathematical Sciences, New York B. L. van der Waerden

Mathematisches Institut der Universtt t Zurich

AMS Subject Classifications (1970): 28 A 65, 60-02, 60 G 17, 60 J 60, 60 J 65, 60 J 70

ISBN 3-540-03302-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-03302-5 Springer-Verlag New York Heidelberg Berlin

'^'

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. Q by Springer-Verlag Berlin Heidelberg 1%5, 1974. Library of Congress Catalog Card Number 74-231. Printed in Germany. Offset printing: Julius Beltz, Hemsbach/Bergsu. Bookbinding Konrad Triltseh, Wurzburg. 2141/3140.54321

DEDICATED TO

P. LEVY WHOSE WORK HAS BEEN

OUR SPUR AND ADMIRATION

Computer-simulated molecular motions, reminiscent of 2-dimensional BROwNian motion. [From ALDER, B. J., and T. E. WAINWRIGHT: Molecular motions. Scientific

American 201, no. 4, 113-126 (1959)1.

Preface

.rte

ROBERT BROWN, an English botanist, observed (1828) that pollen grains suspended in water perform a continual swarming motion (see, 'L7

for example, D'ARCY THOMPSON [1: 73-771). .'.

0.°°

L. BACHELIER (1900) derived the law govering the position of a single grain performing a 1-dimensional BROwxian motion starting at

aER1 at time t=o: 1)

Pa[x(t) E db] = g(t, a, b) db

(t, a, b) E (0, + oo) x R',

C+1

where g is the source (GREEN) function

g(t, a, b) _

2)

of the problem of heat flow:

- 2 aa' au

(t>0).

I

A

au at

3)

yin t

BACHELIER also pointed out the MARKOVian nature of the BROWNian

path expressed in

P.[a]L Sx(t,)
4)

b,

bs

=f f ... as

a,


b an

A

0
5)

P. l at x(s) 5 bl = 2 r

J y- da

t > 0, b z 0

0

(see BACHELIER [1]).

A. EINSTEIN (1905) also derived 1) from statistical mechanical considerations and applied it to the determination of molecular diameters (see, for example, A. EINSTEIN [1]). BACHELIER was unable to obtain a clear picture of the BROwNian

motion and his ideas were unappreciated at the time; nor is this sur-

Preface

VIII

prising because the precise definition of the BROWNian motion involves a measure on the path space, and it was not until 1909 that E. BOREL published his classical memoir [1] on BERNOULLI trials. But as soon as the ideas of BOREL, LEBESGUE, and DANIELL appeared, it was possible to put the BROWNian motion on a firm mathematical foundation; this was achieved in 1923 by N. WIENER [1].

Consider the space of continuous paths w :I E [0, + oo) - Rl with coordinates x (t) = t v(1) and let B be the smallest BOREL algebra of VII

subsets B of this path space which includes all the simple events B = (w : a:< x (t) < b) (t Z 0, a < b). WIENER established the existence of non-negative BoREL measures P. (B) (a E R', B E B) for which 4) holds; among other things, this result attaches a precise meaning to BACHELIER'S statement that the Brownian path is continuous.

P. LEvY [2] found another construction of the BROwxian motion and, in his 1948 monograph [3], gave a profound description of the fine structure of the individual BROWNian path. LEVY'S results, with several complements due to D. B. RAY [4] and ourselves, will be explained in

chapters 1 and 2, with special attention to the standard BROwxian local time (the mesure du voisinage of P. LEvY) :

t(t, a)

-

measure (s: a 5 x(s) < b. s S t) VII

6

bJ,a

2(b - a)

Given a STURM-LIOUVILLE operator 0 = (cd2) D' + cl D (c= > 0) on

the line, the source (GREEN) function p = p(t, a, b) of the problem 8u

at

7)

(t > 0)

= (5S u

shares with the GAUSS kernel g of 2) the properties 8a)

o

p

00

f p(t,a,b)db=1

8b)

R3

8c)

p(t,a,b)= f p(t-s,a,c)p(s,c,b)dc

t>s>0.

R1

Soon after the publication Of WIENER's monograph [3] in 1930, the associated stochastic motions (diffusions) analogous to the BROwNian motion (0 = D2/2) made their debut; the names of W. FELLER and A. N. KOLMOGOROV stand out in this connection. At a later date (1946),

K. ITo [2] proved that if 9)

1 ci (b) - c, (a) I + I 11c,

- V c, (a) I < constant x I b - a I

,

Preface

IX

then the motion associated with (SS = (crf2) D2 + c1 D is identical in law to the continuous solution of r

a(t) = a(0) + f c1(a) ds + f cY(a) db

10)

0

0

where b is a standard BROWNian motion. W. FELLER took the lead in the next development.

Given a MARKOVian motion with sample paths w : t -. x (t) and probabilities Pa (B) on a linear interval Q, the operators

HI:/--* f P.[x(t)Edb]/(b)

11)

constitute a semi group :

(tzs),

H1=HI, H.

12)

and as E. HILLE [1] and K. YOSIDA [1] proved, (t > 0)

H1 = e1G

13)

with a suitable interpretation of the exponential, where 0 is the socalled generator.

III

D. RAY [2] proved that if the motion is strict Markov (i.e., if it starts afresh at certain stochastic (MARKOV) times including the passage

times ma = min (t : x (t) = a), etc.), then (SS is local if and only if the motion has continuous sample paths, substantiating a conjecture of W. FELLER; this combined with FELLER'S papers [4, 5, 7, 9] implies that the generator of a strict MARKOvian motion with continuous paths (diffusion) can be expressed as a differential operator 14)

(

05 u

lim )(a)-bl,a

u+ (b) - u' (a)

m(a.b] p,.

where m is a non-negative BOREL measure positive on open intervals and, with a change of scale, u+ (a) = bilm (b - a)-1 [u (b) - u (a)], except at certain singular points where f 3 degenerates to a differential operator of degree :5.'1. E. B. DYNKIN [1] also arrived at the idea of a strict MARKOV process, 4'W

derived an elegant formula for CS, and used it to make a simple (probabilistic) proof of FELLER's expression for (SS; the names of R. BLUMENTHAL [1] and G. HUNT [1] and the monographs of E. B. DYNKIN [6, 8]

should also be mentioned in this connection. Our plan is the following. BROwxian motion is discussed in chapters 1 and 2 and then, in chapter 3, the general linear diffusion is introduced as a strict MARKOVian

motion with continuous paths on a linear interval subject to possible

X

Preface

annihilation of mass. (3S is computed in great detail in chapter 4 using probabilistic methods similar to those of E. B. DYNKIN [5]; in the socalled non-singular case it is a differential operator

u+(b) - u+(a) - f udk ((53 u) (a) = li

15)

m

m(a,

(.. b)

b]

with u+ and m as in 14), where now k is the (non-negative) BOREL measure that governs the annihilation of mass (such generators occur somewhat disguised in W. FELLER [9]).

Given (s3 as in 14) and a standard BROwNian motion with sample paths w : t --+ x (t), if t is P. LEvY's Brownian local time and if f -1 is the

inverse function of the local time integral f (t) = f t(t, e) m(de),

16)

then the motion x (f -1) is identical in law to the diffusion attached to (0S, as will be proved in chapter 5, substantiating a 'suggestion of H. TROTTER; B. VOLKONSKII [I] has obtained the same time substitution in a less explicit form.

Given 0 as in 15), the associated motion can be obtained by killing the paths x (f -1) described above at a (stochastic) time m. with conditional law

P.[m, > t x(f-1)] = e I

17)

as is also proved in chapter 5; in the special case of the elastic Brownian motion on [0, + oo) with generator (5S = D2/2 subject to the condition

yu(0) = (1 -y)u+(0) (0
0,sS

f = f t(t,e)2de=measure 0

is identical in law to the classical re/letting Brownian motion x+ = Ix I, and 17) takes the simple form x (f -1)

000

r

18)

P.[m. > t x+] = e 1Y

t+(t, 0)

(t+ = 2t),

mow.,.-

substantiating a conjecture of W. FELLER: that the elastic BROwNian motion ought to be the same as the reflecting BROWNian motion killed at the instant a certain increasing functional e($+ fl [0, t)) of the time t and the visiting set 3+ = (t : x+ (1) = 0) hits a certain level. Details about the fine structure of the sample paths of the general linear diffusion with emphasis on local times, will be found in chapter 6. BROwxian motion in several dimensions is treated in chapter 7, and in chapter 8, the reader will find some glimpses of the general higherdimensional diffusion.

Preface

XI

are

Acknowledgements. W. FELLER has our best thanks, his ideas run .:.

through the whole book, and we shall think it a success if it pleases him. We have also to thank R. BLUMENTHAL and G. HUNT who placed

at our disposal their then unpublished results on MARKOV times, and H. TROTTER with whom we had many helpful conversations. This book was begun at Princeton and the Institute for Advanced

n`...

Study (1954/56) with the partial support of the Office of Ordnance Research, and continued at Kyoto (1957/58) with the aid of a Fulbright grant, at Hanover, N. H. and Cambridge, Mass. (1960) with the support of the Office of Naval Research, and at Stanford, Calif. with the support of the National Science Foundation (1962/63); to those institutions

and agencies, also, we extend our warmest thanks. Finally, we must thank the staff of Springer-Verlag for their meticulous labors and cordial attitude to what must have been a difficult job.

Kyoto, Japan, and Cambridge, Mass., November 1964 K. 10

H. P. McKean, Jr.

The present edition is the same as the first except for the correction of numerous errors. Among those who helped us in this task, we would particularly like to thank F. B. KNIGHT. K. I., H. P. McK. September 1973

Contents pace

Prerequisites

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I

Chapter 1. The standard BROwxian motion .

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7

HINCIN'S proof Of the DE MOIVRE-LAPLACE limit theorem

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The standard BROwxian motion

10 12 19 22

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The standard random walk . . . . . . . . Passage times for the standard random walk .

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...p-

P. LEVY'S construction . Strict MARKOV character Q°G

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7.

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vow

Passage times for the standard BROwNian motion . . . . . . . . 25 Note i : Homogeneous differential processes with increasing paths 31 1.8. KOLMOGOROV's test and the law of the iterated logarithm . . . . 33 P. LEvv's HOLDER condition . . . . . . . . . . . . . . 1.10. Approximating the BROWNian motion by a random walk 1.9. ._.

Chapter 2. BROWNian local times

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P. Lsvy's local time

r+°

The reflecting BROWNian motion .

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36 38

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40 42

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45 48

t+ as HAUSDORFF-BESICOVITCH 1/2-dimensional measure

Note 1: Submartingales

E2'

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Elastic BROwNian motion t+ and down-crossings . .

0

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ran

°a6q

2.1. 2.2. 2.3. 2.4. 2.5.

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Note 2: HAUSDORFF measure and dimension 2.6. KAC's formula for BROWNian functionals . 2.7. BESSEL processes . . . . . . . . . . . . 2.8. Standard BROWNian local time . . . . . . 2.9. BROWNian excursions . . . . . . . . . .

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52 53 54 59

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63

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75 79

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81

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2.10. Application of the BESSEL process to BROwxian excursions 2.11. A time substitution . . . . . . . . . . . . . . . . . .

Chapter 3. The general 1-dimensional diffusion Definition

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MARKOV times

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Matching numbers . . . . . . . . . . . . . . . Singular points . . . . . . . . . . . . . . . . . . Decomposing the general diffusion into simple pieces GREEN operators and the space D . . . . . . . . . . . . . . . . . Generators . . . . . . . . . . . . . . . . . . . . Generators continued . . . . . . . . . . . Stopped diffusion . . . . . . . . . .

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83 86 89

0.a 0'a

3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9.

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91

92 94 98

O00

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tea.

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100 102

Contents

1-I

XIII

page `°.

+V+

Chapter 4. Generators

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°.:

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ova

QS at an isolated singular point ...

4.5.

Q>QQ>

4.1. A general view . . . . . . . . . . . 4.2. (l as local differential operator: conservative non-singular case 4.3. G as local differential operator: general non-singular case . . 4.4. A second proof . . . . . . . . . . . . . . . . . . . . . .

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4.6. Solving Wu = a u . . . . . . . . . . . . . . . 4.7. Q$ as global differential operator: non-singular case 4.8. 05 on the shunts . . . . . . . . . . . . . . . . 4.9. (J as global differential operator: singular case . . .

105

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105

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116 119 125 128 135

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t36

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142 '.O

OVA

4.10. Passage times . . . . . . . . . . . . . . . . . . . . . . . . 144 Note 1: Differential processes with increasing paths . . . . . . . 146 4.11. Eigen-differential expansions for GREEN functions and transition densities . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 .

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4.12. KOLMOGOROV'S test .

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164

Construction of sample paths : a general view . Time changes: Q = RI . . . . . . . . . . . Time changes: Q = [0, + 00) . . . . . . . . Local times Subordination and chain rule . . . . . . . . Killing times . . . . . . . . . . . . . . . .

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164 167

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5.8.

FELLER's BRowNian motions IKEDA's example . . . . . .

5.9.

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Chapter 5. Time changes and killing . 5.1. 5.2. 5.3. .in

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191

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196 200 201 206 209

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Time substitutions must come from local time integrals

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Mme

InU:

5.11. Shunts with killing . . . . . . 5.12. Creation of mass . . . . . . . 5.13. A parabolic equation . . . . . 5.14. Explosions . . . . . . . . . . 5.15. A non-linear parabolic equation

Chapter 6. Local and inverse local times

6.1. Local and inverse local times . . . . . 6.2. LEvY measures . . . . . . . . . . . . 6.3. t and the intervals of [0, + 00) - 3 . . 6.4. A counter example: t and the intervals 6.5 a t and downcrossings . . . . . . . . .

6.5b t as HAUSDORFF measure 6.5c t as diffusion . . . . .

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of [0, + oo) -8 .

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6.5d Excursions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Dimension numbers . . . . . . . . . . . . . . . . . . . . . . 6.7. Comparison tests . . . . . . . . . . . . . . . . . . . . . . . Note 1: Dimension numbers and fractional dimensional capacities 6.8. An individual ergodic theorem . . . . . . . . . . . . . . . . .

Chapter 7. BROwNian motion in several dimensions. e+1

7.1. 7.2. 7.3.

171 174

176 179 186 188 190

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5.1u. Shunts .

161

r-.

Ova

5.4. 5.5. 5.6. 5.7.

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b0'+$

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212 214 218 220 222 223 223 223 224 225 227 228

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232

Diffusion in several dimensions . . . . . . . . . . . The standard BROWNian motion in several dimensions .

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Wandering out to 00 .

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Contents

XIV

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Excessive functions .

NEWTONian capacities .

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Gauss's quadratic form

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237 243 245 247 250 253 255 257

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nor

Application to the spectrum of d/2 Potentials and hitting probabilities .

?6.."""

. . . 7.10. WIENER'S test . . 7.11. Applications of WIENER'S test . 7.12. DIRICHLET problem . . . . . . .

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7.13. NEUMANN problem

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7.14. Space-time BROwxian motion . . . . . . . . 7.15. Spherical BRowxian motion and skew products . . . . . . . . 7.16. Spinning . . . . . . . . . .

V'1

7.4. 7.5. 7.6. 7.7. 7.8. 7.9.

page

GREENian domains and GREEN functions

7.17. An individual ergodic theorem for the standard 2-dimensional .

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Chapter 8. A general view of diffusion in several dimensions . 8.1.

Similar diffusions .

8.2. 8.3. 8.4. 8.5. 8.6. 8.7.

0 as differential operator Time substitutions . . . Potentials . . . . . . .

. .

277 279 283 286 288

000000'

N++

BaowNian motion . . . . . . . . . . . . . . 7.18. Covering BRowNfan motions . . . . . . . . . 7.19. Diffusions with BROwNian hitting probabilities 7.20. Right-continuous paths . . . . . . . . . . . 7.21. RIESZ potentials . . . . . . . . . . . . . .

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Boundaries . . . . . . . . . . . . . . Elliptic operators . . . . . . . . . . . . FELLER's little boundary and tail algebras

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Bibliography

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List of notations

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Index

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296 302 303

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. . . 315

Numbering : 1.2 means section 2 of chapter 1; 1.2.3) or problem 1.2.3 or diagram 1.2.3 means formula or problem or diagram 3 of 1.2; 3) or problem 3 or diagram 3 means formula or problem or diagram 3 of the current section. R. BROWN [1: 2] means page 2 of the item R. BROWN [1] of the bibliography.

boa

Problems accompanied by some indication of their solutions are placed at the end of each section; these often contain additional information needed below and are an essential part of the exposition. Rio

Diagrams do not pretend to photographic faithfulness; for example, the BROwxian path is often pictured as if it had isolated zeros, which is not at all the acual case. A list of notations is placed at the end of the book. NIA

..'-

Warning: positive means >0, while non-negative means ?0; it is the same for negative (<0) and non-positive (S0). a A b means the smaller of a and b, a V b the larger of the two.

Prerequisites The reader is expected to have about the same mathematical background as is needed to read COURANT-HILBERT [1, 2]. Besides this, he

should have mastered most of W. FELLER's book on probability [3] plus the topics listed below (see A. N. KOLMOGOROV [2] for a helpful

outline and some of the proofs). Algebras. Given a space W, a class A of its subsets is said to be an algebra if ,.a

WEA

1)

A - B,AUB,Af1BEA incase A,BEA.

2)

A is said to be a Borel algebra if, in addition,

U Bn, n Bn E A in case Bn E A (n Z 1) .

3)

nZ1

nZ1

Borel extension. A class A of subsets of W is contained in a least Borel algebra B. B is the so-called Borel extension of A. Probability measures. Consider a non-negative set function P (C)

defined on an algebra A. P is said to be a probability measure if 1) P(W) = 1 and

P(AUB)=P(A)+P(B)

2)

A,BEA, AnB=O.

P is said to be a Borel probability measure if, in addition, P(Bn)t0 (nt+oo) 3a)

or, what is the same,

3b) P( ,IU iBn) _nZ1 P(B,)

Baf1B,, = 0(n <m), Bit EA(n?1), U BnEA.

ski

Kolmogorov extension. Given a BoREL probability measure P on an algebra A, there is a unique BOREL probability measure Q on the BOREL extension B of A that coincides with P on A : Q (B) inf X P (A.), f'

i

where the infemum is taken over all coverings U A. of B E B with nZ1

An E A (n

1) ; the triple (W, B, Q) is said to be a probability space.

ite/KcKan. DWusion paucesies

I

2

Prerequisites

Measurable functions. Given a space W and a BOREL algebra B of its subsets, a function f : W -i [- oo, + oo] is said to be measurable B (or BOREL) if the ordinate set /-'[a, b) E B for each choice of a < b. Integrals. Given a probability space (W, B, Q), the integral or expectation of a non-negative B measurable function / is defined to be

E(f) = f / dQ w

= "t*eo121 lim T l2-"Q[/-1[(l - 1) 2-", l2-")]

Q[/-1(+0o)] = 0 QU-1(+00)]

= +00

> 0-

E, applied to such non-negative functions, satisfies E(f) 0 1) 2) E(1) = 1

E(l1 + l,) = E(/1) + E(/,) E(/,) t E(/) /" t /

3)

4a)

E(/1)<+00,

4b)

E(lim f") S limE(f"). E(f, B) = E (B, f) is short for f f dQ. 5)

Products. Given probability spaces (W1, B1, Q1) and (W=, B=,Q=), the class A of finite sums A of disjoint rectangles B1 x B. (B1 E B1, B= E B,) is an algebra and Q(B1 x Bs) - Q1(Bi) xQ,(B:)

can be extended to a BOREL probability measure on A; the Product Q1 x Q2 is the KOLMOGOROv extension of this Q to the BOREL extension

B- B1 x B, of A. Fabini's theorem. Given a B1 X B= measurable function / from W1 X W= to [0, + 00) ,

f / dQ1 x Qt = f dQ1 f l dQ, = f dQs f l dQ1 W, 1,

WaxWS

W1

Infinite products. Given probability spaces (W", B", Q") (n

1),

All: A=Bx W"+1X W,+2X etc., BE is a BoREL algebra and

Q(A) o-- Q1xQ,x ...xQ"(B) is a BOREL probability measure on the algebra A = U A"; the infinite 'C!

"a1 product x Q. is defined as the KOLMOGOROV extension of Q to the BOREL nZ1

extension x B. of A. "Z1

Prerequisites

3

Independence. Given a probability space (W, B, P), the BOREL algebras B1, B2 C B are said to be independent if P(B1 fl B2) = P(B1) P(B2)

B1 E B1, Bs E B2;

!'t

the BOREL algebras B (n z 1) are independent, if, for each n ? 1, B. is independent of the BoREL extension of U BI; the sets B. E B (n Z 1) l$n are independent if the algebras Bn = [0, Bn, W - B,,, W] (n Z 1) are such; the B measurable functions 1n (n Z 1) are independent if the BOREL extensions Fn of their ordinate sets /nl [a, b) (a < b) are such; the B measurable function / is independent of the BOREL algebra A C B

if the BOREL extension of its ordinates sets is independent of A; etc. E (/1 /_) = E (/1) E (/a) if /1 is independent of /_. Kolmogorov 01 law. If A,, (n ? 1) are independent subalgebras of B, if B. is the BOREL extension of U Al, and if B E fl Bn, then

nal Ian P(B) = 0 or 1. Strong law of large numbers. Given independent, non-negative, P(nlim lt+oo n

1) with the same distribution,

kin

/k=E(/l)1=1. ..a

B measurable functions / (n

Borol-Cantelli lemmas. Given events B. E B (n ? i), if !z1

P(B!) < + oo, then P( fl y Bll = 0; if X P(B,)

nilllan

l21

gay

and if the events are independent, then P[ fl

oo

U B1] = I.

nZ1lan

11

Conditional expectations. Given a BOREL subalgebra A of B and a non-

negative B measurable function /, the conditional expectation E (/ I A) of / relative to A is defined to be the class of A measurable functions g

such that E (g, A) = E (/, A) (A E A); two such functions differ on a null set E A. E (/ I A) is the RADON-NIKODYM derivative of Q (A) =

E(/, A) with respect to the restriction of P to A. E(/ I A) (applied to non-negative /) satisfies 1)

2) 3)

5)

E (l, + 12 I A) = E (/1 I A) + E(/21 A)

E(/.I A)tE(/I A)

1nt/ AsJA,

`ti

4)

E(/IA)?o E(IIA)=1 E(E(1I As)IA,)=E(/I A1)

and 6)

E(E(/ I A)) = E(/);

in addition, 7)

E (I I A) = E (l)

1

4

Prerequisites

if / and A are independent, and

E(e/I A)=eE(/I A)

8) .t7

if e is measurable A. In all this, it is understood that E(f I A) = (z) /1 is short for the statement that /2 /1 for some f, from the class E(/ I A). Conditional probabilities. Given B E B, the conditional expectation E (e I A) of its indicator function e is the so-called conditional probability P (B I A) o l B relative to A; the rules for its manipulation are evi+O'

dent from the preceding article. Gaussian distributions. A class of measurable functions / is said to be Gaussian and c e n t e r e d if each choice of x = (/1, /, , ... , /d) and (yi , . . ., ya) E Rd, the inner product y x is GAUSS distributed with mean

E (y x) = 0, i.e.,

b

P(a5v.x
7)

.f OX-0

a

where Q is the quadratic form Q = Q (y) = E [(y x)2] = Z E (/; /I) y: yt ;1

and the integral is interpreted as the unit mass at c = 0 if the determinant I Q I = 0. Suppose that the determinant ( Q I + 0; then one can invert the FoURIER transform E (eir'x) (= a-Q!E) to obtain the probability density of x :

p(x) = (2n)'

2)

Rd

e

(Za)"' Y IQ I

-+.

where Q -1 is the inverse quadratic form. Under the norm VE (/_)(J=), the functions / span out a HILBERT space, and in this geometrical picture, statistical independence and perpendicularity are the same.

1. The standard Brownian motion 1.1. The standard random walk Consider a person walking in the integers according to the following

rule: start at time n = 0 at s (0) = 0, ±i , ±2, ... and flip a true coin ; if tails comes up, step at time n = I to s(i) s (O) - i ; if heads

comes up, step to s (1) = s (0) + 1. Coming, thus, at time n - 1 to s (n - 1), flip the coin ; if tails comes up, step at time n to s (n) = s (n - 1) - 1, if heads comes up, step to s (n) = s (n - 1) + 1; etc.

We need a few definitions. W is the space of sample paths

f1, f2,...;

i)

S, (w) = s (n, w) is used in place of sw = s (n) if the path w needs to be emphasized.

B is the BOREL algebra generated by the sets

(w:s(n)=l) n?O, 1=0,±1,±2,...

2)

P, (B) is the probability that the sample path s (n) : n ? 0 lies in B E B as a function of its starting point s (0) = 1 = 0, ±1, ±2, ... ; for example,

PP[s(n) =k]=(n+)2-".

3) Nip

where

n+ = 2 + 2 (k-1), (,) = nl/(n - n+)!n! for n+ =0,1,.,n, and (nn) = 0 otherwise. D = [W, B, Pt : I = 0. f 1, ±2, ...] is the standard random walk. Given m Z 0, 4)

(1)=1]

1=0,±1,±2.... i.e., if s (m) is known, then the next position s (m + 1) is independent of the past s (n) : is < m ; in other words, D is a Markov process.

6

1. The standard BRowxian motion

D is in fact a strict MARKOV process as will now be explained.

Define B. = B [s (n) : is 5 m] to be the smallest subfield of B including all the events (a s (n) < b) (n 6 in, a < b). let m = m (w) _ 0, 1, 2, ... , + oo be a Markov time in the sense of

(mSm)EB.

5)

(ma0),

let B. be the field of events B E B such that B fl (m ;g m) E B. for each m a 0, and let w* be the shifted path s, (w+,) = B. should be

thought of as measuring the past s(n) : is S m, and the statement is

that if m < +oo, then, conditional on the present s (m), the past m and the future s (n + m) : is a 0 are independent. the future being a standard random walk starting at s (m) : s (n) : is

6a)

P. [w, E BI Bm] = P,(m)(B)

m < + oo, BE B,

i. e.,

6b)

P.[A, wm E B, m < +oo] = E,[A fl (m <+ 00), P,(m)(B)]

AEB.,BEB; in other words, if m < +oo, then the walk starts from scratch at time is = M. Here is a simple proof of this. Given m = 0, 1 , 2, ... , the constant m = m is a MARKOV time, Bm = B. = B[s(n) : is S m], and 6a) is just P.[w*, E B I s(n) : is < m] = P.(m)(B)

7)

BE B,

which is evident on iterating 4) ; this is the so-called simple MARKOV property. Consider the general MARKOV time m and an event A E B.. Because the indicator of A is a BOREL function

e[s(1), s(2), ..., s(m), s(m), s(m), ...] of s(nA m) (n ? 0), e[s(1), etc.,] = e(s(1), s(2), ..., s(m), s(m), s(m), ...], on (m = m), and since (m = m) E B,,,, A fl (m = m) E B., and therefore 8)

P,[A,wn,EB,m<+oo] = mao = XE.[Afl (m= m),P.(wmEBI = Z E. [A fl (m= m). P, (m) (B)j Mao

as desired.

= E.[A fl (in <+ oo), Ps(m)(B)],

BEB

Bm)]

1.2. Passage times for the standard random walk

7

Consider the spatial shift w -> w + 1:

s(n,w+1)=s(n,w)+I

(n ?0)

and the reflection w -i -w: s(n, --w) = -s(n, w)

(n z 0);

using 3), it is easily seen that 9)

Po(w + l E B) = P,(B), Pi(-w E B) = P_l(B)

BE B.

Problem 1. Sketch the RADEMACHER functions

e,(t)=-1 (0St< ) =+1 (4St<1)

(05t
(21t) being the fractional part of 2"t, and check that, relative to the classical BOREL measure on [0, 1),

(n=0) ICI

s(n)=0

is a standard random walk starting at 0. 1.2. Passage times for the standard random walk MARKOV times of special importance are the passage times

mk=min(n:s"=k)

1)

k=0,±1,±2,...

We will use the strict MARKovian character of the standard random walk to compute the generating functions 2)

Ej(ACmt)=I&"Pl(mk=m)

1 ,k=0 ,±1 ,2 ,

.

0
M

Because Mk (w + 1) = Mk _, (w) and mk (-w) = m_k W, it is clear f ;om 1.1.9) that 3)

El[ama(w)111

Eo[am_a(w)]

= Eo[ama_t(w)I = F.o[ama(w+l)]

=

Eo[amk(-w)]

= E0[&na(w)].

Using the strict MARKovian character, it is also clear that, for k > 0, 111k = m + mk(wm) with m = m, so that 4)

Eo (ama) = Eo

= Eo

lam1+ma(w+m) [aM1

I Bill

= Eo (am-) Ei (ama) = Eo(am1)

Eo(ama_1)

= Eo(am')k;

1. The standard BROwNian motion

8

in addition, mk = 4 + ntk (ml) , and so E0(am=) = E0 (a(i+mi(f0+ )1

5)

= a E0 [E0 (amk(1-) I B1)] = a EO(E., (am ))

SIN

= 2 E-i (am') +

E+1(amk)

= a E0 (ams+1) + 2 E0 (ams-1) .

Putting k = I in 5) and using 4),

y$-2a-1y+1 =0

6)

y=E0(ams),

0
Solving 6) gives y = E0(am1) = a-1(1 f y1 - -&2).

7)

9.:9

and here the + sign is excluded owing to E0(am') < 1. Expanding 7) in a MACLAURIN series, 8)

rill

E0(ama)

(2n - 2)! 2-2n+1,

nl(n - 1)I

and thus PO (m1 = 219) = 0

(2n - 2)1 2-in+1 P0(m1 = 2n - 1) = n!(n-i)! Using 3) and 4), it follows that

9)

n=1,2,3,...

E1(amt) = yIZ-kI.

10)

Because y 1' 4 as a t 1,

Pr(mk < +oo) = 1,

11)

and, using the fact that s,: n ? 0 starts from scratch at a passage time, it is seen that the walk visits 0, ±i, ±2,. .. each an infinite number o/ times.

Problem 1 after K. L. CHUNG and W. FELLER [1]. Consider the number tZn of integers k = 1, 2, ..., 2n such that the greater of s;_1 and sk is >0; prove that nao

Vi-- a=Ijl+]l1 -fls

0
2

V RI" E 0[ahs , sIn = 01 _

and deduce the so-called arcsine law:

P0[ts,=2k1ssn=O]=(n+1)-1, k=0,1,2.... In, n?1.

1.2. Passage times for the standard random walk

9

[Given 0
III`

e:n =

k

_

n

ksn

Eo(al+t, in = 2k) Eo(&4*-'t, s:"-2k = 0)

= 2k) esn-ak for n

All

III

+ask) Po(m

Zni(1

1;

thus, if 0 < ft < 1, then

/ =na0 Zfl'"e:n = 1 + Z j(i +a'")P'"Po(m = 2n)i. na1 and using the formula

' as" Po (m = 2n) = 4 - 1 - &2,

nZ1

it is seen that 041 P1 -

YT-

Ps).

Po (t,n = 2k, s,* = 0) is the coefficient of a2k Pen in /; thus, for k - n, 2I

P0(t2 = 2k, s,n = 0) is the coefficient ofa=kp$"+2 in a2 , and since that coefficient is independent of k, the proof is complete.] Problem 2. Use 40) and the successive passage times: ml = min (n : sn = k) m2 = min (n : n > ml, sn = k)

ma= min (n:n> m', S.

k)

etc.

to evaluate the generating function Ea"+1 PI (s. = k). n

rn2 c

1 Pt(s"

= k)

_naoI an+1mat I Pt (mm = n) = a Y El (am-)

mat

= amat Z E1 (amt) E0(am)m-1 = a yIl-'I [1 - E0(am)]-'

with m as in the solution of problem 1; now use the evaluation a-1 [1

- Eo (am)] = a-1(1 - a y) =&-I y1 - as = i (y-1- y).]

1. The standard BROWNIan motion

10

1.3. Hindin's proof of the de Molvre-Laplace limit theorem Our next topic is the ingeneous proof of the DE MOIVRE-LAPLACE limit theorem : b

n?n P. (a S sn) <

1)

bl J

ll

=f a

` v- ` 2-

1'F

due to A. YA. HINCIN [1: 3-5] (see W. FELLER [3:133-137], H. TROTTER [2], and problem 2 below for different proofs, and 1.10 for a bigger ..4

and better limit theorem). "''

Define

t

u,.(t, a) = Eflrna] U(n-f sins])]

for / E C (R1)

? 0, a E RI

and n z 1 *. Because u,, (t, a) = u, (m/n, l/yn) for end

(''

2)

m=[nt] andl=[Vna], `yam

3)

a)

= E,U(n-1 Sm+i)]

= El [Es, [f (n- i s,.)] 'L'

)+--u(t1 ! ynt

2 un(

t yxt

)

= 2 un(t,a - yn) + 2 4a)

4b) 5)

is x

[".(I+

a) from both sides of 3), it is found that 'L1

and subtracting

a)

un(t, a)

- 2uR (t, a) + unt, a - yn J = 2 I n!$ X IU (t, a + y_) n b) _ /([f HINCIN'S idea is to compare un with the solution u = 14(t, a) = e -(b- 8121 /(b)

J

y2n t

db

Rl

of the continuous analogue of 4) :

au -I 8=u 8t - 2 aas u(0+ , ) _ /

6a) 6b)

and to conclude that 7)

lira

Rt+00

[a t], [V n a], [V n b] indicate integral parts; otherwise [ ] has its usual meaning.

1.3. HIN6IN'S Proof of the DE MOIVRE-LAPLACE limit theorem

11

Consider with HINCIN the function u' = u + n-113 t assuming / E C4 (RI), note the bounds 8a)

a= f

=

aas

ail =

8b)

4

d3 //d b$ db S c1 < +oo

y2n t

a '_

(' -Yin--t a d4 //db4 db Z -c, > -oo

J

and choosing n > (c1 + cg)°, let us check the estimates

Qm:un(m,

n

`n yn 1

for mZ1.

1 / yn

000

Q0 is automatic, and supposing that Q. holds and using 4a), 6a), and 8), one finds 1

9)

'

1

n

yn/I

/m 1},n1)

1

= 2 un n ,

t un/ n, m lynl )

+

VII

t J lynf\+ lynl\ Zu (n,

82u m I Su'(nM yn+ 2n 8a2 ln m I 8u m + 1

1

(n yn

U*

M+ 1

Mn
n

(n yn - n-ua + n-sts cI 1

vl 1-f- 2n-sCS --n-413+n312Cl 1

1

y-J+n-31$[cl+ cs -- nlt° n

l n

at

1

1

yn

in other words, Q. implies Qm+I , and this completes the induction. Now

5u'\m", n/l m=[nt],l=[yna], jl III

10)

and letting n t + oo, it appears that

lim un S u;

11)

nt+co

the proof of lim un Z u is left to the reader as well as the proof of 1) nt+m

using 7) for / E C4 (RI).

1. The standard BRowxian motion

12

On the basis of 1), it is natural to conjecture that n-1/2 s[n t] (t Z 0) approximates some stochastic (BRowNian) motion for which 6a) takes

the place of 4a). 1.4 and 1.5 are devoted to the construction of this BROwNian motion, 1.10 to the discussion of the approximation. Problem 1. Check that t

lim P0 main-1/2 s ne f s5t [ l

of+cc

z a]

-

t,a?0

a e-a'/2e dO y2n 8'

1-f o

-4m

(see problem 1.10.1 for another proof). (Because min (0: n-111 s(ne) ? a) = n 1 m

that

,

it is obvious

[Y"°]+1

=

P°lmaa n-112s[nel Z a] +0'

l

f e...at PO [n

P°[n-1

m[yea]+I

S t]; also, using 1.2.7),

m[ya]+ l E dt] = EJe

-aa'm [yaa]+1)

0

a

= (e'/" - Vesa/" -

as n t + oo ;

now use the formula e-y2aa

=j ,J

e-at

a

yin P

e-°'I2tdt

0

(see A. ERDELYI [1 (1): 245]).]

Problem 2. Prove the DE MOIVRE-LAPLACE limit theorem using STIRLING'S approximation : nl=y2nnn+I/2e-n+e/12n 0<0=0(n) <1,n>1.

[Bound P° [a S

S (n)

yn

<

bl

for 0 < a < b < + oo first and then

1

estimate the tail P0[s (n)/1 > b] using CEBYSEV.] 1.4. The standard Brownian motion

Consider the space W of continuous functions w : t -+ xt = x (t) from [0, +oo) to R1* and introduce the class C of subsets 1) C = xt 1(B) = xt,lr,...e (B) t = {tl, t=, ... , t*) ,

0
2)

* xx (w) = x (t, w) is used in place of x, = z (t) if the path w needs to be emphasized.

0 * B (R*) is the algebra of BoRzs. subsets of R*.

1.4. The standard BROWNian motion

13

C is an algebra; in fact, -I«

W = xt1(R"), W - xt 1(B) = xt 1(R* - B)

3)

1«« t,1!

4)

and

x:1(B1) U xt 1 (B,)

5)

=

x:1 (B1 U B,) , ...

mob

i.e., x,1 B (R*) is an algebra, and to complete the proof, it is sufficient to note that two such algebras x-' B (R") are contained in a third. Consider, next, the GAUSS kernel -(a - .Yhe

2nt

and the set functions Pt(C)

7)

t>0,a,bER1

awl

g(t,a,b)=

6)

f

B

g(ti,0,b1)db1g(t,-t1,bl,b,)db,...

g(t" - t"-1, bn-1, b") db" BEB(R*),

n?!.

C=x-1(B),

«'b

...,

P, is a probability measure on xt' B (R") ; indeed, since xt' (B1) = x, 1(B,) implies B1= B,, Pt is well-defined; since xt' (B1) (1 x, 1(B,) = 0 implies

B1 fl B, _ 0 , it is clear from 5) and 1) that Pt is additive; and, since +00

r 'db 2n

..:

000

+00

8)

f g(t,a,b)db=2 .1 0

- 00

f

+ 00

+ 00

C41

d a f d b e-6''2 e-b'IQ

0

nl2

0 +OD

i

f d9 f e-''12rdr 0

0

I, which completes the proof. Pt coincides with P, on x;' B (Rr)* if 9 C t; indeed,

P, (W)

+W 9)

f g(t-s,a,c)g(s,c,b)dc -00 +00

/'

J

-00

y2n (t - s)

V2a t

j/2n s

=g(t,a,b)

* m is the number of points in 8.

dc

t>s>0,a,bER1,

1. The standard BRowNian motion

14

and taking, for example, B, E B(R"), B2E B(R"-1),

C = xl,[,l,t,...tw(B1) `

one finds 10)

g(t1, 0, b1) g(tz - t1, b1, b2) X

J

x

ba)EB, xg(13-12,b2,b3)g(tt-13,b3b4)...g(1"-1"-1,b"-1,b") X

x db, dbz db3 db4 ... db"

f

boo

g(t1, 0, b1) db1g(tz- t,, b1, bz) dbz x tom

(b1. b b,.... b,) E B,

b3, b4) d b4 ...

X f g (t3 - tz, bz, b3) d b3 g (14

g (t" -

bA_,

,

b") d b.

b, E R'

= f g(t1, 0, bl)

dbl g(19

- t1, bl, b2) db2g(t4 - tzi bz, b4) dbz .. .

B,

... g (t. - t,-1,

b"-1, b") d b.

Define P (C) = Pt (C) for C E xi 1 B (R") . P is a probability measure on C, and, as we will now prove, it is extensible to a Borel probability measure on the Borel extension B of C. Consider, for the proof, C E C such that C, ) Cz) ... and

P (Q

c1 > 0*; it is to be shown that 11 C is non-void. "Z 1 Suppose, as we can, that

11)

C" = xtoi B"),

where

t" = B" E B (R"'), m = the number of points of t", n a 1, and, noting the estimate 12) P ( <max . 1 x,, - x,l > C2 2-"/3] o

.-'12sl.%Etw

1»W

M11o
a,.:,Et, 00

r.'

o«.#Is2-* fit.Is E to

26"18

e,2-x!3 7S'

S 3n

e-bS12(h-,)

2

e-42012

db

TX(,-SS 1)

-l **

C2

cl, c3, etc. are positive constants. «00

+00

fe'/2*db a r e-01211 db = a estimate.

see problem 1.4.1. for a finer

1.4. The standard BRowNian motion

15

choose cz > 0 so great that

2' 3n 25n/6 e-ell"/a-1 nZ1 C2

13)

G1

3

Choose, also, compact B',' C Bn C B,, such that a-.

P(xt' (Bn -

14)

c13-"

cad

and 15)

max

x t ' (Bn) fl (w : 0

«.-+,52-

I x,, - x.,

ca 2-"l$) = xto (Bn).

81.f2 E to

and note that

P(ms xto (Bm))

16)

Z P (Cn) -"3 s "P (xto (Bm - Bm))

c12-3=6>0.

s

nP

(x7+ (BM - Bm))

fl xt"; (B',) is therefore never void, and, since B; was compact, we

m$n

can select a point (x (s) : s E t) E R°°, t = U t, such that

n z 1.

(x (s) : s E tn) E B;1

17)

But then 0 < S2 - S1 S 2'n, Si. s$ E t, , n Z 1 , and it follows that x (s) is uniformly continuous on each bounded sub18)

I x (s2) - x (s1) I S c2 2-n/3

set of t; in fact, if s1,s2Et,

19)

0<sa-s1<2'm,

0<s1<s2<m,

then 20)

I x (sa) - x (s1) I < c3 2-m13

c3 = 4 ca T 2-"/3.

nZl As to the proof of this, take n such that s1, sz E to+," and consider S1

= k1

21)

2-n-m S2 = k2 2-n-m such that

s1 < si < sz < s,,

si

- s1 <

s2 - s2 < 2-n-m;

and, using 18) and the terminating expansions s' = k 2-t - 2-91 - 2-P1 - etc. m S 1 < p1 < pz < etc. 22) SY = k 2-' + 2-Qi + 2-Qt + etc. m S I < 41 < 4z < etc., estimate as follows: 23)

Ix(sz) - x(s1)I

S

Ix(SI') - x(S1)I + Ik(SI') - x(SI')I + Ix(S!)

-

< C2 2-(m+n)/a + 2 Y C2 2-(m+i)I3 + C2 2-(m+»)/8 C3

2-m/3.

x(s2') I

1. The standard BRowNian motion

16

x (s) (s E t) can now be extended to a continuous function on Co. + co), and since this function is a member of n (B.) c n C, , the proof nZ1 x21 is complete. Now P can be extended to a BoREL probability measure on the BOREL extension B of C; with this extension, the triple [W, B, P)

is called standard Brownian motion starting at 0. P is the so-called Wiener measure.

Given a E R', the same proof justifies the extension of the set function 24)

Pa (C)

=1 g(tL, a, b,) db1 g(ts - t1, b1, b2) db=

C = x='(B),

BE B(R"),

n

to-1 ba-1, bn) db,

1

to B.

iii

Because g(t, a, b) = g(t, 0, 1 b - a

=P..a(B) BE B, where w + a is the translated path x (t, w + a) = x (t) + a and -w is the reflected path x (t, -w) = -x (t) . Pa(B)

25)

Because 26)

Pa [x (0) = a] = Po [x (0) = OJ = lim l im Po [lx (t) < n-'] "-it- 1/2

* i02 Jr

gays v-2n

db = 1,

0

.-.

P. (B) is to be thought of as the chance that the event B occurs for the Brownian path starting at a E R'. D = (W, B, Pa : a E R'J is the standard Brownian motion; it is a collection of individual stochastic processes, one to each point a E R', knit together in a certain manner as we will now explain. With the help of 24), it is clear that 27)

Pa [x (t) E d b lx (t1) , x (t2) ,

=g(t-tn,x(tx),b)db

... , x (tn)] =

[x (t - tn) E d b]

t>t,Z...Zt=Zt1,n? i,a,bER',

000

or, what is the same, 28)

Pa [x (ts) E d b JBt,] = Pe [x (t$ - t,) E d b]

c = x (t1), t2 ? tl, a, b E R',

where B4 = B [x (s) : s S tl] is the smallest BOREL subalgebra of B

including all the events (w: a S x(s) < b) (s S t,); this is the so-called simple MARxov property. 28) exhibits the knitting together of the individual BROwxian motions; it states that the Brownian particle starts from scratch at time tl; in more

17

1.4. The standard BRowxian motion

precise language, it states that, conditional on the present b = x (tl) , the /uture x (t + tI) (t ? 0) is a Brownian motion starting at b, independent of the past x(t) (t < 1I); it is this starting afresh which the adjective Markovian describes.

Consider the operator 03 u = u"acting on D (0) = C2 (RI). Given / E C (RI), the (bounded) solution of

aufat = 0 u

29)

u(0+, ) = / is

u =u(tra) = Ea[/(xi)] =

30)

RI

e

2n t r f db,*

J

+00

and its LAPLACE transform u = f e-at u dt (a > 0) is the (bounded) solution

0

+00 e+1

u = u(a a) = E.

31)

J

r

e-°tl (xi) dt = RI

0

fdb**

Y2

of

(a - () u=/:

32)

in operator language, if et 0 / = is (1, ) (t > 0) and G. / = u (a > 0) , +00 :Z7

then et0 maps C(RI) into C°° (RI), (a/ot - 0) etc = 0, Ga= f e- I et*dt 0 maps C (RI) 1 : 1 onto C2 (RI), and Ga'= a - ('i. ( is said to generate the Brownian motion. G. is the so-called GREEN operator of the BRowxian motion ; see 2.4 for applications. Problem 1 after Y. KOMATSU [1]. Check that

as + 4 + a]-I

VII

2[

ea'l2 f a-b'12 db S 2[ a2 + 2 + a] -I

for

a

o.

a

[Writing g_ for the under-estimate, g+ for the over-estimate, and g

-

Cow

.M.

for the modified error integral in the middle, a s'mple computation gives g'_ _> a g - 1, g' = a g - 1, g+ S a g+ -1; therefore (g - g _ )' S S a (g g-), (g+ - g)' S a (g+ - g), and using g-, g, g+ S a-', it follows that g - g_, g+ - g cannot become negative.) See I. PETROVSKY [2: 241-244). + co

I

r2a

e-=e

y2nt

dt; see A. EitasLYI [1 (1) : 146 (27)).

0

I0IMc1ean, Diffusion processes

2

1. The standard Baownian motion

18

"5-

Problem 2. Polliimtx(i/t) = 01 = 1. 1+0 Problem 3 after P. LEvY [3: 246]. Consider a standard BROWNian motion starting at 0; the problem is to show that

t=0 t>0

a(t)=0

A

tx(1/t) and

t? 0

b (t) = c x (t/c1)

(c > 0)

ion

are likewise BROwxian motions starting at 0. [Both these motions are GAUSSian with mean 0, and E0[x (s) x (t)] = E0[s x (1 /s) t x (1 /t)]

= E. [c x (s/c') c x(t/c')) = s A t .] t4'

Problem 4. The BROWNian motion is differential in the sense that

its increments x[ai, bi) = x(bi) - x(ai) (i S n) over disjoint intervals [ai, bi) (i 5 n) of (0, + oo) are independent. + c x", check that Problem 6. Given / = co + cl x + c= x2 +

u(t, a) = G/ = V n

t"(6"l)(a) x!

O

is the solution of au

at = c u

u(0+. ) = 1.

and deduce, for the standard BROwNian motion, i.'

n 1. Eo(xt") =1.3 ... (2n-1) t' Problem 6. Give a direct proof that 31) is the bounded solution

of 32) and use this to deduce the formula +00

e-0-u+/srdt

e- j2,x

y2nt

2a 0

of the footnote to 31). '1c

Problem 7. Give a proof that almost all BROWNian paths are nowhere differentiable. R. PALEY, N. WIENER and A. ZYGMUND (1) discovered !ti

...

this; the proof suggested below is due to A. DVORETSKI, P. ERDds, and S. KAKUTANI [3].

[If the BROWNian path is differentiable at some point 0 S s S 1, then

lx(!)-x(s)l
I and some m

nzm

(!Z1

1. But this event is part of the event

U mal U Sam n O
19

1. S. P. LEVY'S construction and I

P°

n

[

k

o
S(n+2)P°Llxln)I<

=(n+2)( 7: (i<-;,

a-4'19

f2n

k- 1 n

)I<

n)] yl

nl,a

)8< constant i o

d

j/n

(nt +oo).l

1)

1.5. P. Levy's construction P. LEVY [3: 15 - 20] gave another construction of the standard BROwNian motion; the use of HAAR functions below is adopted from Z. CIESIELSKI [1].

Bring in the HILBERT space L2[0, 1] with an orthonormal bas's defined by the HAAR functions

0S1

f0(t)=1

1a) 1 b)

fk 2-"

(t) = + 2("' I)/= (k - 1) 2-" S t <

1

k 2-"

=-2("'I)m k2-"St<(k+1)2-" kodd <2",n? 1. 7..

Given a standard BROWNian path x(t) (t S 1) with formal derivative x' (= white noise), the formal FoURIER coefficients I

go= f fox* dt=x(1)

2a)

0 I

2b)

k odd < 2", n> I

gk 2-" = f f k 2-w x* d t 0

= 2(n-I)12[2x(k2-") - x((k - 1) 2-") - x((k + 1) 2-")] should be Gaussian: a

f

P(g < a) =

_ db,

e-4'19

y2n

-00 rI

E (gk 2-" gJ 2 ..) - .l fk 2

/52d( = 0

(k 2-" + j 2-'")

0 44))

and hence independent (see prerequisites), and the integrated expansion i

c

S)

go f /ods

I

ALd

koddG2"42-442- ds

2

20

1. The standard BxowNian motion

should converge to x (t) (t S 1) ; the idea is to use 5) to define the BRowNian motion.

Consider for this purpose, the algebra B[0, i] of BOREL subsets of [0, 1] with the classical BOREL measure P attached, let 0 5 u S 9 be expanded as u=2-1u1+2-2us+2-3U3+...=.UIUSUs...,

a

let h be the function inverse to f e-b'12 d b f 2n, and defining - CO

V1 = u1 US u4 n[7 ...

Vp = 0 U3 ub US ...

V2 = 0 0 ua u9 ...

V4=000.Ujo... eic., let

go = h (v1)

911, = h(v9) g1/s = h (v5)

9114 = h(v3)

g3/4 = h (v4)

gs/s = It (v3)

gs18

= h (v7)

g718 = h (v8)

etc.,

and note that the g's are independent with common distribution 3). Consider now the sum t

t

em = go f fods + T X gk2-of n$m oddk<2"

6)

fk2-. ds.

Because 7)

rAt2-wds = 2-(n+1)/2 1 odd k < 2"

~`a

t

(1

as is clear from a picture,

Men - en-1II* S

8)

so that +00

9)

P [I I en

- en-1

2 .2-" lg 2n] < 2n

.-1 e-612

y2-t

<2

_ynig2

Using the first BOREL-CANTELLI lemma and the estimate

,F * 11111 = max ! f (t) I.

t51

2,2 mlg2mC lg2

V2,2-nlgr,

1.5. P. LEvY's construction

21

it is immediate that e converges as n t + oo to a continuous function e°, and that 10)

nt +oo] = i P[(I e,, - e"II S lg2 Because e was GAussian, so is eoo ; since E (e") was 0, so is E (e.);

and to complete the proof that [e.: 0 S t S 1, B, P] is a BROwxian motion, it suffices to check that E [ec° (s) e°° (t)] = F_o[x (s) x (t)],

11)

i. e.,

t

0

r

"`'o

t

f to f to +

12)

c =S odd2; /tk2-a/1k2-a

Li

0

,-I

d-- ...,

!+i

But this is obvious from the PARSEVAL relation. Problem 1 after P. Li:vY [3: 19]. Use the BOREL-CANTELLI lemmas

to find the lower bound of the values of c for which lie, - e"-l l l < < c 2-"lg2" as r-.

[c=1].

Problem 2. Use the result of problem I to check that lim sup

P

e°° (t,) - e°° Cu) l? yElg(I /E)

l0t54<4 1

1=1

.-.

(see 1.9 for more information on this point). [e°° (m2-") - e°, ((m - 1) 2-")] + j [e. (m2-*) - eO° ((m + 1) 2= e"(m2-") - e"-1 (m2-") for m = 1, 3, ..., 2" - 1; therefore m ax [e° (m 2-") - e°° ((m -1) 2-")] e" - e"_ 1

now use the answer to problem 1.] 'ti

Problem 3. WIENER'S construction of the standard BROwNi.an mo-

tion; see R. PALEY and N. WIENER [1: 140-162]. Let g", n:>- 0, be independent and GAUSS distributed: a

f

VII

P(g" S a) =

°°

e-b'12db. V2n

Then

x (t) _

go +

'

n

sinkk t gk

Z-E 11 Y

"Z 1 k-2"-

Iv'

converges uniformly in 0 S t S x with probability I and is a standard BROWNian motion.

[Put Sm, (5) =

S'1 si kk t M

gk

tmn = o S x. j sm" (t) j;

1. The standard BROWNian motion

22 then

+2

S

n-1

(E(tmn))2 s E(tl2n)

-i;Ifn-

"

k + 2 T' t-1

i-1

i

-+n

(E(I

n-- i M

n-1 1

g,g,+1 I. 1(1'}'

g1Kj+i

In

M2

!

f-m 1(9 +) I /)) n-i-1 I

i

S I k, + 2 "-m-1 i V (-M M S

.^i

2 n-1S 1k in max, n_ Z1ekrA

tz

+ 2(n - m) ( is m'm

E(t,n.l,n) s V3 m-114,

E[ I

+oo,

t!"-',!"l :9 X

Pl(n2

+o01 = 1,

and therefore our series converges uniformly with probability i, so that almost all sample paths x (t) (t S x) are continous. It is now enough

to check that E[x(t) x (s)] =

t+

n

sZ

sinki inks = sAt.l

1.6. Strict Markov character

Consider the standard BRoWNian motion D, let w, be the shifted path xt (w;) = xi (w) (t z 0), and, as before, let B, be the smallest BOREL subalgebra of B including all the events (w : a:< x (t) < b) (i 5 s). D is MARKOVian in the sense of

P. [w,EBI B,]=Pb(B)

1)

b=x(s),BEB

(see 1.4.28)), i.e., the Brownian traveller starts from scratch at each time

t=sZ0.

BRowxian motion enthusiasts are familiar with the fact that the Brownian traveller also starts afresh at certain random times, such as the passage time ml = min (1: x (t) = 1) *.

G. HUNT [1] discovered the complete statement of this feature of F..

the BROwxian motion.

Consider a MARxov time 0 S in = m (w) (S + oo), i.e., let 2)

(m<1)EB,

and define Bm+ to be the class of sets B E B such that 3)

'Bfl(m
tZO.

min (I: x (t) = 1) = +00 if the path never meets 1. * (m < 1) is short for (w: in < t).

1.6. Strict MARKOV character Bm+

23

is a BOREL algebra and (m < t) E Bm+ for each t > 0; in addition

Bm+)>ofl B[x(tA(m+e)):tz0],

4)

as will be seen below. Bm+ is to be thought of as measuring the Brownian

path up to time t = m+. HUNT'S statement is that, conditional on the present position b = x (m), the future path x(t + m) (I ? 0) is a standard Brownian motion starting at b, and this Brownian motion is independent o l B.+; in the language of conditional probabilities,

m < +oo, b = x(m), B E B,

P.[wm E RI Bm+] = Pb(B)

5 a)

i. e.,

5b)

P.[A,w+mEB,m<+oo]=E.[Afl(m<+oo),Px(m)(B)]

AEBm+,BEB; in other words, i f in is a Markov time and i f m (w) < + oo, then the Brownian traveller starts afresh at time t = in (w) . Before proving 5), it is convenient to examine the MARKOV times m and the algebras Bm+ in a little detail.

Given t ? 0, m I is a MARKOV time, as is clear from 2), and B,,,+ = Bt+ - n B,; indeed, if B E Bm+ and if s > t, then B = B a>t

n (nl < s) E B,, while, if B E fl B,, then 1>t

s>t, sSt,

Bn(m<s)=B

=0

and B E Bm+. VII

Given MARKOV times ml S m2, if B E Bm,+, then B n (ms < t) = B fl (m, < t) n (m2 < t) E Bt for each t ? 0, i.e.,

COa

Bml+ C B....

6)

CMS

Given MARKOV times in and m, ? m2 ? etc. I in, if B E n Bm"' n'>l

_/M

then B n (m < t) = U B n (m, < t) E Bt for each t z 0, i. e., using 6), nZ1

n Bm"+ = Bm+.

7)

nal

Coming to the proof of 4), if m is MARKOV, then m A t is measurable Bt

for each t ? 0 as is clear from 2), and, using the fact that x (t As, w) is a B [0, + oo) x Bt measurable function of the pair (s, w)', it follows that x (M / i A s) is measurable Bt for each s ? 0. But then .

(w:x(mns)
1. The standard BROWNian motion

24

(t ? 0). i. e., (w : x (m n s) < b) E Bm+ for each s ? 0, and, using 7) and the fact that m + e is MARKOV for each r> 0, n B [x(t n (m + e)) t Z 01 >0 is found to be a part of n B(m+4r)+ = Bm+, as stated in 4). a>0

Coming to the proof of 5), it is enough to show that, for each Bmeasurable e (0 S e S 1), E.[B, e(w+,), m < +oo] = E.[B, E=(m)(e), m < +oo] B E Bm+.

8)

Because the tame functions a (w) = / [x (tl) , x (Q. ... , x (Q)

9)

0
10)

t' 0,

and 11)

r E.(e) _ ,I g(ti, a, b1) g(t2 - t bl, b,) ... g(1, - t.-t, b.- 1, b.)

jr

/(bl, b2, ..., b,) dbl db2 ... db.

E C(R'), and so, in this special case,

12)

E.[B, m < +oo] _ "fim E. (B, e(w.,,,), m < +co] + Go = xhi

k,

m" = 2-"([2"m] + 1) (n 2:

1)

l E.[B n(m. = k 2-0), e(w; 2--)]

"

t+ 00

Z1 E.[B n((k - 1) 2-" S in < k2-"), E:($-")(e)] VII

_r

CC

= Afm kV E.[B n((k - 1) 2-" S in < k2-"), e(wL2--)]

_ "timo E. [B, E. (m-) (e), m < + oo]

= E. [B, E. (m) (e), m < + oo], as desired.

Problem 1. Give an example of a MARKov time such that

(mSt)EB, Give also a MARKOV time for which this is false. [m, = min (9: x (t) = 1) ; mo. = inf (I: x (t) > 0) .] * B n ((k - i)2--S m < k 2-") E

now use 1).

t1-> 0.

1.7. Passage times for the standard Bxowxian motion

25

Problem 2. BLUMENTHAL's 01 law (see R. BLUMENTHAL [I]). B0 is

trivial in the sense that BE B0 .

P. (B) = 0 or 1 [P. (B) = E. (B. P. (B I B0+)) = E. (B. P. (o) (B)) = P.

1.7. Passage times for the standard Brownian motion The most important MARKOV times are the passage times:

aER'.

ma=min(t:x,=a)

1)

P. LEVY [3: 221-223] has shown that [ma : a ? 0, PO] is the onesided stable process with exponent 4 and rate V2, i.e., it is differential and homogeneous, with law 2)

f b - a e-cb-a)'12 ds

PO[Mb - M. S t] = Po[mb-a S t] = 0

2n

b?a,tZ0

(see note 1).

Beginning with the proof of 2), if a z 0 and if f is the indicator function of [a, + oo), then, using 1.6.5 b) and the fact that e-a m = 0 if m = + 00, +00

3)

f

ewe P.(xi

z a) dt

0

W

fe

Eo

(x1) dt

0

+m

=Eo

f e_aef(xi) dt)

(m= ma)

M

=EO[e-'- EO( ('tee-atf[x(t,wm)]dtI Bm+)] 0 _ell

EO(e am.) E. f e

/(x1) dt

0

+00

Eo(e-am.) f e-"1Pa(xg L> a) dt

=

0

= (2a)-' EO(e-"m-) + 00

=

(2a)-1

f

e-°`iPO

(mnEdt)

0 +00

e--' PO (ma S t) dt, 0

0 m is measurable Bm+

1. The standard BROwNian motion

26

proving the celebrated refkction principle of D. AxnitE: 4)

PO (ma

t) = 2 PO (xi ? a) +cc

=2

6f=jdb f yen:

1,aZ0.

a

,M,o

2) is now immediate from the formula: +*0

t

SIN

Ya

2 f y2E!L db= af

a/

1, a 2 0,

e-a1128ds

0

and, at no extra cost, it is found that P. (mb < + oo) m 1. Because of 4),

+00 r E0(e-am-) = 2a f e-atdt r +00

5)

e-a+/ss

db

y2nt

J

a

0

f

+00

=2a

y2a

db

a

=

e-yQaa

using the footnote to 1.4-31); more directly, it is clear from 2) and the first integral in 5) that g E0 (e-am-) is the solution of 6)

9" = tag,

g (O) = 1,

g(+ 00) = 0,

and solving this gives 5). To complete the proof that ma (a L> 0) is a one-sided stable process, it is now sufficient to check its differential character. Given 0 < a < b, 7)

mb = min (t : x (l) = b)

=m+min(t:x(t+m)=b)

m=ma<+o0

= m + mint : x(t, ze) = b) = m+mb(wm), '.'

8)

co'

so that, using 1.6.5 b) and the fact that e-a m = 0 if m = + oo, E0[e-asm a-w

= E0

ll

14--al m- E0 (e ''m'(a"`) I Bm+)J

= E0 (e-N-m-) Ea (e-a,'"')

= E0(eim-) E0 le = E0(e-&-m-)

E0[e-a+(me-m-)).

1.7. Passage times for the standard BxowNian motion

27

A similar computation shows that for

a1,a2,...,an>0 and 0
Eo[e

9)

Eo [e at mgt] E0 [E as

gnat)]

. . .

)] ,

E0

which completes the proof.

.-.

Coming to the actual sample paths ma (a z 0), it is clear from +m -a 1 10)

E. (e-a me)

o'

y2nl

JJJ

=e-Y2aa=e

0

az0

that 00

...

ma = f l4,([0, a) x dl),

11)

0

where 12)

p(da x dl) = the number of j u m p s o f m a g n i t u d e 1 E dl that ms (b_>0)

suffers for b E da

is the POISSON measure with mean d a x v2n

P

(see note 1) ; in parti-

...

cular, mb is a sum of positive jumps. Note that (0, a) is used in 111 instead of [0, a] as in note 1; this is because ma is left continuous. The maximum function t- (t) = max x (s) (t z 0) is the inverse ,fit function of the passage time process m; it is continuous, and, as such, it looks something like the standard CANTOR function (see 2.2 and problem 5 for additional information on this point). Problem 1 after P. LEvY [3: 211]. Give a proof of the joint law PO[xtEda,maxx,EdbJ

- (._jr)"2(2b - a) e2b)'12tda db 2

[Given b-'> a, +co

VII

f e-tdtPo[xt Sa,mnxx,ZbJ 0 oo

= EOlm*b f e a!f(xt)dt1 +00 W-0

=

E0(e-.2m*)

Eb a

= e-Viab

f e-"t f d) _

1

y2a - 00

1

1 0

dc,

t > 0, 0

b, b

a.

1. The standard BRowNian motion

28 c>1

where / is the indicator of (- oo, 4]; thus, +00

f e-"idtPolxiEda, ax xsEdb, =

2e-y,2"(2b-a) dadb;

0

now use 2) and 6).]

1'+

Problem 2. Use the result of problem 1.4.3 to give a new derivation of the passage time law 5). [Given c > 0, min (t : c x (1/c') = b) = CS mb/C and mb are identical in law, and, using the homogeneous differential character of mb (b ? 0), it follows that Eo (e-O'm6) is an exponential a-bo(o), whereg (cza)/c = g (a)

(c > 0), i. e., g (a) = g (1) ya (a > 0) ; now compute the constant g (1).] ...

Problem 3. Check the arc sine law of P. LEvy [3: 216] : that, for the

greatest root b- < t of x(s) = 0,

't

('

1

P. (b_ S s) =

=? arc sin j s!

dI

J

t Z s Z 0;

0

compute also the distribution of the smallest root b+ > t of x (s) = 0 conditional on 6+00

[Po(b- Ss) =Po[mo(wf) > i - s] = f Po[x(s) E dal Pa[mo > t - s] -00 +00

=2f

a

da

y2ns

0

-

+ CO

a

t-:

y2:2is

e-.0121 d l

=a

('

t

dI

f

0

By a similar computation, Po[b+ E d b 16- = a]

(t-a)j(b-a)Sdb,2(b>t).] Problem 4. Use the formula

(27t i)-1

I+i CO

CT7

i

f

e" ei' da for the inverse LAPLACE

1 -1 00

transform to check ei'

+00

= JI e'"' y22t dt. o

(11-n)

Diagram I

_

[Working on the branch j =1'N etm/2 (a = , a I e1 ') , if I' is the closed curve pictured

1.7. Passage times for the standard BROwnian motion

29

.ti

..r

in the diagram, then (2n i) -1 f eta a-Y2n da = 0, and, letting n t + oo, it results that r 1+i0 _ (2n i) -1 f eta a-r2a da 1-ioo

-00

o

f eta+02H da + (2n i)-1 f eta-iV2jal da

_ (2n i)

o

CO

+00

+00

_

r J

e-ta sin

da = y

j

r

f

e-ta' cos

V a da =

'

e-1/2t

yens

1

J

0

0

«»,

a^_

Problem 6. The set 8 of roots of x (t) = 0 is a topological CANTOR set (closed, uncountable, of topological dimension 0, having no isolated point) 1; its LEBESGUE measure is 0. [8 is closed because the BROwNian sample path is continuous, of LEBESGUE measure 0 because +00

E0 [measure (8)] = f dt P0[x(t) = 0] = 0, 0

and of topological dimension 0 for the same reason. Because

t + mo (wi) E t as a function of t, lijmE0f a a(t+n,o(w+))]

10

= i oEo[e-k2al=(t)I] = I

implies that Po I l m (t + mo (wf)) = 0] = i , and, seeing as t + m0 (wi) E 3 t+O

Given t! > tl,

if it is < + ao, 8 is infinite. Define m = t1 + m0 P0 [x (s) has just one root between t1 and t2]

S PO (m < t2, l j ( + m0(w;+m)) > 0] S Po[l;o m0(w4) > 0] = 0, 80

i.e., 8 has no isolated point ; it follows from this that 8 is uncountable, and the proof is complete.]

Problem 6. Given a < < b, E.[eame, Mb

< ma] _

a)

sinhy2a(b - a) [e_a In , ,, ma < Mb] _ sinhy2a(b Et sinhy2a(b - a)

PE[mb <ma]=b ba -a EE

[ea m. A 1116]

PE [ma < mb] = b ` a

_

1 See ALEKSANDROV-HOPF [1: 45].

cosh V2& d coshy T a (b - a)/2

,

1. The standard BROWN Ian motion

30

and El[eam,nme]

=

cosy2ad

a < n=/2(b - a)2,

Cosy20, (b - a)/2 44A

where d is the distance between

and the midpoint of [a, b].

(Given a < < b, if m = man mb, then e-V aU - a)

= EE Le

a ma] =

Ee [e-a m, ma < mb] + E! [e-a m, 111b < ma] e- Y2 a (b - a)

and

= Et [e

a m+] = EE [e-a m, ma < mb] e- Tr i (b - a) + E f [e-a m, Mb < ma] ;

now solve for EE [e-°` m, ma < mb] and EE [e-a m, ma < ma] .]

Problem 7. Prove (b - a)'

Pa(xt E db, mo > t)

=

[e

se

(b

-e

a)' sr

V

Idb a b > 0.

[Use problem 1.]

Problem 8. Show that, for the standard BROWNian motion,

PP[x(t) E db, t < mo A Mil = X (-)"g(t, a, b")db InI?o

t>0, O
where g (t, a, b) is the usual GAussian kernel and b; = b + 2 x,

b:n-i = -b + 2n. [Given a BOREL subset B of [0, 1 ] , if B. = (b.: b E B), then

0
P4[x(t)EB,t<m]

m=moi\m,

P. [x (t) E B", t < m] n

P. [x (t) E B,.]

-X (-)" P. [x (t) E B", t ? m] ; n

now use the symmetry of the standard BROWNian motion to show that

P.[x(t)EB,,,tZ m] n

= f P. [mo < in,. m0 E ds] In Zo (-)" PO [x (t - s) E Bn] -{-

1

0

= 0.

[x (t - s) E

P. [m, < mo, m, E d s] InI 9:O

1.7. Passage times for the standard Baowxian motion

31

PQ [x (t) E d b, t < m] can also be expressed as g' (t, a, b) db, g' (t, a, b) being the fundamental solution 00

2 Y e-n'n`""2 sinn;r a sinn x b n-1

of s

du

I

8t

2 db=

u(t, 0+) = u(t, 1-) = 0, (see 4.11) and this can be transformed into InI

' (-)" g (1, a,

with the

o

aid of one of JACOBI's theta function identities.) Note 1. Homogeneous differential processes with increasing paths. A stochastic process with sample paths p (1) (t z 0, p (0) = 0) is said to be differential if its increments p [tl , t=) = p (t2) - p (t1) over disjoint intervals [t1,12) are independent, homogeneous if the law of p [tl + s, .-.

12 + s) is independent of s(;=>O), and increasing if p (tl) S 1f (ts) (tl S t=) .

P. LEvy [1: 173 -180] showed that if, in addition, p (t +) = p (t) (t ? 0), then 1) E[eav(')] > 0, where +OD

(a) = m a + f [1 - e"t] n (d l)

2)

o+ +OD

m ? 0, n(dl) z 0, f [1 - e'] n(dl) < +oo; o+

+°+

n (dl) is the so-called LEvy measure of the process. Corresponding to this formula, LEvy [1: 173-180]1 decomposed the sample path into a linear part plus an integral of differential Poisson processes : +00

P (t) = m t+ f l p ([0, t] x dl)

3)

0

e

o+

p (dl x dl) being the Poisson measure with mean dt x n (dl), i.e.,

4,

is

Poisson distributed: B

v°x

4) P [p (B) = n] = n e e n z 0, P = f dtn (dl), BE B([0. +00) x (0. + oo)),

that p (U

"al

o°$

't7

di//erential in the sense that the masses p (B) attached to disjoint B E B [0, + oo) x (0, + oo) are independent, and additive in the sense X .p (B") for disjoint B,, B8, etc. E B ([0, + oo) x (0, + co). n2:1

1 See also K. ITO [1].

* P[V(B) = + oo]

I

if

P = +oo.

32

1. The standard BROWNian motion

is just the number of jumps of p(1); tt S t < is of 1+1A

p([11, to x [11, 10) ...

magnitude 11 S 1 < 1$, as is clear from 3). Here is the proof of 1) and 2).

Given a > 0, the differential character of p implies that e(t) _ = E [e- 11(l)] is a solution of

0<e51, A

a(t)=e(t-s)a(s) t?s,

5)

proving

a?0, 1Z 0.05Tp<+oo.

e(t)=e'!r(a)

6)

Because Ltd

(a) G t,(a) = -t-1 saa tr = -1 E[p(t) e-a'(l],*

7)

8)

w' (a) = ll ym t-1 f 1 e-411 P (p (t) E d t)

is the I.APLACE transform of a non-negative BoREL measure m (dl) on [0, + oo) , and using ip (0) = 0, it is seen that a

a

+m

+ao

9) Ye(a)=fo (P)d/3= fdii f Gaim(dl)=am(0)+ f [1-eat] 1-1m(dl); 0

0

0+

0

in other words, +00

y'(a)=am+ f [1 -eat]n(dl)

10)

o+

+m

with m=m(0),dn=1-1dm (1> 0), and 1(1-e-t)dn S o(1) < + oo 0+ as desired. Coming to 3), if p (dt x dl) is the POISSON measure with mean

dt x n (dl), then +00

p (t) = t m+ f 1 P ([O, t] X dl)

11}

o+

is differential and homogeneous; in addition, .N+

12)

E

-a f :y((o.t)xd0 e-011" E e 0+ +00

-a f 2-"[2"zJv((o:c]xd!)

= Cain lim E e e+ *t+00

-lf

= e-alfx lim e 0+ fit+ao +00

e-6' is a bounded function of p Z 0 if a > 0.

(i-s'as-"[s"t])»(d!)

1.8. KOLMOGOROV's test and the law of the iterated logarithm

33

and it is immediate from this that 11) and 3) are identical in law as stated.

Call p (t) : I z 0 one-sided stable if to each a > 0 corresponds a = fl (a) > 0 such that a p (t1E3) : I z 0 is identical in law to p (t) : t a 0; in this case, 13)

e sPC'Yir(') = E[e-P(P(aY)t)] = E[e--YV(t)] = E(e ='('B(Y))]

= E[e-'('B(a)g9(Y))] = e-1p(a)A(Y)Y(1), fl(ay) =P(a)P(Y),

14)

and, if tp (1) > 0 (p m 0 in the opposite case), then p (x) is a constant multiple c = y (1) of P= a` with a> 0 because yr E t and e S 1

because v' = m +

J*

l e-*1 do E +.

a is the exponent of the stable +00

process; the constant is its rate. p=ctife=1; p=mt+ f lp([O,t]xdl) cedt °.+ and n(dl) = 1'{1 _ e) t'+. if r < 1 1.8. Kohmogorov's test and the law of the iterated logarithm

Given a standard BltowNian motion starting at 0 and a positive function h E C (0 , 1], the event (w: x (f) < h (t) , 11. 0) E Bo, and an

application of BLUMENTHAL'S 01 law implies that PO [x (t) < h (t), 14 0] = 0 or 1; h is said to belong to the upper class if this probability is 1 and to the lower class otherwise. A. HINCIN's celebrated local law o/ the iterated logarithm [1: 72-75] states that h (t) = (1 + e) V2 t 1g2 i ** belongs to the upper class if E > 0

and to the lower class it e < 0; i.e., 1)

(1)

P° i o

x

V21lga t

=1

1'

as will be proved below.

Kolmogorov's test states that if h E t and i/ t-212 h E + /or small t > 0, then h belongs to the upper or to the lower class according as !' t-a/s h e-)''121 dt converges or diverges. o+

* See P. LisvY [1: 198-2041 and S. BOCHNER (1] for additional information on stable processes. ** 192 = lg (Ig) It6/McKean, IKlfasion processes

3

1. The standard Baowxian motion

34

ono

:"1

1) is a special case of this; see problem 2 for additional examples. KOLMOGOROV'S proof is unpublished, but I. PETROVSKI [1] gave a proof based on PERRON'S ideas about the DIRICHLET problem (see also P. ERDts [1] and W. FELLER [2]). ((9

We will use the passage time distribution

t

hV

Zir/L

S t) =

PO (ma

2)

J

al'/s' d s

2x s$

aZ0

(see 1.8.2)] to establish the cheap half of KoL-

/.

I MOGOROV s test: that =f n E t and t-1/t ac * for

small

Di'Sram I

then

t,

the convergence of the

integral

f t-Al' h e-A'l't dt implies Po [x (t) < h (t) , t4 0] =1 ; 0+

.a)

the other half will be proved in 4.12 using the elegant method of M. MoTOO [1].


Given 0 < a = tl
that h E f for

such

t S b, it is clear from the diagram that h (t) for some a S t S b] 3) P0 [x (t) S P0[mA(a)S a] +

.

Pot,,,-1 < mA t tw

a

-f _

0

S tm

h(a) y2 z s

r h(tw-1) a-he--1)1/28dt

mX 2: 2 J

X

tw_I

i'

and, as the subdivision becomes dense in [a, b], this goes over into 4) PO [x (t) ? Is (t) for some a S t S b]

f

a/A (a)'

0

Because t-1/2 h E

-1128dt

2AN

D

h(t)

+, y2rN

-4(8)'128

a

dt.

a

for small t, the convergence of f t-$/' h e-1'/2t dt 0+

VII

implies that lint-1/2 h = + oo, and letting a 4 0 in 4), one finds that 10 5) Pa [x (t) ? h (t) for some 0 < i S b] a

J

2XN

0+

But this overestimate decreases to 0 as b 4 0, so the proof is complete.

Consider the special case h (t) = (1 + e) V25 lg2 j (e > 0) ; then h E t for t< e-1, t-lh h E 4 for t S 1, and f t-8/2 h e-A'/Qt dt < + oo, 0+

1.8. KOLMOGOROV'S test and the law of the iterated logarithm

35

proving

'

_x()

{ Po liio

6)

S1=1;

V211g2

the proof of 1) can be completed as follows. Given 0 < e < 1, and ignoring the exceptional class of BROwNian

paths for which i m x (t) /VV lg2 l > 1, if 840

An:x(e") <(1

3yE)y2En-1926-

- x(?') <0 - aye)

2e lg2E-" -{-

as n t + oo, then Bn : x (E")

y2E"+I192E-n-1 2

as n t + oo. But 1- PO (Bn) = 7)

f

e-W

constant

2db 22c

J

n-o

ylgn

BVS lg, s-" nt+oo.9='-ys

f -a <1, and, using the independence of the events B. (n

of the sum 8)

P. (U 1n 2:

lgn

nZ2

it is found that

( Am] S Po (U ( B]

man

1) and the divergence

lln2: man

lim Po (n B,,,] = 0, ==,,t +OO

loran

proving 9)

1 = Po [x (e") ;>= (1 - 3 I e) y2elg2e-" i. o., n t + oo]

S PO lim t{o

1/211g2

i

3y

1) is now immediate on letting e 1.0.

Problem 1. Prove that if h belongs to the upper class then l+mt-112h = +00. 8

o

110

lim Two

-CO

e'"1=db = lim PO [x (t) < h (t)] = 1. yZn a-o 3*

36

1. The standard BeowNian motion

Problem 2 after P. ERDOs [1]. Use KOLMOGOROV'S test to show that MIN

h (t) = V2 t 11g. +

+ 2 lga t + Ig. j + ... + lg.-1 t + (f + e) Ig.

]

ti..

belongs to the upper or to the lower class according as e > 0 or e S 0; show also that t 3 t 1 ls* h (t) + 2 Ig. 1 -}-% 2 29 Il Ig. lg+

-I-

-r

I

belongs to the lower class. Problem 3. Use KOLMOGOROV's test to prove that if h E C (1, + oo) (11

is positive and if t-' h E 4. and t-'t' h E t for large t, then + 00

P° [x (t) < h (t) ,

t t + oo] = 0 or I according as f t-312 h e-A'12 t d t

+ oo.

Hinern's global law of the iterated logarithm:

P.

=1=1

x (t)

lim

t+m )121 Iga t

is a special case of problem 3; see A. HINCIN [1]. [Use the fact that [t x (111) : t > 0, P°] is a standard BROWNian motion

starting at 0; see problem 1.4.3.] 1.9. P. Levy's Hiflder condition K. L. CHUNG, P. ERDOs, and T. SIRAO [1] showed that if h E C(0, 1] ,N,

'""

is positive and if h E t and t-'t' h E 1. for small t, then P°

-1)

max

IO:S4g'
l x (t,J - x (tl) l < h (e) , e 4. 01 = 0 or 1 according as

f t-7/4ha e-''/2tdt. +oo; o+

2)

,

then h E t fort

e-', t-'t' h E I

ft'7I2hae-h+12edt=2}2(1+g)a fllgtl'lst(l+al''sdtz+oo a50, o+

0+

establishing P. LEvY's HOLDER condition : P°

.~V

3)

VII

for tS1,and

i

C'.

for example if h (t) = (1 + 6) V21 lg

lira

t, t3-'4.0

0its
I x(ta) -- x(IJI

l`/2e1g1 r

= 1 = 1.

e

t)

(n g 2).

1.9. P. L*vv's HOLDER condition

37

LEVY'S proof of 3) [1: 168-172] is a model of its kind and is presented here.

Because I - s < Po[m x [x(k2-")

4)

llk:S

2*

-

if 0 < 6 < 1, then x((k - 1) 2-")] S (1 - 6) 2 2-" log2" j e-V!sdb 24

= 1rr

< I1 -

J

y2n

(1- 6)}'2)g2" 2 w(1-d) ]20

n

n t + oo

l0

and, letting 61o, it follows that ix(it) - x(t1)I

lim

P0

5)

2e ig

o Ir

I

z1=1.

E

P°

6)

Ns'

Consider, for the proof of

lira 1L 5-1 = 1,

1-r,-110

0Stl«51

V2elge1

and a number 0 < 6 < 1, and using the

.'7

the function h (e) = j/2e lg estimate 7)

1/

max wd h (12-")_l I x (92 2-") - x (,12-") L> 1 + 6 l J-J.-JI52 J

Po l 05J1<Ja52w

e-+'Isdb

S 2(1+6)n X2

y2

(1+6) Y2 ]We

<2

d(1+6)n

n t + 00,

conclude from the first BOREL-CANTELLI lemma that I x (92 2-n) - x (712-") < (1 + 6) h (9 2-n) I

8)

0511<5,< 2n.

9 =72-91 5 216,

nZn(w),

where, for later convenience, n (w) is chosen so as to have 2(n+1)6-1 > 2, 9a) 2-"(1-6) < e 1,

9 b /)

and 9 c)

' h (2-"') = h (2-")

m>n

for n Z n (w).

Ma1 V 2a"

1g 2w

lg2

w

< 6 h (2-(1 + 1) (1-d))

1. The standard BRowNian motion

38

Given 0 S ti < t1 S 1 such that e = is - t1 < 2-"(10)(1-0), choose is L> n(w) so as to have 2-("+1)(1-°) S e < 2-"(1-6) and consider the expansions 10a)

ti=j12-"-2-",- 2-"-etc.

n
10b)

is = js 2-" + 2-"" + 2-1% + etc.

n < mi < m2 < dc.

Because

and

8) justifies x(to - x(11)

!1)

s x (j1 2-") - x (t1) (+ I x (ti) - x (js 2-')l + I x (js 2-") - x (ji 2-") < 2 N (1 + 6) h (2-1N) + (1 + 6) h (j 2-*), M>"

and, seeing as h E t for t S E S 2-"(1-°) < e-1 [use 9b)], 9c) implies ! x (ts) - x (ti)

12)

< 2(1 + 6)

6h(2-("+1)(1-a)) + (1 + 6)

h(j 2-")

<(1+48)h(e), and 6) follows on letting 610. Problem. Use the CHUNc.-ERDOs-SIRAO test to prove that if I.+

M4/

h(t) = 2t[lg i + .2I& I + igs i + ... + lg.-1 I + (1 + 6)1g" a ] , Y then

P01056<651 max x (t,) - x (ti) I< h (E) , e 4 0= 0 or I according as 6

0

and compute this probability for h (t) = V 2t [lg also.

j + Z 1g$ _ +"X lg i

J

[ans. 0] 1.10. Approximating the Brownian motion by a random walk

Given a standard BROWNian path x(t) starting at 0 and an integer

t > 1, introduce the exit times 1)

eo=0,e"=min(t:Jx(t)-x(e.-i)I =

(n 1)

I.M. Approximating the BROwNian motion by a random walk

39

and let

(n z 1) .

st (n) = Jrlx

2)

Because the BROWNian path starts a fresh at each time e, it is clear that [st (n) : n L>.- 0, PO] is a standard random walk. F. KNIGHT [1] proved that if xt(t) : t :!g I is the polygonal line joining

the points

t = k/l, a = st(k)/j( as in the diagram, then P.[lim max I xt(1) - x(t)I = 0] = 1; 3)

VII

t It

the proof follows.

(k S 1),

co tS I

Given k 1 1 S t < j (k S 1), an application of P. LEVY's HOLDER condition implies that for large 1, 4)

- x (t) I

I xt (t)

s Ix,(t)

- xx(k/l)I + I xl(k11) - x(k/l)I + I x(k11) - x(t)I ...

s I x (ek) - x (ek -1) I+ I x (ek) - x (k/l) I+ I x (k/1) - x (t) I Owe

8 = Jek -

V

< 1-1/3 + constant x y8lg- (1 /8)

k 1

C1.

provided, for example, that el < 2, and so, using the BOREL-CANTELLI

tea'

lemma, it is enough to prove that the time lag mat

I

ek - j

I

is small;

for example, it would be enough to check that I

1-1151

..c

is the general term of a convergent sum. But ek is the sum of k independent copies of el, and (see problem 1.7.6) 5 a)

ED (e1) = 1-1

5 b)

ED (I e1 - l-1 12) = constant X 1-2

5 c)

ED (I e1 - 1-1 4) = constant x l-4.

Because of 5 a), et - kll is a martingale so that (ek - k/1)4 is a submartingale, and now Doos's submartingale inequality (see note 2.5.1) implies the desired bound: 6)

P° [max I ek k:SI

-

I

1

>

V 12 11

fo

'--

ek- 1 >

p1=

C

3 2

1-1/51

S 1415 ED (I ei -1 14) < 1415 x constant 1-2

= constant x 1 615.

I

-3-2-1 0 1 2 3 4 £71t - 1 1-+ Diagram I

2. BROWNian local times

40

KNIGHT has used his method to prove the existence (but not the smoothness) of the standard BROWNian local times of 2.8) and to discuss

the time substitution of 5.2). DONSKER's extension [1] of the DE MOIVRE-LAPLACE limit theorem

follows at no extra cost: if / is a Borel /unction on the space o/ continuous

paths x(t) : t 5 I and if it is continuous in the topology of uniform convergence at all but a negligible class of Brownian paths, then 7)

lm Pofa_/(S(

» . ao

ll

\`

=P0[a where [n t] is the greatest integer

])t51)
/(x(I):t S 1)
preted as the polygonal interpolation between the points (k/n , s (k)l yn) : k;5

S n, and PO refers to the standard random walk in the first instance and to the standard BROwNian motion in the second. Problem 1. Use 7) to give a new proof of the result of problem 1.3.1. Problem 2. Use 7) and the evaluation PO [measure (I : x (t) > 0, 1

1) < 0] = n sin'' y8

0S0S1

[see 2.6.13)] to prove the arcsine law for the standard random walk: itl mo PO l

n # (k: s (k) > 0. k S n) < 01

=

sin-:L

ye

(see ERDds and KAC [1], P. LEVY [2], and 2.6.17)).

2. Brownian local times 2.1. The reflecting Brownian motion

Consider the standard BROWNian motion D and recall the associated differential operator 0 = D2/2 acting on D(6) = Cs (R1) (see 1.4). Consider also the reflected BROWNian path x+(t) = Ix(t)1

1)

t Z 0,

let Bt = B [x+ (s) : s -< t] be the smallest BOREL subalgebra of B including all the events (a 19 x+ (s) < b) (s t), let D+ be the stochastic motion

[x+ (t) : t Z 0, B, P. : a ? 0], let (33+ be the differential operator 03 acting on the domain 2)

D(0') = C=[O, +oo) n (u: u+(O) = 0)1** * # (Q) means the number of objects in the class Q. u+(0) = lira e-1 [u (e) - u (0)]. 4+0

2.1 The reflecting BROwNian motion

41

and introduce the fundamental solution

g*(t,a,b)=g(t,-a,b)+g(t,+a,b)

3)

t>0,a,bz0

of au/at = 03+ V. A..11

Given t > s z 0, 4)

P. [x * (1) E d b I B,]

=P.[x+(t-s,w;)EdbI B,] =Pz(,) [x (t - s) E db]

a,b?0,

=g+(t-s,x'(s),b)db

and, using the fact that B; C B,, it is found that D+ is a MARKOV process.

D` is the reflecting Brownian motion; it stands in the same relation to (53' as the standard Brownian motion does to (53, i. e., the fundamental solution g+ of a u/d t = (53' u is the transition density of D+, just as the

fundamental solution g (= Gauss kernel) of au/at = (33 u was the transition density of D. Given a standard BROwNian

t

path starting at x (0) = a Z 0, intro-

duce, next, the new sample paths x- (t) = x (t)

5)

I

XS

zat=

tSm

= t- (t) - x (t) t > m, where m = mo min (s : x (s) = 0) and t- (t) = max (x (s) : in S s S t),

let B- = B [x- (s) : s S fl, and let D- be the motion [x- (t) : t >- 0, B, 0].

PQ:

D-

is identical in law to the

-X/4,f z

reflecting Brownian motion as will Diagram t now be proved'. Because x- (t) = x+ (t) up to the MARKOV time t = m, it is enough to check that [x- (t) : t Z 0, B, PO] and [x+ (t) : t --> 0, B, Po] are identical in law.

But, using 6)

P0(x(t) E da, t- (t) E db) = (2/n 13)1/2(2b - a) e-(26-a)'/21dadb I P. Liivv [3: 2341.

t > 0, a S b, b ? 0,*

* P. Livv [3: 211 ]: see problem 1.7.1.

2. BROWNian local times

42

it is clear that VII

7)

Po [x - (t) S c f B,] .:..'

= Po[t-(s) - x(t) 5 c,max o$t-,[x(6, w;)] - x(t)

c I B,I inn

r-'

= Po [x- (s) - [x (t) - x (s)] S c emSaxl [x (e, w;) - x (s)] -

- [x (t) - x (s)] s c I BI

f

d =t-s

P0[x(A)Eda,t-(A)Edb]

x-(8)-a$e

b-age

-oo
f

db

z-(,)

+00

+2

J

-_ (s)+c

0

Ix-(s)+bl$e

b-c

z-(+)

r(s)+e

2f-

b

f da + f db f da)/

x(s)-e

0

=

+ao

b

2nd

V

mod, (2b

+oo

(' J

(' J r(s)

NIA

rbZ0

r(s)+c

,-byZd

vine

- a) e-(2b-a)`IQA

db

db

i> S.c? 0

S c]

and since B; C B,, the result follows. 2.2. P. Levy's local time

Consider the visiting set 8+ - (1: z'(1) = 0)

(1: x (t) = 0).

$+ is a topological CANTOR set of LEBESGUE measure 0 (see problem 1.7.5).

P. LEvy [3: 239-241] discovered that it is possible to define a new time scale t+ = t+ (t) on ,3+, affording a non-trivial measure of the time the BROWNian traveler spends at 0; this t+ is local time. P. LEvY gave several

different definitions of t+ (see 4, 5, 6, 7)]; of these, the last is the most direct, but also the most difficult to establish. Consider the sample path x- (t) : t Z 0 and recall the decomposition of the passage time ma = mint : x (f) = a) (a ? 0) : +00

1)

ma = f

([0, a) X dl),

o+

where , (da x dl) is the POISSON measure with mean da x (2n 13)-11s dl (see 1.7). Because p ([0, a) X [1, + oo)) is differential in 1, it follows from 2)

Eo[p([O, a) X [e, +oo))] = a X 21ne

2.2. P. LEvy's local time

43

and the strong law of large numbers that Y(Lo.a)x[e,+oo))_a,a_> 0 =1 p 3)

[

2fne

and, substituting t-(t) = maxx(s) for a, since the difference of lSt P ([0, t- (t)) x [e, + oo)) and the number of flat stretches of the graph of t- (s) : s S t of length ze is not greater than 1, it is seen that 4)

2x the number of flat stretches of

Po [lipo

t-(s):sSt

oflengthze

=t-(t),tz01=1.*

4) shows that t- = t- (t) is a function of 8- = (t : x- (1) = 0); in addition, it is clear that t- grows on ,3- and is flat outside. Because D+ is identical in law to D-, there is a corresponding function t+ = t+ (t) of 8+ = (1: x+ (t) = 0), growing on 8+ and flat outside. V is local time; V 2-1a t+ is the mesure du voisinage of P. LEvv [3: 228].

Because t+ and t- are identical in law and the flat stretches of the graph of t+ are just the (open) intervals .,,(n 1) of the complement of g+, it is clear from 4) that 5)

Po [!im V"e x the number of intervals 8. C [0, t]

0

2

t+(t),t011,**

of length z=

and a little algebra converts 5) into 6)

PO [urn 1/iLs x the total length of the intervals

& C [0, t]

o t length < e

= t' (t), t

01

-= 1;***

thus t+ is a function of the fine structure of $+. 7)

P. LEvy [3: 239-241] also proved (2e) -1 measure (s: x; S e, s S t) = t+ (t), t ? 0] = 1. PO l

lo

c71

Here is his proof. Given 8+, the excursions e w [x (t) : t E 8.] are independent, and,

since 9 enters into the distribution of e via its length 1.3. 1

alone, 1

* P. LfivY (3: 224 - 225)

' P. I.frvv [3 : 224 - 225]. *** P. Lfivv [3: 224 - 225] .

2sJn = E. [f l p (CO, 1) x d l)] = the expected

sum of jumps of magnitude a figuring in tn,. 1 P. Livv [3: 233 - 236]; see also 2.9.

2. BRoWNian local times

44

the strong law of large numbers indicates that, for a 0. measure (s : x, S e, s s' t) 8a) measure (s: x, S e, s E& n [0, t)) can be replaced by Y E0 [measure (s: x; S e, s E 8b)

I,g+]

3 C(0,1) sac [0, t) +00

= f l (1 - e-'''It) p([0, t+(t)) x dl), 0

where p (di x dl) is the POISSON measure for the (one-sided stable) inverse function of t+, p ([0, t+ (t)) x [l, + oo)) being within I of the number of flat stretches of t+(s) : s S t of length Now use 4) to conclude that +00

f 1(1 - e-2 0/1)

9)

p([0,

t+(t)) x dl)

0

+00

= f p((0, t+(t)) x (1, -boo)) d[I(1 0

+00

f

t+ (t) d[I (1 - e - 21 0 0 ) ] = 2e t+(t)

i 0.

0

The difficult point of this proof is the jump between 8a) and 8b); although the meaning is clear, the complete justification escapes us (see, however, 2.8 for results of G. BOYLAN, F. KNIGHT, D. B. RAY and H. TROTTER that include 7) as a special case). Problem 1.

(ti. to fl $+ + 0,

o s ti < t2

bid

t+ (tz) > t+ (to

III

for almost all BROwNian paths. [Because t+ is identical in law to t-, PO [t+ (t) > 0, 1 > 0] = 1, and, if the statement were false, then e - PO [t+ (t=) = t+ (ti), (tl , to n 8+ + 0] would be >0 for some tt < 12. But e = P0[t+(12 - m, wm) = 0, ..+

m=tl+ MO (wi,)
any I> 0. [x (s) = t- (t) for 3 different times s S t implies that t is flat on an open interval that meets 8-; now use the result of problem 1, and the fact that t - is local time on 9-.] * P. Lgvv (-3: 238).

2.3. Elastic BROWNIan motion

45

Problem 3. Use the result of problem 1.7.1 to deduce the joint law P0[xI(t)Eda,t*(t)Edb]=2 a + b e-tb+6>'1tdadb a,b?0. 2nts Problem 4. Use the fact that t+ and t- are identical in law to prove ...

that the HAUSDORFF-BESICOVITCH dimension of $+ is a1/2 for almost all sample paths (see note 2.5.2 for the definition of HAUSDORFF-BESICOVITCH dimension).

[Because t- satisfies the HOLDER condition

0 St1 Sis 51,

t-(12) -t-(t1)
is-11+0

a.'

(see 1.9), if [nil, rito : n z I is a closed cover of ,3* fl [0, 1] and if each nti - wt1 is small enough, then t* (1) S X t* (nts) - t' (a11) <

nal

< n21 z v3 cnt: - ntl) I Ig (wts - A)

VII

f"1

IN'

now use the definition of dimension number and the fact that PQ[t+(1)>0]=1.] A. S. BESICOVITCH and S. J. TAYLOR [1] proved dim ($+) S 1/2 and, 1-7

in a different place, S. J. TAYLOR [1] proved dim (3*) a 1 /2 (see 2.5 for a proof of dim ($+) < 1 /2 and also problem 2.8.21. 2.3. Elastic Brownian motion

Given y > 0, we use the reflecting BROwxian local time t+ of 2.2 to construct the elastic BROwrtian motion associated with the differential operator

0'=aDs acting on the domain D(c5') = C=(o. +oo) fl (u:yu(0) = u+(o)) Consider, for this purpose, the enlarged sample space W x [0, + oo) of points (w, mom) and the corresponding BoREL algebra B x B [0. + oo),

and extend P. to B x B [0, + oo) according to the rule 1) P.(m , > I I B) = e-Yt*tt) Adjoining to [0, + oo) an additional point oo, define 2) P,. [m." = 0] = 1 and introduce the stochastic process D' with sample paths 3)

x' (t) = x* (t)

(t <

E.0

* u* (0) = lien a-1 [u (e) - U (0)]. e-0

00

(i g m..)

.

I -> 0.

2. BRowNian local times

46

D' is a (simple) MARKOV process as will be proved in a moment ; in addition, its transition density g' (t, a, b) :

g'(t,a,b)db =Pa[x'(t)Edb] Ea [x+ (t) E d b, e-Y t+ (t)]

= Ea[e-Yt+(t) I x+(t) = b]g+(t, a, b) db

t > 0,

a, b

0

is the fundamental solution of the heat flow problem

au=! asu

4a)

2 ab= y U(" 0) = u+ (t, 0) 8t

4 b)

t>0,

b>0, 1 > 0,

i. e., D' stands in the same relation to @* as the reflecting Brownian motion D+ does to W. D' is the so-called elastic Brownian motion. Because t+ is flat off 8+ and P. [maa E d t I B, M. > t] = y t+ (d t) , D' can be described as the reflecting BROwrian motion killed at b = 0

at the rate y t+(dt) : dl. Beginning with the simple MARxovian character of D', if e (0;5 e:< 1) (;'

is a BOREL function on the space of paths: I -+x(t) E Q (t S s, s, and if e' = e evaluated at the path x' (t) : t S s, then for a, b 0 and VII

Q = [0, + oo) U oo), if e+ = e evaluated at the path x+ (t) : t

s S 1, 5)

Ea [e*, x' (t) E d b]

= Ea[e+, x+(t) E db, t < ma,] = E. [e+, x+ (t) E d b, e-Yt+(t)]

= Ea[e+, x+ (t - s, w;) E db, e-Yt+(8) e-rt+(t

= Ea[e+ e-Yt+(,) Ez+(,)[x+(t - s) E db, a Yt+(t-')]] = E. [e+ a-Y t+ (+) Pz+ (,) [x' (t - s) E d b]]

= Ea[e+, s < m., Pz+(,) [x' (t - s) E d b]] = Ea[e' P.-(,)[x' (t - s) E db]],*

and the proof is complete. As to the identification of D' as the elastic BRowNian motion, it is enough to show that UI Q0

6)

+ OD

u = E. [ f e t l (xt) d t= E. f e-ate-Y t+ l (xe) d t * P00 [x (t - s) E d b] = 0.

c> 0, l E C (Rl)

2.3. Elastic BRowNian motion

47

is a solution of 7a)

u E D (W) = C2 [0, + oo) n (u: y u (0) = u+ (0))

(a-(')u=t

7b) and to invert the LAPLACE transform.

But me

8)

u (a) = E. f e ° t / (xt) d t + 0

+ao

+ Ea f eanl. f e ae e-rt+('

l

)

(m = m0)

/[x+(t, wm)] dt]

o

(

1

- Ea I f e-at / (xi) d t] h'1

Ea f e-at t (xe) d t]

l0

+

me

+ E. [e-11 m"] u (0) I.

r +00 e-a t

Ea I f

+ao

l (xi) d t + E. [ea m,] (u (0) - Eo I f eat / (x,+) dt )

0

-+°° a r2alb-al+e-Y2-. lb+al

V-

0J .+.

f

goo

+ e-fta a U (O)

/(b) db +

-2

e tab /(b) db

0

from which it is immediate that

au- 2 u"

9)

and, using the fact that [x- (t) = t- (1) - x(t) : t Z 0, PO] is a reflecting BROwNian motion with local time t- to compute u+ (0) from 8), one finds + CO

10) -u+(0) + 2f OF-b/(b) db 0

= y2au(0) =y2a Eo[ f

+b

e-rt-(t)/[t-(t) -x(t)] dt]

e-c't

0

+00

CO

e.-atdt f e ybdb faa2. 'CI

= 2oc

,J

0

-co

0

+oo

2n Is

b

= V2-,X f e-Yb db f da

2e-y2a(2b-a)/(b

-00

o

t00

v2ar

2

2Jr e - 26 f 0

L.

db,

- a)

e

ab-/tt(b - a)

2. BRowNian local times

48

u+(0) _ +2Vr2

11)

+oo

/' e_1sab/(b) db

. 0

=yu(0) as desired. 2.4. t' and down-crossings

Consider the number of times d, (1) that the reflecting BROwxian path x+ crosses down from e > 0 to 0 before time t and let us check that

Po[hme d,(t) = t+ (t), t -> 01 = 1

1)

as P. LEvY conjectured [6: 171], supposing for the proof that

(2e)'1 measure (s: x+ (s) < e, s S t), i o as was mentioned in 2.2 and will be proved in 2.8. t+ (t) =

2)

Consider for the proof, the successive returns r, < r2 < etc. of x+

to the origin via e > 0, i.e., let to = 0, tl = m, + nto (w+ ) , and 3)

+ tl (wiw_1) (n > 1) , and note that for 0 < 6 < da (mi) +I < de(ti (wre-,), w k-j ASde(mJ+l

the summands being independent, identical in law, and independent of d, (ml). Because of Po [da(rt) = n] =

4)

(1

- e) e

(n

-

0)

and the above independence, 5)

Eo [a(da (mi) + 1) (d: (m,J]

= 6(d1(mi) +1) Eo[do(ti) +1] = a(de(m,)

+1) [ 8 (1 - a)+1J

= e(da(ml) +1)

(a< 8),

i.e., [e (d, (mt) + 1) : e < 1, Po] is a positive backwards martingale, an since 6)

E0[e(d,(ml)+1)]=e I'9X (I -e)*C+l =i,

DOOB's martingale convergence theorem (see note 2.5.1) tells us that lime d, (m,) exists. $40

2.4. V and down-crossings

49

++"

Next, this limit is identified as t+ (m1) with the help of 2).

Using d in place of d, for e = 2-", it is desired to estimate d" = 2"-1 measure (s : x+ (s) < 2", s < MI) - 2-"d"(m1) = 2"m2+* if d"(ml) = 0

7)

= 2"-1 I [measure(s:x+ < 2-", rk-I S s < tk) kgr"

2-9n+1)

i/ dn(ml)=mz1.

Here the summands are independent, identical in law, and independent of d" (m,) = m, with common mean CO

f dsP0[x+(s) <2-",s
8)

0

+00

= E0 (m+-.) + f d s P2-. [x (s) < 2-", s <m0J 0

= 2 , 2-s" **

and common variance E0[(measure(s : x+(s) < 2-", s < rI)

9)

-

= constant x 2-4"

2-En+1) sl J

as a simple application of the scaling x+(t) -> 2-" x+(22" t) verifies, so 10)

E0 (d s)

=

23"-y

PO (d. (nil) = 0)

+ 22"-$ E. [(measure (s : x4 (s) < 2-", s < rI) - 2-2n 41)2]

ZmPO(d"(m3)=m)

M-1

=

22,s 2

2-4n -2-A*** 4

+ Ps-2-constant X 2-4n Z (1 - 2-")m 2-n m-1

< constant X 2-", .r.

and combining this bound with 2) and the BOREL-CANTELLI lemma, it 'CS

results that limed, (m,) = f+ (m,) e10

.

But now, selecting t ? 0, if t < m1, and if r = min (s: x+ (s) = 0, s a t), then 11 a)

t+(m1) = t+ (t) -}- 0

+.+

or

t' (ml (wi) , WI+)

11 b)

d,(m1) = d. (t) + 0

or

'C!

de(m1(w,), w+)

* in denotes the reflecting BttowNian passage time to a. ++ See problem 1.7.6 and 1.7.7. It6/McKesn, Diffusion processes

*** See problem 1.7.6. 4

2. BROWNian local times

50

IAN

according as t z m, or not, with an error of I at most in 11 b) ; thus, PO [Ii ms d, (t) = t, (t) I t < m11 = I (t ? 0),

12)

L

440

and to complete the proof of 1), it is enough to consider d, and t+ between

the successive passage times of x* to I via 0 and to use the fact that d, is an increasing function of time, while t+ is both increasing and continuous. 2.5. V as Hausdortf-Besicovitch J-dimensional measure 1

Perhaps the most striking feature of the local time t+ (t) is the fact

that if U kn is the covering kZ1

of 8* and if 2)

I k*3n I

is the diameter of k$" , then

Poln+o0 V-122.

t*(t), t ? 0l

"St

1k kzI

k$n=[(k-1)2'",kT"]fl8

1)

J

.4w

t+ is therefore similar to the HAUSDORFF-BESICOVITCH 7J-dimensio-

nal measure A l; in particular,

A ($ l [o, t)) s VII

Po [

3)

/

t+ (t), t z o, = + .

III

...

proving that the HAUSDORFF-BESICOVITCH dimension number of is , and, using problem 2.2.4, it results that the dimension number :.:

of $ is not smaller than and therefore = J. Given t2 ? tl ? 0, if 61 (as) = min (max) 8 fl [t1, tk], * then, as will acs

be proved below,

Eo[t+(t2)-t*(t1)I x;:s511,31,as,x1:sZt2]

4a)

=0

a1=52 NIA

n 2

t1 < 52

(bs - b1)

Because

C. = B[x+(12'"), min i8n, Max t$n. l Z 1]

is a subalgebra of

B[x;:s(k-1)2-", min k$n,maxt8n,x,:sk2-n], 4a) implies ..x

R'1

Eo [t* (k 2-") - t* ((k - 1) 2-") I C"] = a~"

4 b)

Y 2 I k8n!

k Z 1,

See note 2 for the definition of the I-dimensional HAVSDOR" measure.

* 31 =bx=0 if $fl[r1.to=0.

2.5. t' as HAVSDOREE-BESICOVITCH }-dimensional measure

51

and, noting that C. increases to B' = B [xt : t Z 0] as n t + oo, the martingale convergence theorem of J. DooB (see note 1) Implies that,

for each separate t ? 0,

1 Ik8nI1 = I Eo[t+(k2'x) - t+((k - 1) 2-)J Cx]

5)

k2 *$t

=

k2-%t

EO [t+ ([2a t] 2-x) I Cx]

- Eo [t+ (t) I B+] = t+ (t)

n t +00-

2) is immediate from this because

and t+ (t) are increasing

k2-"st functions of t and t+ is continuous.

27;

and t+ (t,) - t+ (tl) = t+ (t2 -11,

Coming to the proof of 4 a), since

III

w) are all measurable B[x+(s) : tl S s S 1,], the conditional expectation ..1

figuring in 4a) is a BOREL function of tl < a1 = s1 < 52 = s2 < t:, x+ (t1) = a, and x+ (ts) = b alone : e = Eo [t+ (ts) - t+ (tl) I x+ (t1) = a, 61 = s1, bs = s2, x+ (ts) = b]. try

6)

Actually, t+ (t2) - t+ (tl) = t+ (t2 - 81, wit) , and since the BROwNian path begins afresh at time bl = 1l + mo (wi,) , 7)

e = Eo [t+ (t2 - s1) 13, = ss - sl, x+ (ts - s1) = b]

with 33 = max (S: s < t2 - s1, x+ (s) = 0). Now consider the conditional BxowNian motion Dab = [x+ (S): s -<- t, P. b (B) = P. (B I x+ (t) = b)] for a, b ? 0 and t > 0 and run it backwards, i. e., consider the backward motion Dab = [x+ (t - s) : s t, Pab]. A brief

examination of the conditional law permits the identification of Dab with Dba. Because t+ (1) is the same for the backward and forward motions, 7) becomes 8)

e = Eb [t+ (t2 - sl) I MO = t2 - s2,

x+ (12

- SO = 0].

Using the same idea that led from 6) to 7), 8) can be simplified to 9)

e = Eo [t+ (s) I x+ (s) = 0]

s = s2 - s1 = 52 - 61,

and this can now be evaluated by means of the joint law of t+ and x4 (see problem 2.2-3): CO

Col

b= a-a'Is'

2

db

0

10)

21y2ns

4*

2. BROwxian local times

52

completing the proof apart from small technical gaps which the reader is invited to fill.' Note 1. Submartingales. Consider a non-negative submartingale eA: it Z1 relative to the BoiEL algebras B, C B2 C B3 etc., i. e., let 0 S e. be mea-

surable B. with E (e,,,) < + oo and E (em1

a eA (m ? n), and let

us prove the martingale convergence theorem of J. DooB U: 3241: that it y = sup E (eA) < + oo, then nZl

P (e = ofm eA exists) = I and E (em) S y . 00

``

Consider, for the proof, 1>0, n , = nun (n : eA = 0) , ns = min(n : is > n, , e 1), ns = nun (n : is > ns, eA = 0). etc., let # denote the number of times em : m < n crosses up from 0 to I (i.e., let # be the

number of integers ns, n4, ns, etc. < n), and let e = e, + Z ti (ei - et_,) , i-8 where ti = 1 if i lies in [0, n,] U (ns, ns] U (ns, na] etc. and ti

otherwise. (ti = 1) is the event that eA : it < i has not completed its current down-crossing; this is measurable Bi-1 so that 1)

i-Q

E(ei-ai-1l Bi-,))a0,

and, using

n
a=ef

2)

n,S<ns

=eA,=OSeA

ns5n<ns nsSn<ns VII

=e,,,+en-en,5en-1 =e,,,+e,,,-e,,,=-e,,,SeA-1

VII

e,,,+e.,--e,,,-}-eA--eA.SeA-21 n4Sn
ne finds

Se - #1,

e+'

1E(#) SE(eA).

3)

Given 1s > 1, >_ 0 and using 3) with the non-negative submartingale

(en - 1,) V 0 : it z I in place of eA : n > 1 and 1s - 1, in place of 1, it is found that i/ # is the number o/ up-crossings o/ em : m 5 is from 1, to 1s, then 4)

coy

E (#) (12 - l,)-' E [(eA - 11) V O] S (1s -- l,)-' E (eA) Consider the case y = sup E (es) (= 1 tm E (e.)) < + oo and make

nal

n t oo in 4) ; then one sees that e,: n

A ao

1 crosses up from 1, to 1s just

1 G. LOUCHARD [private communication] suggested this use of the backward motion improving on the clumsy computation of the original proof.

2.5. V as HAUSDORFF-BESICOVITCH f-dimensional measure

53

...

a finite number of times, and this implies the existence of e = lim eA. "t w E (e,,) S y is clear. A special case is the result of P. LEvY [1: 128-130]: that if e is non-negative and summable, if B, C B2 C B3 etc., and if B is the smallest BOREL algebra including U B., then n'z l

..r

5)

ntoo

B).

Here, en = E (e I Bn) : n ? 1 is a non-negative martingale so that the existence of e = lim en is clear, and since of oo

sup E(en, en Z 1)

nal

= supE [E (e I Bn), en Z l] nZ1

= supE[e, en Z 1] nZ1

S E [e, sup en 6)

1110

nZ1

ll

1 t oo,

JJ

E (e, B) = l E(em,B)=E(e00,B) BEBR,fZ1,

completing the identification of e and E (e I B). Doob's submartingale inequality [1: 3141: P (max ek z 1, S l-1 E (en) k$n

7)

is contained in 3) as a special case, but here is a direct proof. Consider the event At that ei < l (i < k) and ek Z 1; then At E Bk,

AjnAk= 0 (i
E(en, U Ak/ _ X E(en, AL) k-1 k-1 it

IVY

/

_ E E (E (en Bk), Ak) k-1

zl

E (et, At) k-1

n

k-1

P(Ak) = l P(maxek kin Z 1).

Note 2. Hausdorff measure and dimension (see F. HAUSDORFF [1]). Given a continuous function h (t) E t (t Z 0) with h (0) = 0, the outer Hausdor// h-measure of a linear BOREL set B is defined to be A (B) = lien inf X h (ln) ,

.0

nal

the infimum being taken over all coverings U A. of B by means of nZl closed intervals of length l = An I < e (n z 1). A is a CARATHEODORY outer measure, but, in general, R' is not the sum of a countable number of BOREL sets of finite measure.

2. BROwNian local times

54

Given 0 S a S 1, the A associated with h (t) = t°` is the Hausdor//Besicovitch a-dimensional measure Aa; the numbers aim (B) =

dim' A =

0

it A°(B) < +oo

sup (a : A ° (B) = + oo)

_/

A °(B) = + o0 A 1(B) = + oo it

1

inf(a:Aa(B) < +oo)

it A1(B) < +oo

coincide, and their common value dim (B) is the Hausdor//-Besicovitch dimension number of B. A' is the classical BOREL measure. A1921%3 is of particular interest : the dimension number of the standard CANTOR set ...

Kc[0,1]-l3'

T2

9)-(9'

9)-etc.

is 1g2/1g3 and AIg 2/!Q3 (K fl [0, t]) is the CANTOR function.

2.6. Kac's formula for Brownian fanctionals Before the deep properties of BROWNian local times can be proved,

a new method for computing probabilities is needed; this is embodied in a formula of KAC [1]. Given a piecewise continuous function k ? 0, let f(t) = I(t, w) be t

the additive BROwNian functional f k [x (s)] d s, so named because of the addition rule ° 1)

I(t)=f(s)+I(t-s,w;)

(t? S),

and let (i be the differential operator 2a) (@* u) (a) = iu"(a±) - k(a±) u(a) applied to the class of functions u E C (R') such that b

2 b)

b

4 [u' (b) - u' (a)] - f k u d4 = f u dt a

(a < b)

a

for some u (in W u) belonging to C (RI). KAC'S formula states that for a > 0 and / E C (RI), too

u = E.f f e-«t e-tcn/(x) dtl Lo

is the bounded solution of 4)

s

ICI

3)

j

(a- (t3')u=/.

KAC'S formula suggests that heat flow with cooling is the same as BROWNian motion with annihilation of particles, indeed, if k (b) ds is

the chance that a particle is killed in time ds having come safe to l (x) stands for l [x (t)] in such an integral.

2.6. KAC'S formula for BROWNian functionals

55

'C!

the place b = x (s), then its chance of surviving up to time t a 0 conditional on the path x (s) : s S I is fl [1 - k(x(s)) ds]* = CIO; 5)

.; t

hence its chance of coming safe from a into db after a time t is 6) g'(t, a, b) db = Ea[e-t, x(t) E db] b] g(t, a, b) db, = while according to KAC'S formula, g' is the fundamental solution of

au/at = W u "(0+, ) _ /,

7a) 7 b)

which describes heat flow with cooling. As to the proof, if it is as in 3) and if G. is the standard BROwxian GREEN operator of 1.4 8)

(Gal) (a) =

f

e-y

l(b) db,

a

R

then

Gal - u

9)

,00

= E f e'at (1- e-i)/(x) dt f0

f e "t f

o^+

=

t(-10;) f(ds) /(x) dilll

e

+ f I(ds) f e-atei(t-.,w;) +00

=

/(x) dt

,

too

,

f e-a'I(ds)+ Bate-t(t.' )l[x(t,w:)]di] +00

E. f e-"'ku(x)ds to

=Gaku, the interchange of temporal integrations in lines 3 and 4 being justified by t

f

10)

e-t(t-''10")df

S 1.

0

Using (x - (53) Ga = 1, it follows at once that u solves 4) ; also, the boundedness of u is clear, and since a solution of (, 5 v = a v is convex obi

fl is meant to suggest a continuous product.

,.5r

2. BaowNian.local times

56

or concave according as it is positive or negative and so unbounded or aaaaaaa 0, is is the bounded solution of 4).

GREEN functions provide the classical method for solving problems

such as 4) ; the facts are indicated below in a form suitable for applications.

Given a > 0, (SS' g = a g has two independent solutions 0 < g1 E t and 0 < gs E I, their Wronskian B = g'1 gs - g1 g''2 is constant (>0), the Green /unction

(a S b) VII

G (a, b) = G (b, a) = B-1 g, (a) gs (b)

11)

VII

satisfies 2a f G d b S 1, and is = 2 f G/ d b is the bounded solution of 4). VII

Here is a rapid proof. Choose / (a) = 0 (a S b) and / (a) > 0 (a > b) in 3) and let gab = Ea [e-am1 a-t (m')] (a < b). is satisfies

u(a) = E. °"'

12)

'~g

+00

e-'/(x)dt)

=gabu(b)

(a < b),

1

r'7

and since u (b) >0 and (a- (SS') u = / = 0 (a < b), it follows from the multiplication rule 13)

(a
got gib am a t(m)] EE[e am*

= Ea[e

(t(m)+t(m.(Ji4J.'°m)])

Ea(e-a(m+me(wm)]

that

(m = MI)

C

= Ea[e-`mee-t(m')] = gab

14)

gl(a) =bt11ma gab/gOb

is a positive increasing solution of 4S' g = a g; the same method can be used to make positive decreasing solution. B satisfies b

15)

Bibs =f [gs@*g1-g1?*gt]df=0 (a
so it is constant (> 0) , 2ocB f Gdb 16)

= 2a [gi(a)f g1 db + g1(a) f gs db 00

= 2 [gs(a)f

CtS' g1 d b + g1(a)

s

+oo

S gs (a) f gi' d b + g1 (a) .f g;' d b -co

SB

a

/Ys gsdb,

2.6. KAc's formula for BROWNIan functionals

57

as desired, and now a routine differentiation shows that u = 2 f G J d b is a bounded solution of 4), completing the proof.

KAc (1) used his formula to prove P. LIv 's arcsine law for the standard BROwxian motion [2: 323] OR

17)

PO [measure (s : x (s) Z 0, s S t) S 9] = n

f

ds ys (f - s)

0

sin-' 1 9 / t

0: ! ! 5 ;t .

Use the additive functional f = measure (s: x (s) 0, s :!g t) based on the indicator function eO of [0, + oo) and let (SS_ (s} - 8 eo. KAc tells us that +oo

a,p>0

u=E.(f e°1e-afdt)

18)

0

is the bounded solution of

(a- 0Y)u=1,

19) so

20)

u(0) = f e-011 dt f e-a' PO[f(t) E ds) 0

0

+W

o

=2B-1 [gz (0) ,f g1 d b + g1(o) f g: d b, -W

,

o

'L!

and solving, it is found that 21 a)

g, (b) = eYQ"a

(b S 0)

21b)

g2(b) = e- s(a+P)a

(b ? 0)

B=+ 2(a+fl)

21c)

u (0) = I a-(OT fi) . V) is now immediate from the formulas 21 d)

+W

a-y' di r-

22 a) 0

vn t

t yy

and +W

22b)

t

e-°`e f

1

}Ia (a _+P) - J 0

e-p& _

a 0J V(t - s) s

dsdt.

A beautiful application of KAC'S formula is to the WKB approximation. * See problem 1.7.3 for a different aresine law also due to P. LAvv.

2. BROWNian local times

58

Given k tending to + oo at the ends of the line, the spectrum of ® acting on L' (R', d b) is a simple series of eigenvalues 0 > yl > y, > dc.

- oo, and under a simple additional condition the WKB approximation states that SIN

I 1 (2 z)-1 area [(a,b):-_+k(b)
23)

y.>y as y 1- - oo. KAc's idea is to express the trace T eys as S al f fk (z (s) + 11 ds

E. [e a

24)

I x(t)

= 01 dl x (2n t)'112

)i

and then to check that 24) - (2.n t)-h1= f e-41;0) dl as t--,. 0 (see KAC [1] R,

for a complete proof and additional applications). Problem 1. Check the evaluation +00

f e`af E0 [e-0measure(s:z(s)2; 0,s$e) I X(t) = 0]

dt y2n1

o

- V2a + y2(a + P) 2

a,P>0

and use it to prove P. LEvr's law [2: 323] PO [a

:91-1 measure

(s : x (s)

0 , s S t) < b j x (t) = 0] = b - a 1.

[2 [ 2a + V2 (a + #)]-I is the GREEN function of 19) above evaluated at a = b = 0; now invert the LAPLACE transforms.] I Problem 2. E0 [e 6 messnre (s: z(s) 0. s$ m,)] = (P > 0) cosh 2" [Use the additive functionals e0 and el based on the indicator functions e0 and e1 of [0, + oo) and [1, + co], let +00

u = E. l 0 (' and note that

eo(x(t)) dtl (a, P, y > 0),

If

( m,

lim

lim u(0) = E0I f e -P to eo(x (t)) dt

E0(e-Pss(ml))].

0 r^,

KAC tells us that u (0) can be expressed as

oaf

+00

2g1(0) B-1 f gs d b 0

(09 g = a g, (? = 0 - eO - y e1) ;

2.7. BESSEL processes

59

the formulas are VI!

(b S 0),

& (b) = ey2ab

(bz1,a-a+A)

gs(b) = e' Zt°+YIb

=cosh

(1-b)+V

(OSb<1),

III

B = [V + 2 (a+ Y)] cosh I1 + [ 2a(1 +y/a) + % _2a] sinh I, e-V2(V+Y' B u(0)/2 = sink 20 + j/_+ y cosh 2a - t r-,

y2(a+y)

y2a

a

20

2.7. Bessel processes

We prepare the so-called BESSEL process with a view to a better understanding of Bxowxian local times (see 2.8) and Bxowxian excursions (see 2.9 and 2.10).

Consider the standard d-dimensional Bxowxian motion D = ... , xd (t)) : t Z 0, Ps] , in which the coordinates of the sample path are independent standard 1-dimensional [E (t) = (x1 (t) , x2 (t) ,

BRowNian motions and P.(B) is the probability of B as a function of the starting point X (0) of the d-dimensional Bxowxian path. Given t > s, if B, = B [E (0) : 0 S s], then I)

P.[E(t) E db I B,] = Pa [IC (t - s) E d b] (2n(t - s))a/s

a=(a1,as,...,ad),

a = E (S)

db

b=(b,,b,,...,bd),

b - al _

(bl-a,)2+(b:-az)s+...+(bd-ad)',

Ayr

...

db = db, db,... dbd

III

because of the independence and MAxxovian character of the coordinate (1-dimensional) Bxowxian motions. Because the d-dimensional GAUSS kernel g (t, a, b) = (27r t) -(d/2) e- I b - 412/29 is the fundamental solution of s = 2 4 , 2 (ab; +

s

ab;

t ...

:

`-l

at = (3 u

}- abj)

o,,

D is a simple Markov process, standing in the same relation to 0 = d/2 as in the 1-dimensional case (see 1.4; for additional information about the d-dimensional Bxowxian motion, see chapter 7).

2. BROWNIan local times

60

Coming to the BESSEL process D * = [r(t) = Ij(t)I : t 0, Pj (= the reflecting Bttowxian motion if d = 1). it is clear from 1) that

fort>s,

P-10) < b I BO = Pa [r (t - s) < b]

2)

r

e-I b- ass/acs-"

(2x (t - s))'1'

J

IbI
a = it (s) db

is a function of r (s) = I I (s) alone, and noting the inclusion R, a B[r (0) : 0 S s] C B, , it is found that I

3)

P.[r(1)EdbI R,] =P.[r(t-s)Edb]=g+(t-s, a, b) db a = r(s) = 101,

where 4)

g- (1, a, b)

f

d

(2n t) 2 e-Ibo-a(1,o,...,o)fI2t do ba''*

sd-1 t)_

(2n

a'+bs

d 2

e

2221

f

ab o e

t

do bi" **

Sd-i

=t''e

.b. 29

214-1( rb)b4-1 t>0,a,b>0***

(ab)1

2

is the fundamental solution of 8u

5 a)

1

8=u

at = 2 (8b= + li o ba-'

alt

au

5 b)

d - 1 8u b

ab)

b> 0

=0

(see problem 1) ; in brief, the Bessel motion is a simple Markov process, standing in the same relation to the Bessel operator 6a)

(SS+ = 2 (d b= +

b

1

db)

he

radial dart of ( _ (1 /j) d

acting on the domain

6b) D((3S*)=C[0,+oo)fl(u:C *uEC(0,+oo),hobau'(b)=0)t as in the X-dimensional case (see 2.1). o E Sd -1, the unit sphere. d o is the element of surface area on Sd -1.

* 0 is the angle between o and (1, o. .... 0).

*a I(&n,-) is the usual modified BESSEL function.

t limb'' u'(b) = 0 is automatic if d a 2; see 4.6. 6+o

2.7. BESSEL processes

61

6a) suggests that the d-dimensional BESSEL process r should be a standard BROWNian motion b subject to a drift at rate (d - 1)/2r,

i.e., that it should be a solution of r (t) = b (t) +

d

t

2

1 f d s/r (s) ; this 0

U)^

NIA

conjecture is correct for d z 2 (see H. P. MCKEAN. TR. [3] for the proof and additional information), and it is also true for d = 1 if t VII

lim (2E) -1 measure (s : r (s) < e, s

1) is used in place of d 2

(see H. P. MCKEAN, JR. [6]).

I

fasir(s) °

...

A. YA. HINCIN'S local law of the iterated logarithm PO[e1m

7a)

I(I =1=1 1

0 y2

>)g=

(see 1.8) and P. LEvy's HOLDER condition 7b)

P

I(s)l lim ° 15-11840 y2dlgI/6

1=t ]

(see 1.9) also hold for the d-dimensional BROWNian motion as one can

easily prove, using the fact that the projection of the d-dimensional

Flu°.w

motion on a line in Rd is a standard t-dimensional BROWNian motion and noting the obvious estimate III = supI (gr, e.) I for I E Rd and a

8a)

8b)

P ° [el+o tTIM-

P°

[

I

'V r('

81g' 1/a

D

't7

ea1 countable dense set of points e (n 1) on the surface of the unit ball. Because the angular distribution of the large changes in I(t) (t S 1) is isotropic, the same laws hold for the BESSEL process as well:

)i - 1 l = 1

(t ? 0)

Ir) - r(S)1- = 1l = 1.

-81-640 y2 d 1g 1 /d

J

...

A final point needed below is that the d 2 2 dimensional BESSEL path does not touch r = 0 at a positive time : 9)

P° [r (t) > 0, t > 0] = i ;

it is enough to discuss the 2-dimensional case. Consider for the proof two circles I a I = a and l b = b > a centered at 0, take a point E in the shell between them (a < I < b), draw about

it a small circle I g - 11 = c contained in the shell, and let e be the exit time min (t : I I (t) - I I = c) for BROWNian paths starting at

2. BROWNian local times

62

is invariant under rotations about E, it must be the uniform angular distribution do on Ig - 11 = c, and since the BRoWNian traveller starts afresh at time e, the 1 (0) =I I. Because Pj [I (e) E" d>)]

radial function 10)

p(III)

= P:(ma< mb) mr = min(t: I1(t)I = r) = min(t: r(t) = r)

satisfies

P1tma(wt) < mb(We )] = f P(ItI) do,

11)

i.e., it is harmonic (dp = 0) in the shell. But the projection argument used above shows p (a+) = i and P (b-) = 0, so

a
b/r

12)

P {r) = lgb/a lg

and now 9) follows from P. (mo+ < + O°)

13)

= btimo loPE(ma < mb) igb/r = bt+. lim ado lim lgb/a

(III = r > 0)

=0 (see problem 3 below and also problem 4.6.3 for other methods).

Problem 1. Give the details of the evaluation of g+ (t, a, b) in 4) and also a direct proof that g+ (t, a, b) is the fundamental solution of 5) (see KARLIN and McGREGOR [5: 9-11] and problem 4.11.2 for more information about this kernel).

Problem 2. Give complete proofs of 7) and 8) using the indications

of the text. Problem 3. Give a direct proof of 9) for d z 3 similar to the DvoRETSxY-ERDds-KAKUTANI proof that the 1-dimensional BROWNian path

is non-differentiable (see problem 1.4.7). [ P. (r (t)

= 0 has a root at some time

SPo

u nn nSu

( {ii

S nt+oo lim n (3ns[lgn\d/2 = 0.]

1

(kl

l n/I

t < 2) 3

<

lgn//

2.8. Standard BxowNian local time

63

(:3

2.8. Standard Brownian local time Given a standard Bxowxian motion x and a point a E R', let xa be the reflecting BROWNian motion xa (t) x (t) - a , let to be its local time, and let t (1, a) m t (t, a, w) be the standard Brownian local time ...

at a: t(t, a) = 0

1)

t < ma

t? M..

_ ita(t- ma)

is non-negative, continuous, increasing, and flat outside the visiting set $. = (t: x = a) for each separate a.

t (1, a)

As will be proved below, t can be modified so as to be continuous in the flair (t, a) E [0, + oo) x R', and with this modification,

t(t, a) = Jim measure (s: a(b x(s) < b, s

2a)

t)

t z 0, a E R',

b4a

b

2 f t(t, $) d$ = measure (s: a < x(s) < b, s S 1)

2b)

t Z 0, a < b,

a

and

sSt,aER',

t(t, a) = t(s, a) + t(t - s, a, w,) .an

2c)

for all but a negligeable class of Brownian Paths, but first a provisional

result in the direction of 2a): measure (s : a;5 x
t)

t(t,a) > I

J

tz0,e=b- a, a
Considering the case a = 0 only (the general case is left to the reader as an exercise), it suffices to check the bound 4)

D (t) = Eo[(2e)-' measure (s: 0 s x < e, s g7- t) - f (t, 0)1']

G constant x e Y and then to use CEBYSEV's inequality. Now 5 a) Eo [t (t, 0)'] Eo [t- (t)'] = t/4

(2e)-" Eo [measure (s: C s x < e, s s t)'] VII

5 b)

A,

r

t

a

_ (2e)-' 2 1 ds f d oI da f db J 0

do = t t yo(t-a) rds as f (s-Q4 = tf4 a s

t

S

4n 0

2n s-a

o

1

B,

2. Baowxian local times

64

and V11

2(2e)' Eo [t (t, 0) measure (s : 0 S x < e, s S t)] VII

5c)

I

VII

=8-'f dsEo[to(t),0 Sx(s) <e) 0 aft

VII

=e-1 f dsEo[tO(s),05x<s] 0

I

+ e-1 f ds E0[0 S x(s) < e, E:c,l(to (t - s))] 0 +00

a

e

=s 1 rds rda f bdb y ±b e-cam°>'IQ J

0 c

('ds +e_1 J 0

2n $'

J 0

0

a

f

eeda (a e'tdax yix02 2xs t-,

0

0 +ao

x 2 f bdb 0

y2x(r - S - Q>

= C.

A = A + B - C bounds D above and is an increasing function of t +`=

as can be seen from its LAPLACE transform +00

6)

a

e-a'Adl = J

o

46a= f

(1 - e-lrsia)(2+ e-Ysaa)da,

0

so +CO

VII

D5ASi

7)

f

e F e/8, ... ...

.-./

f3'

and this completes the proof. Now it is permissible to identify t (t, a) with VII

lim 2' 1 measure (s : a S x (s) < b, s S t) , b-a+2-" bla tad

and 2 b) follows, first for each separate t > 0 and a < b, and then for b

all t > 0 and a < b at once, because f t (t, t) d t is monotone and a

measure (s: a

x < b, s S t) is monotone and continuous in the triple

See problem 2.2.3.

2.8. Standard BROwNian local time

65

(1, a, b) E [0, + oo) x R2; in particular, supposing that t (1, ) is a continuous function of E, 2a) and 2c) follow and also 8)

P0 f h

o

t) = t+(t), t

(2s)-' measure (s : x+ < e, s

0] = 1

as needed` for the down-crossing lemma of 2.4 and confirming 2.2.7).

H. TROTTER [1] gave the first proof that t(t, ) is continuous on (0, +oo) xRI;rhe obtained It(r, b) - t(t, a)I = 0, 9a) Po lim

b -a

t z 0] = i

yb lg 116

4O

and

Fin

Po

9b)

e

aE

1

t(t, a) - t(s, a) =0=1 (lgt/6) id

.+.

...

using KAC'S formula, and later E. BOYLAN (1) simplified the estimates with the same result. H. P. McKEAN, JR. [4] next obtained 10a)

Pole+m 1 t

and

< 2 yt(t, 0)1 = 1 J

l

Po

b aa-fc

1t(t, b) - t(t, a)I S 2 Y F RI tct, a, = 1 yalgl/d '"w

lOb)

0) 1

(t, 6)-7(1

't'

...

'id

a44

oho

ti;

+U+

'ti

for each separate t 0, using a stochastic integral for t due to H. TANAKA, and a little afterwards D. B. RAY [4] proved that 10) is best Possible (i. e., that the = sign holds in both the inside inequalities). RAY observed that if x is a standard BROWNian path starting at 0, stopped at

ti.

-.,

its passage time m, to 1, and if # (t, [a, b)) is the number of times it crosses down from b to a < b before time t ml, then for 2" z j > k > 0, # (ml,

[(k - 1). 2-", k2-")) is the sum of # (ml, [(k-1) 2'",j2-")) +1 independent copies of # (mfg-,,, [(k - 1) 2-", k2-")), these copies are also inde-

ate

web

.-.

pendent of # (m,, ((j - 1) 2-", j2-")) and so [# (m,, [(k-1) 2-", k2-")) : 0 < k S 2", P0] is a MARKOV chain. But according to the down-crossing lemma of 2.4, 2'" #(m,, ((k - 1) 2-", k2-")) should tend to t(m,, a) (k = [2" a], n t +oo), so [t(m,, a) : 0 S a S 1, Po] should also be MARKovian, and indeed, RAY proved that it can be expressed in terms of the 2 and 4-dimensional BESSEL processes of 2.7 as follows:

a S b - t min x (t) ,

t (MI, a) = 0

11 a)

;SMj

11 b)

Po (bSc)=

11c)

t(m,,a)=r4(a-b)2/2

Ite/McKan, Diffusion processes

1

1

c50 ,

c

bSa50, 5

2. BROwVNian local times

66

where rs is a 4-dimensional BESSEL process starting at 0, independent of b, 0 < a S 1, t (MI, a) = r, (1 - a)2/2 lid) 4+.-

where r, is a 2-dimensional BESSEL process starting at 0, conditioned

to land at r,(-b) at time 1, but otherwise independent of ri and of b. The strong laws (i. 0) 1 = 2 it(m1, 0)l = 1

Po (lu It(m',va

12a)

192,16

lL

JJ

and PO

(m,, a)1

a-a

a{o

g/t

O-0

12b)

= 2 sxt(mt, a) = I I

OTC

follow at once from 2.7.8) and the above description. F. B. KNIGHT [2] obtained a similar result using the inverse local time t-1(t) = 'ti

tad

.-.

min (s : t (s, 0) = t) in place of m, ; this proof is left aside, but the reader is invited to prove 1i d) in problems 5 and 6 below. 10) (with = in place of S) and an improvement of 9b) are proved below using a second result of D. B. RAY [4]; this states that if e is an exponential holding time with conditional law P.(e > t I B) = e-112 then for each a, b E R', the conditional local times (t (e, ) : E R', Pab (B) = P. (B I x (e) = b)] can be described as follows : Vii

HIV

...

,`M Ayr

VII

261

S a /\ b ,

Pab (c < ) = e-uanb - F)

13 b) 13 c)

Scmiinx(t), is C

t(e,E)=0

13 a)

t (e, E) = e-21 r* (e2t - eE0)2/4

c S S a /s b,

where r4 is a 4-dimensional BESSEL process starting at 0, independent of c, 13 d) t (e, ) = e-'E r, (elf - e=aAb)s/4 /or between a and b, 60i

where r, is a 2-dimensional BESSEL process starting at r4 (ela^b- e21 but otherwise independent of r4 and c,

t(e, ) = e*E ra(e-s" - e-si):/4

a Vb

VII

13e)

S d,

where ri is another 4-dimensional BESSEL process starting at 0, condiy,0p

tioned so as to make t (t, $) match ate = a V b ±, but otherwise independent of d w mis(ax x (0, r,, c, and the old rs, 13f) 13g)

Pab (d > ) = e-t<E-aVb)

t(e, E) = 0

? aVb, ? d.

Here the MAxxovian nature of [t (e, f) : f E R', Pab] can be traced to the fact that stopping at an exponential holding time brings to the

2.8. Standaid BROwNian local time

67

distributions a LAPLACE transform which changes a) convolutions into simple products and b) the GAUSS kernel into a GREEN function which

is also a product. The first step is to check that if f is the additive functional based on a non-negative piecewise continuous function /, then Eab[e-I(e) It(e, ) = 0] =

14a)

20--l

t-

x

X [G (a, b)

for a not between a and b, and

)G G(a,GE(f.

(t > 0)

Eab[e-I(e) I t(e, $) = t]

14b)

b)

=elb-114 If-al G(a, E)G(E, b) G (E. E)'

VII

for all , where G(a, b) = G (b, a) = gl (a) g. (b) (a S b) is the GREEN function of 1/2 - (3Y ((3S = Qi - /). KAC tells us that if e_,pb is the indicator of a < b, then u(a) = Ea[e-I(e), x(e) < b]

15)

_ J E. [

+CO

f

e-t/z a-'(t) a-a4 b (x)

dtl

0

= f G(a, c) dc, c
and since +00

16

P[x(e) E db] = Z Jr e-'lz dt e-"db _0

it follows that 17)

t-Ib_al

a

Eab[e-1(e)] =

db12,

Vin--t

Ea[e-I(e)I z(e) = b] = 2elb-al G(a, b).

Given E E R' and y > 0, add to / the indicator function of [E, rl) (p7 > t) times y/2 (17 -1) and let rj 4. ; then the modified left hand side of 17) tends to Ea b [e-I (c) - y t (e. E)] in view of 3), while the modified GREEN

function tends to G (a, b) - y G (a, E) G (E, b) (1 + y G ing the formula 18)

prov-

Eab[e-r(e)-yt(e.h] G

y>0.

=2elb-al[G(a,b)-y 1(a yG(EE)J G., the modified GREEN function, satisfies G. (a, b) = G (a, b) - (y/tl - E) f G. (a, c) G (c, b) dc; E

making r)1. E and solving gives the stated result. 5*

2. BROWNian local times

68

As y t + 00, 18) tends to Eab[e_f(e),

19a)

t(e, ) = 0]

= 2elb-al [G(a, b)

-

G(a. 6)G(f:. b)

and subtracting 19a) from 18) and inverting as a LAPLACE transform in y gives 19b)

Eab[e-f('), t(e, E) > t]

=

2elb-aI G(a,

E)G(E, b)

G(J, E)

e-4100- 1);

in the special case / = 0, 19) becomes 20a)

Pab(t(e, ) = 0) = 1 - elb-al-Ib-fl-If-al Pab[t(e, $) > t] = elb-al-Ib-11-11-al e-28,

and 14) follows. Because f (e) = 2 f t (e, ) / d $, it is enough /or the proof that the conditional local times are Markovian, to check that the additive /unctionals f : based on

/_=/ (rJ $) =0 (r/>J)

and /+ = 0

=l

(,<)

(r1? ) are independent, conditional on t (e, ), and since the BROWNian motion is

invariant under translations and reflection about its starting point, it is permissible to take $ = 0 and a < b. Using 12) and the fact that Pab[t(e, ) = 0] = 0 for $ between a and b (see 20a)], the problem is now simplified to checking that if Gt are the corresponding GREEN functions, then 21 a)

G(a, b) - G(a, 0) G(0, b)/G(0, 0) 00-6) . [G-(a,b)-G-(a.O)G-(O,b).

(0

-

I

21 b)

21 c)

G, (a, b) - G, (a, 0) G, (0. b) ] G+ (0. 0)

G(a, 0) G(O, b)/G(0, 0)'

- ial+Ibl e and

X

+'n

X[

-

G_(a, o)G_(O. b)

G,(a, o)G,(o. b)

G- (0, o)=

G, (O . o)s

G(0, 0)-1= G-(0,0)-1 + G+(0, 0)-1 - 2.

But G_ is the product of

22a-)

g, (r1)

g, (0)

(r1 < 0)

gi (0) sinhrl

a < b)

(rl

0)

2.8. Standard BRowxian local time

69

and

22b-)

g1(0) g2 (r1) - g1(rl) g2 (0) + gi (0) g2 (rl) - g1(r1) g: (0)

(r1 < 0)

(1720)

en

z

a`.

divided by their WRONSKlan

23-)

g1(0) +gi(0),

while G. is the product of

22a+) 'L7

and

(r/ < 0)

en

& (11) g2 (0) - gi (0) g2 (7)) + gi (0) g2(11) - g1(rl) gs (0)

22b+)

g2(0) cosh, + g$0) sinhr7 92 (17)

(112: 0)

(ri < 0)

(11 ? 0)

divided by their WRONSKIan

23+)

g2(0) - g2'(0).

and now 21 c) follows on substituting from 22) and 23). 21 b) is obvious

because both sides solve 00 u = u/2 as functions of a or of b on either side of 0. 21 a) is left to the reader. Using 14), Eab [e-710, h ) I t (e, ) = t] is now computed for J
and t = 0 and for

< S a V b and t > 0: just substitute

2 e-Ib -al

4

(1 +

2)-1 e-Ib-171-In-al*

in place of G in 14) and obtain Eab[e-rt(t. n) I t(e, f)

24 a)

_ ! - e-'aA+-nt = i - e-24M-Et -I.

0

= 0] = 8 + (1 - 6)/d

< rf S a A b

d { ) -= 1 + 2

and

= t]

(t > 0)

"lE-nl/4

< , between a and b

Eab[e-rta, W I t(e, e)

= d -i = d-,

A

24b)

t< 71 S a A b.

Because Pab[t(e, ) = 0] = 0 for between a and b, it suffices for the proof of part d) of 13) to use the MARKOVian nature of the conditional local times and of the 2-dimensional BESSEL process rs and to e-Ib-6I/2 is the GREEN function of 1/2 - 6; see the line above 18) and the footnote attached to it.

70

2. BRowxian local times

check that for

and r/ > between a and b, 24 b) agrees with

Er[e-ye 21)r,(stn-"E)2/4]*

25)

=J

(r = rs(eUE_ e2aAb))

I

2n(e='7-e2?)

e-r'-= n P 12/2 d b

(111=r)

Rs

2e2'1/Y 0291- 21 + 2e=17/y

tog

Spa

,y,

As to t(e, ) for S aAb, it follows from the independence of c and r4 combined with P° [r4 > 0, t > 0] = 1 (see 2.7) that the composite motion r4 [(eU - en V O] is MARKovian, and now a comparison of 24a) and 24b) for J < rj
P0[c?r11 E0[c < 11,

e z EJ** it

= 9 -{-

=B+ and 26b)

E,[e-re-'

e-f(.A+-v) 1 - e-SRAb-C)

2

y0

i(1"_e2E)s/4]

(2n(e"'7 - e=E))= (2e2g/Y)2

=

e-2(ane-n dl/d=(l)

2

(es" - ell f+2e''7/Y)=

(r = r4 (e't - e8°)) a-re

Ibl /4du

(III =r)

-r'/2(esn-62"+ 262h/Y)

completes the proof of 13 a, 13 b, and 13 c).13 e,13 f, and 13 g) now follow

using the MARKOVian nature of the conditional local time and the obvious invariance of its law under reflection about _ (a + b)/2. Using 2.7.8), 27a)

and 27b)

It (C' 6) - t(e, 0)I

PO

l

e+0

V6 ig:1/6

P° [e a10 Italgt/a =2

= 2 t(e 0)1 = 1 1

mRt(e,a)1=1

.o'

E, now stands for the 2-dimensional BESSEL expectation. E, now stands for the 4-dimensional BESSEL expectation, and c is independent of r4 with distribution 13b).

2.8. Standard BRowNian local time

71

can be read off, and it follows (use FUBINI) that 10) holds (with = in place of ;:5) at some positive time. But now the scaling x (t) , cx (t/cs) (c > 0) maps t (t, a) into c t (tics, a/c) and changes the statements of 10)

into the corresponding statements at time i/cs, so 10) holds (with = in

place of $) at all positive times, and an extra result obtained at no extra cost is that Po[t(t, $) > 0, minx(s)

28)

#;58

t > 01 = 1 < f <maxx(s), SS8

as is immediate from Pa [rd > 0, t > 0) = 1

(d = 2,4).

Because of lim 2"-' measure (s: a < x < b, s s t), b-a+2-" I!\

t(t, a) =

29)

b.a

t(s, ) S t(t,

30a)

s S t, E R1;

)

also, in view of 10) and FuBINI's theorem,

0, and since t (s, ) as a function of s is constant near s = t for t * x (1),

b aOIt(tl,b)-t(t,a)J -0.0

30c)

t(t, ) is a continous function o/

r.+

3ob)

for each t = k 2-"

St(t-f-,e)-t(ttZ0,.ER'.

a<E
-+'

Accordingly, for the proof that t (t, ) is continuous on [0, + oo) x R', it is enough to check that Po "limo max t

31)

k<2

(2

"2,w, n9 ") S 194 = 1

SERI

(S,

and to use ASCOLI'S theorem in conjunction with 30a); in fact 31) can be used to improve 9b) to Po

32)

lim e-.-64o

t(t,E)-t(s,E) V lg 1/6

s4 =1

EER'

as the industrious reader will check. 2a) and 2c) follow at once as F'.

noted before.

Because of the HOLDER condition for the standard BRowNian path (1.9), it is enough for the proof of 31) to check that for B > 1g4, 33)

Qn = Pp [max t(2-", , Wt2-") > 0 2-"n, x (2-", mk 2-") > - j132'" is tg 2 for some k < 2")

is the general term of a convergent sum, and this can be done using the standard BRowNian scaling x (t) x (tf cs) and RAY'S description

2. BAOWNian local times

72

of the conditional local times as follows: 34)

Q"

S 2" Po m ax Igo

l

) > 0 2-4/2 n, x (2-n) > -2.2-"'2

(2-n,

=2"Po[$goa t(1,>0n,x(1)>-2yn 2" E. [

5 t(1,

) > 0 n, ez(1)no) e2V* I'+

+00 00 (21)-1

2"+a e2y" Eo Lmax t(1, Sr) > On,

[ g°

r e-Ib-z(1)I dbl o

J

= 2"+2e2l+npo I m t(1, t) > 0 n, e > 1. x(e) > 0, 2'

e2yn po [ms t(e, ) > 0 n, x(e) > 0,

X10

S 2"+se21+"Po[ 5 ra(t)2> 40 n] S 2"+4e2)APo [msai x(t)2 > 0 nJ 2n+be2Vapo

[ m m a x x(t) > V

< constant x 2" e2 y"

J

e-eon

yen < constant x e-e" (o < a < (0 - lg4)/2). Problem 1. Check that f d t/x (t)

li

d t/x (t) exists. ssf

Ix(p >.

+00

[

f d t/x (t) = 2 f [t (1, b) - t (1, - b)] AL and this converges as b.

1

e M 0, thanks to lt(1, b) - t (I, - b)
[Given P < 1/2, 32) implies that t (1, a) - t (s, a) < c (s < t S 1, e= t- s 4 0, a E RI); this justifies

n°&? A'&n[o,1) Zt(1,a)

a E R',

and an application of 28) completes the proof.] Problem 3. Prove that the stochastic process [t (mb, 0) : b is differential and derive the LEvy formula +00

Eo[

t(m..o)]

= (a b + I)-' = exp

- f [1 0

e-11 e- 11b

I

0, Po]

2.8. Standard BRowxian local time

73

[Because t (mb, 0) - t (ma, 0) = t (mb (w,',), 0, wm) (b S a, m = ma) and

t(m,, 0) : c S a is measurable Bm+ the differential character is clear

from the fact that the Bxowxian traveller starts afresh at time m at the place x (m) = a. Consider, for the derivation of the LEvy formula, the function e(1) = e(a,1) ^ Eo[e-at(mt,0)].]

Using t (t, 0) = t (s, 0) + t (t - s, w;) (t L> s) , deduce e (I + e) = e (l) [E1 (e-a' (mt+.' 0), mo < mt+.) + P,(mt+. S mo)] = e (1)

0,05

e+(l) = li1m

1

1+6 e (I -}- e) + 1 +611

e(1

+

6)

-

e(1)

_ -l-' e(1) [1 - e(l)],

and solve this to obtain e(1) = [c(a) I +

1]-1

e(1

c(a) =

`t) > 0.

Because the scaling x (t) -> f -1 x (fl$t) maps mt into f-s mp t and t(1,1)

into fl-1t(f2t,Pl), it is clear that c(a) = flc(a/fl) = flc(1) (fi =a), and, to complete the proof, it is enough to evaluate c = c (1). Differentiating e = [acl + 1]-1 and letting a 1'0, E0[t(mt, 0)] = c

and, using E0[t(m1,1)] = E0[t(m1(wm),1, wm)] = Et[t(m1,1)] = Eo[t(m1-t, 0)]

(m = mt),

one finds 1

1

1

'!t

c=2 cldl=2 f Eo[t(mt,0)]dl=2 f Eo[t(m1,1)]dl 0

0

0 +00

EO 12

f t (m1,1) d it = Eo [measure (t : x (t) z 0, t S m1)] 0

+00

A

= f Po[m1>t,x(t)a0]dt 0 +00

+00

b

rP0[t-(t) < 1, x(t) z 0]dt = f dt f db f da 2 2-

f =1,

2n t'

e- (2b-a)=j2t

n"^

where problem 1.7.1 is used in the last step.] Problem 4. Given b > 0, '[t (t-1(t) , b) : t > 0, PO] is a homogeneous differential process with LEvy measure b-2e-111dl (b > 0) ; here t-1(t) is the inverse function of t (t, 0), i. e., t-1(t) = min (s : t (s, 0) = t).

2. BRowNian local times

74

[Use the fact that t-1 (t) is a MARKOV time and the addition rule t-1 (t) = m + t-1 (t - s, w ;) (t > s, m = t-1 (s)) to prove that t (t-1, b) is differential and homogeneous. As to the LEvy measure, y

Eo f

+00

o

J

III

= Eo [Toe-=(.0)e-4t(.b) t(di, 0)Jl

= Eo f e-.t(t.°)t(di, 0) +00

+ Eo[e-at(mb.°)] Eb .V.

f e-=tu.o)e-P1(t.b)t(dt, 0)J L0

=a-1Eo[1 -e

+Eo[e-=t(me.o)]Eb[e-Pt(m..b)]Y OLD.

and, using the result of problem 3, y is found to be [a + Now invert the LAPLACE transform, obtaining

1)

ire

f

+00

Eo[e-Bt(t-I(t),b)] = e4p(pb+1)-,

= e-4

+V+

as stated. Because f b-2e-t1bdl < + oo, this process has a finite number 0

of jumps up to time t = 1 as is also clear from the fact that the stopped BROwxian path x (t) : t;5 1 cannot come from 0 to 1 an infinite number of times.] Problems 5 and 6 below contain the proof of RAY'S description 11 d) of

Problem 5. Check that E0[a-Yit(m1,E1)-Y,t(m,.E4).. -y,1i

l,Ea]

is the reciprocal of

I + isn Ir Vi 0 - Si) +i
+i
+4n

+Y1Ys

(1 -sit)+... Yn(E-$1)(s-ts)...( n- 1) (1 - n)

and n?1.

for

is bounded between 0 and 1, piecewise linear with corners at $1, ..., ,,, and rises to I at = 1. At a corner it satisfies a matching condition u+ - u- = yu, for if Z is a corner, y the corresponding LAPLACE transform parameter, and e the exit [u

EE [e-Y, t (m,.

2.9. BROwNian excursions

75

time from I rJ - x (0) 1 < 6, then for small a,

u() =

BE[eYt(e,E)

EE[e Yt(r,E), x(e)

= - a] u( - b)

+ E je-Y r, E), x (e) = e + 8] u (e + a) ,

and since t(e, ) depends upon the reflecting BxowNian motion xF =I x - alone, it follows that

u(s) _ [u( - t3) + u(E + b)] E. [e Y1(r,o)]. But PO [lime = 0] = 1 , so letting

0,

640

(u+

- u-)/u = 2l mp6-1 Eo[1 - e -Y10-0)] = c(y),

and c (y) can be identified as y using the fact that the function with 1 corner v (t) = E E [e-Y t (mi, °)] satisfies v+ (0) - v- (0) = c (y) v (0) , v (0) = 3.'

42.

(1 + y) -1, and v (1) = I (see problem 3). Now solve for u and use induction to obtain the stated formula.] Problem 6. Check the evaluation (y > 0)

e-71(1+Y"

Er[eYr,(E)2 ] =

1 +vE

and use it to prove the statement of 11 d) that [t(m1, I - ) : 1, where r! is a 1, PO] is identical in law to rt (E)1/2: 0S 2-dimensional BESSEL process starting at 0.

Problem 7. Prove that Po[t (+oo, ) = + oo] = 1. 2.9. Brownian excursions

Given a standard BROWNian motion starting at 0, let 3 be the (open) interval of [0, + oo) - 8 containing the first number of the list 1

1 23 22 1

3

5

7

9

44444

11

4

3

etc.

not included in 8 or U B,,,, introduce the unsigned scaled excursion m
f)

e, (t) =

* 18.1 is the length of $,,.

0
2. BROwxian local times

76

and the sign

e.=-l or +1 according as x (t) <0 or > 0 for t E 3,,

2)

and note that .3., e,,, and e (n 9 1) are BOREL functions of the BROWN1an path.

o-1

3 is the support of the local time P ; as such, it is the closure o t t-1 (0, + oo), where t-1 (1) = min (s: t* (s) t) is one-sided stable with exponent 1/2 and rate y2. e1, e1, etc. are independent and identical in law, e1 being MARKOVian with

3a) Po[e1(t)Edb]=h(O,O,t,b)db=

(1-t)sb'db

2n=3

0
3b) Po[el(t) E db el(s) = a] = h(s, a, t, b) db j

e-u-m)11(S-n - e-a+.r+r:u-I)

1 - s 3/2 be-i'na-n

2n(t - s)

0<s<1<1.

e1, e,, etc. are independent and identical in law with PO [e1 = -1 ] _ 1) is a standard coin-tossing game. P. [e1 = + 1 ] = 1 /2, i. e. , e, (n 3 and e (n z 1) and e (n 1) are independent. P. LEvY [3: 238] indicated a proof of these facts; here is a new one.

Consider the interval 31 that covers t = 1, the excursion e1, and the sign e1, and let us compute the expectation of 4)

j1[x(s) : s S inf81]j1(81)13(e1)j,, (el)js(waup83)

where a)

j1 [x (s) : s 5 t] =ji[x(s1 A t), x(s1 A t), ..., x(s,, A t)]

0 < S1 < s1 < . < S,,, j1[a1, as,

. .

.,

E C(R" ),

b) j1(81) =js(inf81, sup81),js(s, t) E C(0, +00)1, and 91(81) = 0 for 18, near 0 or + oo, c) j3 (e1) = j3 [e1(t1) , e1(tz) , ... , e1(tn)]

0 < t1 < t1 < ... < t < 1 ,

j3[b1,b2,---, b,]EC(0,+oo)*,

j3=0

outside some compact figure of [0, + oo)A, and

d) 0 S js S 1 is a BoREL function of the BRowNian path. js(w up8) is independent of the other factors of 4) because m = sup31 = 1 + mo (wi) is a MARKOV time and x (t) : t S inf 31, 31, e1,

2.9. BRowxian excursions

77

and e1 are measurable Bm+. As to j1 j, j3 j4, its integral is
cs

"

kZ2' s S (t - 1) 2-"] j, (i 2-", k 2-") X

71 [x (s)

X js

x(t1 d+ i 2-")

x(tzd + i 2-")

v

l L

d

etc.19e (sign x (t 2-"))]

d=(k-i)2-"

J

+00

f Eo [j1 [x (s) : s S (i - 1) 2-"] , (i - 1) 2-'.5 inf 31 < i 2-", 5 kZ2"

"f1

"

{

0

22

jx (i 2'") j E d a] j, (i 2-", k2--*)

f g(t1A,a,b1)g

x x

X

[o,+oe)"

j3

b,

b1

d

IYd

b.

,

yAJ

db db

db

il.

x

1

E

1 [j, (-1) +j,(+1)]

"2

+ OD

x f g((1 -t")d,b", b)dbPb[mo< o .'7

and substituting lid b1, JG b,, etc. in place of b1, b,, etc. and introducing the kernel h of 3), 5) becomes 0

+00

6) "fi+mo

"

i kZ2"

f Eo [jl.[x (s) : s S (1-1)2"J, (t - 1) 2'

inf $1 < i2-R,

x(i2-")1 E da]

+00

f g (A, a, b) db Pb[mo < 2-"] j,(i2-", k2-") o

f h(0, 0, t1, b1) h(t1, b1, is, b,) [0, +a o)"

X

... h(t"-1 b"-1, t", b") x x

b2, .

x e(d, a, b1, b", b), where 7)

e(d,a,bl,b",b) h (0, 0, 11, b1) g (d, a, b)

-s(ab)-1s ab1 )S(

b"b

* g- (t, a. b) = (2x t)-1/2 [e-(a - 0/11 - e-(1+.)h/211.

312 b1 e

(ti

CI-

=Vd g (t1 d, a, ydb1)8-((t-t")d.P b,. b)

(1 - t11

b.e--w1s(1-w)

e-a'(1-t,)12AI,e-b=t"/2A(1-t"),**

** s (b) - b'1 sinhb

2. BRowNian local times

78

Because of the definitions of j2 and j3. a is bounded for the purposes of this integral; in addition, it tends to I as a and b + 0, with the result that Eo[j1[x(s) : s < inf$1]js(81)j3(el)ja(el)jb(w.np8)]

8)

+CO

f E0[jl [x (s) : s S (i - 1)

Wye

= lim

2-'5 inf $1 < i2-",

552 0 ka2"

Ix(i2-")I E da]

+00

x f g- (d , a, b) d b Pb [mo < 2-"] j2 (i 2-8. k 2-") o

x

f h(0,0,11,b1)

etc.

j3db1db2...db"

10,+00)0

x J[jj(-1) +j4(+1)] E0(ja) '^l

VII

= Eo[ji [x (s) : s S inf 81] js ($1)]

x

f h(0, O, t1, b1)

etc.

j3 db1 db2... db"

(0, +ao)w

x k[j4(-1) +j4(+1)] E0(ja). proving that e1 and e1 are subject to the correct law and are independent of 81 and of x (1) :It 81, i.e., independent of 8 and of a", e" (n ? 2) ; the reader is invited to complete the discussion. Problem 1 after J. LAMPERTI [I]. Use problem 1.7.3 to prove that (t) = t - max (s; x (s) = 0, s S t) , Po] is MARxovian with PO [b (t) > b] = n COs-1 b

(b < t)

and

(a<s
Po [b (t) > b I S (s) = a] (d-b)V0 d

2

2(0+a)8J9 x cos-I V d b r

9

-{-

d

+a

Or

0

0

according as d = t - s > b - a or not; observe that b is the same as t - t-1(t+) (t-1 (t) = min (s : t+ (s) = t)) and hence identical in law

to t - m4 (a = t-(t) = 5 x(s)). [Using b- = the biggest root of x (t) = 0 (t < s) and b+ = the smallest root of x (t) = 0 (t > s), it is clear that Po (b+ < t 18 n [o, s]) = Po (b+ < t I b-) = Po (b+ < t i i(s)),

2.10. Application of the BESSEL process to BROwxian excursions

79

and so the MARKOVian nature of a follows from

>t

8(t)t-s+6(S)


_ now use problem 1.7.3.]

r..

mop

Problem 2. Write G, for the BOREL algebra of events B E B whose indicators are BOREL functions of the intersection of the graph of the standard BROwxian motion with the strip [0, + oo) x [0, e) and prove that, the germ Go+ = fl G, is identical to B (8) x B (en : n ? 1) _ ,>0 B (t) x B (en : n 1), up to sets of WIENER measure 0. [Because Go+ is independent of B (en : n ? 1) and the universal

algebra B is identical to B (8) x B (en : n z 1) x B (en : n ? 1), the identification of Go+ with B(8) x B (es : is 1) follows from the obvious inclusion Go+ D B (8) x B (en : is ? 1).]

VII

2.10. Application of the Bessel process to Brownian excursions We saw in 2.9 that the scaled BROwxian excursion el (t) : 0 S t S 1 is MARKOV with

1a) P0[el(t) E db] = h(0, 0, t, b) db = 2

Vn t=(1

b2db

0
.d.

Pa [el (t) E d b el (s) = a]

- t)'

III

h(s,a,t,b)db 2n(t - s)

_

1 - s 312 2n(t - s) J \ f - t)

I1/'() ab

(t-s)Vab

db

I -s 3l2 e-visa-t>

.+I

1 b)

s

db #

0<s
The BESSEL motion of 2.7 can be used to give a new interpretation of el : if r_(t) and r+(t) (0 S t S 1) are 2 independent copies of the radial (BESSEL) part of the 3-dimensional BROWNian motion subject to the conditional law 2a)

Q(B) = P0(B I r(1/2) = b) and if the terminal point b is subject to the law 16

2b)

e-2b'b1db

V2n

then the composite motion 3)

e(1)=r_(t)(0S51

is identical in law to e1. * lus(h) = (2/n E)1I2 sinh f.

2)

=r+(1-t)(2 ti)

b>0,

2. BROwxian local times

80

Consider, for the proof, the BESSEL kernel 2.7.4) in the case n = 3: ., + i

g+ (t, a, b)

4)

Given

9t

a

11/z (a b

I ab

) b'

t>0,a,b>0.

1 > I > s > J and 0 < a, b and using the MARKOVian

character of r+(1 - t)

J), one finds

(t

P. `lr+(1-t)EdbIr+(i-0):2 SOSs,

5)

= g+(1-t, 0, b)

g+(1 -s, o, a)

(t-s))lab I

1Iz

g+(t - s, b, a) db (

a=r+(i-s)

ab \ ( t -s\312

t-

t -t

s

b' d b

= h(s, a, t, b) db. Because e (j) and el (j) are identical in law [see 1a) and 2b)], r.+

5) implies that e (t) (1 Z J) is identical in law to e1 (t) (t ? J), and to complete the proof, it suffices to note that e and el are symme-

trical in law about t = J. Here is an application to the fine structure of the standard 1-dimensional BROwxian path.

A. DVORETSKY and P. ERDOs [1] proved that if 0 S / and if t- /(t) E t for small t, then Po [r (1) > / (t), t 4 0] = 0 or 1 according as f t- EJz / (t) d t Z + 00

6)

o+ +'3

where PO is the WIENER measure for the 3-dimensional BROwNian motion starting at 0*. Now using the definition 2.9.1) of the scaled BRowNian excursions coupled with the fact that

I

I8 I

f t-s/2/(I8sI t) dt = f t-312/dt, 0

0

one finds 7)

4 0, n ? 1] = 0 or 1 according as

Po[x(t, 1

f t-s/z /(t) dt Z + oo, 0+

where P. is now the WIENER measure for the 1-dimensional BRowNian

motion starting at 0 and the 3, (n z 1) are the open intervals of the * See 4.12 for a proof of 6).

2.11. A time substitution

81

complement of .3 = (t : x (t) = o); for example 8)

PO 1Ix(t. wi.fgjI>

1

Ig

t

192 ! ... (ig7) 1

+. , 1. 0 , n Z 1 = 0 or 1

according as e S 0 or e > 0. According to 2.7.8a), PO lira

9)

7

1+o

V

`-

r(t)

=1=

2119s

from which one deduces 10) PO

ol$

i

I, r j

J l=

Ix (t ! $.I ; V211g=

1

lo

Jj/2t1gs__ x(t , w,N) I= 9, n Z 1 = 1,

2.11. A time substitution

Given a standard 1-dimensional Bxowxian motion starting at 0, the decomposition of 2.9 can be used to prove that if f -1 is the inverse junction of f (t) = measure (s : x (s) z 0, s S t), then [x (f -1). Po] is identical in law to the reflecting Bxowxian motion [x+ xI, Po] of 2.1.

Given a Bxowxian path starting at 0, throw out the negative excursions and shift the positive excursions down on the time scale,

-1 +1

f(t) Diagram I

closing up the gaps as indicated in the diagram; the path so defined is x (f -1) , and it is clear from the picture that, for its identification ItA/11cKean, Diffusion processes

6

2. BRowwian local times

82

as a reflecting BROWNian motion, it suffices to show that W = (t x (f -1(t)) = 0) = f (8+) is identical in law to 8+ = (t : x+ (t) = 0). Because f (8+) is the closure of f t-1[0, + oo), where t-1 is the leftcontinuous inverse function min(s : t + (s) Z>= t) of the reflecting BROwxian

local time t+, it suffices to show that f (t-1) is a one-sided stable process

with exponent 1/2. But f (t-1) is t-1 minus the jumps arising from the negative excursions : 1)

1

A

f (t -1(t)) =

an CIO, t-' (I))

') (1 + en) 13 I .

+00

and, using the POISSON integral f l p ([0, t) x d l) for t-1(t) (see 1.7) 0

coupled with the fact that the coin-tossing game en (n ? 1) is independen tof ,$+, it is clear that the effect on the PoIssoN processp ([0, t) x dl) .,y

of ignoring the negative excursions is to cut its rate in half (see +00

problem 1). f (t-') is then identical in law to t-1(t/2) = f lp([0,t/2) X dl)

(t z 0), and the proof is complete.

0

An alternative proof is to note that f (t-1) is differential and homogeneous (see problem 2) and to use 1) to compute Eo [-a f (t-' (t))]

E0

lT

e-a(1 +on)18.1/2

old

2)

l8.CLO, t-'(t))

+0

f )g(

1+28

=E0eO

I

)I p((O.8)xd1)

+00

( 1 + e a1-1)

t

J t

e0 e

2

2

di 2-.P

2a

Time substitutions are treated in 5.1, 5.2, and 5.3; it will be seen that the present substitution is not accidental. Problem 1. Given a standard coin-tossing game en (n 2 1) (P [e1 = -1 ]

= P[e1 = +1] = 1/2) and an independent PoissoN process with jump size 1, rate x > 0, and jump times t1 < t2 < etc. with distribution P [t,, - 1, 1 > t] = e -"t (n a I, t0 = 0) , deduce that e (t) _ is POISSON with rate x/2.

k (1 + en)

3.1. Definition

83

[Given disjoint intervals d,, A., etc. of length d,, ds, etc.,

P [toI (1 + en) = j1, 1(1 + eR) =1s etc.l

I P( 1=n1, I J(1 +en)=11 Zt4Ed,1 =ns, non,

tha:h

%ah

n,
(n d1) e-xd=

n22 +a 2i,

=

n,

1!

(x- d1)

2-n, (n d,) n' a-xd, (n=) 2-n,

a-xa, (" dJ a-Xd,

etc.

1s

s

11

with x = x/2.1

etc.

1s!

11!

J(1 + en) =12, etc.

Problem 2. f (t-1) is differential and homogeneous. [Because of the addition rule t-1(t) = t-1(s) + t-' (t --- s, wm) (t > s, .`Y

CDs,

m=t-1(s)).onehasf(t-1(t))-f(t-'(s))=f(t-1(t-s,w ),w,)(t>s); III

in addition, in is a MARKOV time because in < t means that t+ Z s

at some instant < t. Given t S s, (f

(t-'

(t)) < b) fl (m < a) E B4,

i. e., (f (t-1(1)) < b) E Bm+ (t S s) ,

and the homogeneous differential

character of f (t-1) follows on noting that the Bxowxian traveller starts

afresh at 0 at time t = in.]

3. The general 1-dimensional diffusion 3.1. Definition

"''

Roughly speaking, a 1-dimensional diffusion is a model of the (stochastic) motion of a particle with life time m. S + oo (also called killing time), continuous path x (t) : t < in.., and no memory, travelling in a linear interval Q; the phrase no memory is meant to suggest that the motion starts afresh at certain (MARKOV) times including all the constant times in = s > 0, i.e., if in is a MARKOV time, then, conditional on the present position x (in), the statistical properties of the future motion x (t + in) (t > 0) do not depend upon the past x (t) (t S in). Before stating the precise definition, introduce the following objects: Q a subinterval of the closed line. 0o

an isolated point adjointed to Q. 6*

84

3. The general 1-dimensional diffusion

a sample path w : Co. + oo) -> Q U oo with coordinates x (t) =

Cep'

w

811

x (l, w) = xj = xt (w) E Q U oo and life time mm = in,,,, (w) < + o0

such that

t<m

!-F

x(t)=x(t±)EQ

tZm,,,

= 00

esp., x (+ oo) F_ 00.

W the space of all such sample paths. the smallest BOREL algebra of subsets of W including

Be

(w: a S x(1) < b) for each choice of t s' s and a S b.

BBB+,,.

b=a.

.-.

mod

'.7

0

P. a function from Q to non-negative BoREL measures on B of total mass + 1 such that, for each BE B, Pa(B) is a BOREL function of a E Q, and, for each a E Q, P. [x (0) E d b] is the unit mass at

P the unit mass at the single path x (t) = oo (t Z 0). m a MARKOV time, i.e., a non-negative BOREL function m = m (w) S

+ 00 of the sample path such that (z v: m (w) < t) E Bt (t ? 0). x (I + m, w) (I ? 0). B,+ the BoREL algebra of events B E B such that B n (m< t) E Bt (I Z 0).

-0-

w n the shifted path x (1, wm)

ode

Q, B, P. as above are said to define a motion D with state interval Q, path space W, and universal Borel algebra B. Pa (B) is read : the proba-

...

bility of the event B for paths starting at a; note that P. [lim x (t) = x (0) = a] = P. [m > 0] 1 (a E Q). 8 10 D is said to be conservative, if P. (m < + oo) = 0 (a E Q) . D is said to be simple Markov if it starts afresh at each constant time

t Z 0, i.e., if I a)

P. (wi E B I

Pa (B)

a = x (t) ,

BE B,

or, what is the same, if 1b)

P.(x(t)EdbIB,)=P4(x(t-s)Edb) a=x(s),

D is said to be strict Markov or a diffusion if it starts afresh at each Markov time m, i.e., if 2a) E BI Bm+) = Pa(B) a = x(m), BE B,

or, what is the same, if

2b) P.(x(t+m)EdbI B,n+)=Pa(x(t)Edb) a=x(m), (note that x (m)

tZ0

oo if m = + oo).

Given an exponential holding time e with law P(e > t) = e', if w' is the sample path 3)

x' (t) = 0 (t < e)

= t - e (t ? e)

3.1. Definition

85

and if P. is chosen so as to have P0(B) = P(w' E B)

4a) 4b)

Pa[x(t)so a+t,t?0]M1

a>0,

'ay

then the motion so defined on Q = [0, + oo) is simple Markov but not strict Markov; indeed, m = inf (t: x (t) > 0) is a MARKOV time, Po[m > 0, m(wm) = 0, x(m) = 0] = 1, and, should the motion start afresh at time m, PO [m > 0, m (wm) = 0] would be PO (m > 0) PO (m = 0) = 1, which is absurd. Given a pair of motions D and D', if / is a BOREL function mapping part of Q onto Q' and if f is a BOREL function mapping (w: x (0, w) E into the dot path space such that

/[x(0, w)] = x (o, f w) x (0, w) E f-1(Qo) a E f-1(b) , P. (f-1 B') = Pp (B')

5 a)

B' E B', then If w, Pr, (b): b E Q'] is said to be a non-standard description of the dot motion as opposed to the standard description used before ; as an example, if D is the standard 1-dimensional BROwxian motion, then x+(t) = 1 x(t)I tZ0 6a) 5 b)

x - (t) = x(t)

6b)

t < MO

=t-(t)-x(t) t- (t)

x' (t) = x U-1)

6c)

tz mo

Ssl X (S)

tZ0

f (t) = measure (s : x(s) z 0, s S t) are three separate non-standard descriptions of the reflecting BROWNian motion (see 2.1 and 2.12) ; additional examples of non-standard description will be found in 2.5 (elastic BROwxian motion), 2.10 (BESSEL Goo

motion), and problem 2 below. Given a motion, the choice of description depends upon the problem in hand; for example, the non-standard descriptions 6a) and 6b) were useful for the discussion of the BROwNian local time t (1, 0) (see 2.2) cad

but the proof that a motion starts afresh at a MARKOV time is best done in standard description (see 3.9, 5.2, and the second part of 't7

problem 3 below). Problem I. Prove that 2a) includes BLUMENTHAL'S 01 law :

P.(B)=0 or 1 BEBO., (=flB.). .>o

[m = 0 is a MARKOV time, and so, for BE P.(B) = P.(B, wo E B) = E.(B, P.(wo E B I BO J) = P.(B)2.] Bo+.

3. The general 1-dimensional diffusion

86

Problem 2. Explain the correspondence between the MARKOV time m and BOREL algebra Bm+ of a) the standard 1-dimensional BROWNian motion

b) its non-standard description as a scaled BROwxian motion [x' (t) = c x(t/c') :1

0, P11 : E E R'] .

[Because B; = B [x' (s) : s S t] = Btt., if m' is MARKOV for the

scaled motion, i.e., if (m' < t) E B;

(t

0), then (c-' m' < t) =

(m' < c't) E Bt (t --> 0), i.e. m = c-' m' is MARKOV for the unscaled motion, and All

._Y

B'm.+= (B:Bn w< t) E Bt,t;-> 0) `°.

es Bm+=(B:Bn(m
Given t ? 0, define the stopped path x (s, w;) = x (t A s, w) (s Z 0) and the BOREL algebra B; = B fl (B: w E B iff w; E B) and let us prove

B; =Bt.

1)

Because B; contains (w: x (s) < b) for each choice of b E R' and s S t, B; C Bt; as to the opposite inclusion, if B' is the BoREL algebra B fl (B: (w: w; E B) E Bt, t a 0), then (w: x (s) < b) E B for each choice

of b E R' and s Z 0, i. e., B' = B, and it follows that, if B E B;, then

A

o!°

i.'

B.= (w: wt E B) E Bt as desired. A. R. GALMARINO showed us a proof of the useful fact that a nonnegative Borel /unction m = m (w) S + oo is a Markov time i f f 2a) x, (u) = X. (v) s
M (U) < t

M (U) = MM. Consider, for the proof, a MARKOV time m, sample paths u and v

as in 2), and an instant s < t. Because B = (t v: m < s) E B, = B;, it is clear, using 2 a), that m (u) < s iff m (u;) < s iff m (v;) < s iff m (v) < s, and 3) follows on using 2b). As to the opposite implication, if 0 S m S + oo is BOREL and if 2) implies 3), then m (w) < t iff m (w;) < t, i.e., (w : m < t) E B; = Bt, showing that m is, indeed, a MARKOV time.

3.2. MARKOV times

87

o?:

Using GALMARINO's result, it is a simple matter to show that the class of Markov times is closed under the operations: 4a) 4b) 5a)

ml A m2, m, V M2, m,

5b)

6a) 6b)

mi + m2 (wm,) ,

6c)

m$(zemi)

ml -{- M2-

Consider, as an example, the proof that 6b) is a MARKOV time. Given MARKOV times nil and m2, if in = ml + m2 (wm,) , if x, (u) = XI(v)

(s < t), and if t > m (u), then t > nil (u), proving m1 (u) = nil (p); in addition, X. (ue) = x, NO

7a)

s < t - 0,

0 = m, (u) = m1(v)

m2(u+) < t - 0,

7b)

cad

proving m2(ue) = m2(ve), and adding, in (u) = m, (u) + M2 (ue) = m, (v) + m2 (ve) = in M, completing the proof. As in 1.7, the BOREL algebra 8)

Bm+=Bfl(B:Bf(m
Bill, + C Bm,+

.1]

fl Bin. + = Bm+

9b)

n2:1

m in,

fl B[x(tA(m+e)):1?0];

10a)

>0

B,n+ = fl B[x(t A (m + 6)):t? 0] 411

in fact lob)

M, S mg,

e>0

(see problem 1 below) ; in addition, 10c)

Bm+=Bfl(B:wEB iff w;EB,t>m(w))

as will now be proved.

Given BE B, if wEB iff wiEB for 1> m(w),thenBn(m
As to the opposite inclusion, if B E Bm+, then B n (n< t) E B;, and

* (m<1)isshort for(w:m
88

3. The general 1-dimensional diffusion

using GALMARINO'S theorem in case t > m (w), one finds w E B iff w E B (1(m < t) iff wi E B n (m < t) iff wi E B, completing the proof. Problem 1. B., = f l B [x (t A (m + 0): t 0] (see t 0b) ). >o

[Given BE Bm+, w E B iff w; E B on (m < t) thanks to 10c) . B E B [x (t A (m + e)) : t ? 0] can be deduced from this as in the proof of 1).] Problem 2. Prove that the BOREL extension of Bm+ and B [x (t, w;,) :

Z 0] is the universal BOREL algebra B. [x (t) = x (t A M) X the indicator of (m ;2: t) +

T x (t-(k-1)2-", wa,) x

+ him

"t +O°k2"<1

x the indicator o l ((k - 1) 2-" m< k 2for each t Z 0; now use the fact that m and x (t Am) are measurable Bm+ ]

Problem 3. mm = min (t : X (I) = oo) ,

mo. = inf (9: x (t) > 0). mo = min (1: x (t) = 0), mo+ = Iim min (t: x (t) = e) +o

are all MARKOV times.

t) = U (w: x(k2) = oo) E B1 (I ? 0), i.e., k8-"<1

M.0 is MARKOV ; the proof that mo

is a MARKOV time is similar. mo

is MARxov because mo < t means that the sample path tends to 0 along some decreasing series of times k2-" < t. mo+ is MARKOV because

N+ < t means that the points x (k 2-") : k 2-" < t cluster to 0 from above.]

Problem 4. P.[m > 0] = 0 or I if in is a MARKOV time.

[(m > o) = U fl (m ? n-i) E n ns

1n m

O:3

THAL'S 0 1 law.] Problem 6. (m, < m,) E

B,,-. =

B04

;

now use BLUMEN-

+mZI Bmi+.

[(m, < m,) n (m, < t) = (m, < m, < t) u (m, < t S m.J

=(m,(w,)<ms (wi)
(t?0),

thanks to 1) and GALMARINO'S theorem.]

Problem 6.

«E'

M, (wm,) A m, = in, n m2[Using GALMARINO'S theorem, if m, < in,, then m, m, (wm,) z in, or m, (wM,) < m,, in which

case in, = m, (wm) < m,, contradicting m, ? m=.]

3.3. Matching numbers

89

dam,

Problem 7. Show that, for the exit time m = inf(t:.x(t) 4 0),

tz0

Po[m>t] =e"` 0Sx5+oo, and

x<-{-oo. t10

[Because (m > s) E B,+ (s z 0), m is a MARKOV time and P. [m > I

:-'

+ s] = PO [m > s, m (w;) > t] = Pa [m > s] Po [m > fl, proving .-.

PO [m > t] = e "l. Given x < + oo, PO [m > 0] = 1, and since m;5 m., E Bm+ (see problem 5) to show that it is enough to use (m <

=0] SP0[m=0]=0.]

Chi

Problem 8 (problem 7 continued). B = (I: x (t) = 0) has no interior point in case PO [m = 0] = I (x = + oo). 3.3. Matching numbers

Given a S b S c, if x (O) = a, then ma = mb + ma(w) (m = mb) , and since m is measurable Bm+ and the path starts afresh at time

i = m at the place x(m) = b, 1)

E. (e-me)

Ea

is

= Ea Fe-m' E.

(e-m. (wm) I Bm+)]

= E. (e_-') Eb

therefore monotone in a and b (a S b), proving the

existence of the limits e, = lim E. (e-m') = E. (e-m.+) 2a) 2b)

e, = leim Jim E, (e-mb) = Jim Ea+ (e-ma)

2c)

eg = lbm Eb (rm') = Ea+ (e-m')

2d)

e' = lim Eb(e-m'+),

'+V

mb, i. e., ma+ = inf (t : x (t) > a) or min (t: lien x (s) =a in which ma+ #ft according as x (0) S a or x (0) > a.

We are going to prove that 3a) td-

e, =0 or 1

3 b)

e, = 0 or 1

3c)

e4=0or1

4a) 4b)

e,=ele,Se,

5)

e3 = e, e4

ei

e,es=e,e4.

Given a sample path starting at a, if m = ma+= inf (I: x (t) > a) then, m (wm) = 0 and x (m) = a in case m < + oo, so 6) e, = E. (e_m'+) = Ea [e-m, in < + oo, x (m) = a, Ea (e m

jj

(/QM) I Bm+)J

= ei ,

90

3. The general 1-dimensional diffusion

proving 3 a) ; 3 b) is clear from 7)

e2 = lim b4a Ea+ (e ma) = b m l m E.,

(em') Ee (em`)

= e' lim Ea+ (e-m2) = e2 b+a

3 c) from e4 = bjn Eb

8)

=1s{n lim Eb (e-"4) EQ (e-m.+)

= lim Eb (e_ m +) e4 = e42, bfa

4a) and 4b) are trivial; and, for the proof of 5), it is enough to cite the fact that, in case x (0) > a, ma > ma+ implies + oo = ma > ma+ = m

and to infer that el ea = bJim a i m E. l6 ma

9)

/ mI \mm.) m

hi

t

lim Ec [e-'-+ a

lim lira 1' E [G m4 ame (soma)] bia cia d+& = bli{m ova Ji m EC (e-m4) Ed (e-ma) = es e4. a

We can now prove 10a)

ei = P. (ma+ < + oo) = Pa (ma+ = 0)

lob)

e2=birPa+(me<e)

10c)

e3 = bm Pb (ma < + oo) = 8lin Pb (ma < e)

10d)

e4 =b m Pb(ma+ < + oo) =bm Pb(ma+ <e),

in which e > 0 is chosen at pleasure; indeed 10a), lob), and 10d) are clear from the 01 laws 3 a), 3 b), and 3 c) and the fact that, for a:-!9 b, P. (mb < + oo) and P. (mb < e) are monotone in a and b. 10 c) follows from 11)

0 S Pb (ma < + oo) - Eb (e-m') = Pb (ma < + oo) - Eb (e me+, ma < + oo)

= Eb (1 -

ma < + co) S Eb (1 -

Pb(ma<+o0)=0

i 0 if b j a, e4 = 1

if b>a,e4=0

and

12)

Pb (e S ma < + oo) 5 Pb (e S ma+ < + co) S Pb (e S ma+) j 0

bia, e4=1 if b>a, e4=0. if

SPb(ma+<-+oo)=0

lim [md + ma (wm4)]= lim min (t : x (t) = b, t > ms) =min (t : x (t) -b, t > m,. ) dja d4a = m.+ + ma (wa.+)

3.4. Singular points

91

I".

23 = 8 combinations of the values of el, es, e4 are possible. 3 of these violate 4a), 4b), or 5), leaving 5 possible cases: ei

e.

e,

e4

1

1

1

1

1

1

0

0

0

1

0

0

0

0

0

0

1

0

0

F.,

10+

(the values of e3 follow from 4b) and 5); the blank indicates that all values 0 S e, S 4 are possible). Consider, now, the corresponding left limits el, e2, etc. and let el = ei, e2 = eQ, etc. 5$ = 25 combinations of ei, e2, e4 and ei, a", e* are possible. 3 combinations violate the fact that eiei = 4 implies e- e3 = 1 (see problem 1), leaving 22 possible cases. Problem 1. esea = I if eiei = 1. l4 L

= el = E. (e-mom) = lim E. (e-m.) = lim E. [e-"+ E m°(Wm.+)] cta eta

= Jim b jm E. [e m' -m. (44)]:5i bun Eb (a eta

ea as in 9), etc. ]

Problem 2. Give concrete examples of the 5 possible combinations of el, e2, e,, e4.

3.4. Singular points

Keeping the 01 law 1)

el* =PE(mjf =0) =0 or 1

in mind, the point e E Q is defined to be regular if ei = ei = 1, singular (e E K_ U K+) if eiei = 0, left singular ( E K_) if ei = 0, right singular ( E K+) if ei = 0, a left shunt if ei = 1 and ei = 0, a right

shunt if e-=0 and e+=I, a trap ( E K_ fl K+) if e i = e+ = 0. K_ (= the class of left singular points) is closed from the right in Q, meaning that if the left singular points bl > b2 > b3 etc. 1 a E Q, then a is

left singular also; for, in that case, Pa(mb < +oo) = 0 (a < b) because (a, b) contains points that cannot be crossed from the left. K+ (= the class of right singular points) is closed from the left in Q,

Nib

meaning that if the right singular points bl < b2 < b, etc. t a E Q, then a is right singular also; the proof is the same. K_ and K+ are Borel sets; for example, if the points, al, a2, etc. are dense in K+ and include all its left isolated points, then the fact

3. The general 1-dimensional diffusion

92

that K. is left closed in Q implies

K+=Q(1 f n1

1)J

,r.

Q - K_ U K+ (= the class of regular points) is open in the line (i.e., not just relative to Q) ; indeed, if is regular, then e i = e* = 1

which implies e$ = e8 = 1, or, what is the same, P. (mb < + oo) Pb(ma < -}- oo) > 0 for some a < f < b, and the whole neighborhood (a, b) must then be free of singular points. K_ U K+ (= the class of singular points) is then closed in Q. 3.6. Decomposing the general diffusion into simple pieces

Given a diffusion D, if either P. (Mb < -}- oo) or Pb (ma < + oo) is positive, then a and b are said to be in direct communication; if there d'°

is a finite chain of points c1, c2, etc., leading from a to b, such that c1 is in direct communication with c2, c2 with ca, etc., then a and b are said to be in indirect communication. On Q, (indirect) communication satisfies the following rules: each point communicates with itself ; if a and b communicate, then so do b and a; if a and b communicate and also b and c, then so do a and c. Q therefore splits into non-communicating classes Q* with perfect (indirect) communication inside each class and this splitting induces a splitting of D into autonomous diffusions D* = [W* = W(Q*), B* = B(W*), Ps = Pa:aEQ*].

D* is trivial if Q* is a single point 0, for then

1(t)=P0(m,>t)=P0[x(s)=0,sst] satisfies 1)

f(t + s) = Eo[ma, > s, x(s) = 0, Po(ma, (w,) > t I B,)] = f(t) f(s),

and therefore 2)

P0(m,o>t)=ea1

tZ0,05x<+oo.

When Q* is not a point, it is an interval, for if a, b E Q*, then .'a

the existing communication between them implies that each point of (a, b) communicates with a; thus, [a, b] C Q* and the result follows. Consider the (open) set of regular points in Q*, split it into disjoint open intervals, single out the special intervals that end in opposing shunts, and call the end points of the special intervals and the traps belonging to int Q*, special singular points. Given special singular points c1 > c= > c3 etc. i a E int Q*, Pa (ma+ < + oo) = P.. (ma < -}- oo) = 0,

3.5. Decomposing the general diffusion into simple pieces

93

violating the fact that Q* n (b f,' a) and Q* n (b > a) are in (indirect) communication, and, making a similar argument for special singular points cl < C2 < c3 etc. f a E int Q*, it develops that the special singular

7C'

.+.

points are finite in number on each closed subinterval of Q*. Thus, the series of special singular points cuts Q* into non-overlapping intervals Q' (closed at the cuts), each with perfect (indirect) communication and no internal special singular points. Q' contains either no internal singular points or just one kind of internal shunt : for if Q' contains two opposing internal shunts cr < c2, then there is a maximal shunt ct S c3 < c2 of the same kind as cl and a minimal shunt c3 < c4 S cs of the same kind as c2, and either (c3, c4) is a special interval, violating the choice of Q' or (c3, c4) contains singular points, violating the choice of c3 or c4. Consider, for the next step, the exit time e = inf (t: x (t) D' = [x (t, we) : t z 0, Pa = P. : a E Q'] is, itself, a diffusion (see 3.9) ; moreover, a particle starting in an interval Q' = Qi performs the correvii

sponding diffusion D' = D1, up to the exit time e = el (S + oo); if e, < + oo, it enters into a new interval Q' = Q2 (never coming back .r.

°'4C

to Qj - Q2 at all), performing the corresponding diffusion DZ up to the exit time e2 (S + oo), etc. D* is thus decomposed into the simpler diffusions

D. Up to a change of the sign of the scale on the intervals Q' without internal left singular points, 9 separate cases arise:

3d) . 3g)

0

---i

3 b) ---

3 c) 0-a

3e)

3f) ---0

rat

3 a) 0-0

3h)

i

3i)

ii

and -o mark the end points not belonging to Q', .- and the end points belonging to Q* - (K- n K+) , and j the end points belonging to Q' n (K- n K+), and, in each case, Q' has perfect (indirect) communication and no internal left singular points, so that if inf Q' < a < b < sup Q' , then Pa (ntb < + oo) > 0, i.e., direct comin which

munication prevails on int Q*; indeed, if a E intQ', then 0 < Pb (ma < + co)

for some b < a < c, because, if not, then a is a right shunt, i.e., Pa (ma+= 0) = Pa (ina- = + oo) =1, and Pa_ (ma < -I- oo) = 0 would prohibit

communication between Q' n (b < a) and Q' n (b > a) ; now a simple HEINE-BOREL covering completes the proof.

In the complicated bits below, case 3 g) will often be singled out, the other cases being left to the reader.

94

3. The general f-dimensional diffusion

3.6. Green operators and the space D

Given a diffusion D on an interval Q with endpoints 0 and 1, let B (Q) be the space of (real) BOREL functions / defined on Q U oo such

that /(oo) =0 I1/I1 < +00,

1a)

1b)

and, introducing the Green operators +00

a > 0,

Ga : / E B(Q) -> E. If c °t /(xt) dt

2)

0

let us show that u = G.1 satisfies 3)

u E B (Q)

4a) 4b)

u(a-) = u(a)

PQ[ma- = 0] = I

u (a+) = u (a)

Pa [nta+ = 0] = 1

5 a)

u(a-) exists

bim Ea_ (e-m') = I

5 b)

u(a+) exists

lbi m

6 a)

u (a-) = [1 - k, (a)] u (a)

urn Pb [ma_ < +001

u(a+) = [1 - k- (a)] u(a)

k+(a) = Pa- [ma = + 00] lim Pb[n ,, < +oo] = 1,

6b)

i

= 1,

.

a

k- (a) = P 1. [ ma = + 00] 7a)

u(1-)=0

1*Q,

bmPb[ml_<+001=f

7b)

u(0+)=0

Off Q,

bl o1Pb[ma+<+O0]=1.

3) is clear because P. (B) is BOREL on Q for each B E B. Given a MARKOV time in, since e -am is measurable Bm+, II

u=Ga/=E,[1e_ at/(xt) dt+e-a"'

aat/(x(t,w+))dtj

`''

8)

0

0

= E. I f e-°t /(xt) dtJ + E. e-a n'E. (C&

f

=

I/(x(t,

wm)) dt j Bm+

E[fe/ (xt) dt + E[em u(xm)]; 0

.,.

9)

3-0

4), 5), 6), 7) are going to follow from this. Beginning with 4 b), if P. [ma+ = 0] = 1, then 8) implies

u (a) = l {Ea [ I m to

/ (x) d t, + E. (em) u (b) = b m u (b) J

3.6 GREEN operators and the space D

95

C"'

in which the existence of the limit is part of the assertion; as to 5 b, 1, then 8) implies if lim bla

r

m

10) lm u (c) = m m l E[fet/(x.) dt , ..n

o

+ Ee e="u () o+au b (- ) b()= lim

toL6b),

if limPb[ma+ < +oo] = 1 and if

as desired; and, as

bya

Pa+ [ ma < + oo] = 1 - k_ (a) , then (see 3.3.10 c)] Ea+ (e'a m.) =1 - k_ (a) ,

and, using 8) coupled with u (oo) = 0, it is found that m.+ l

b m u (b) = p m (Eb f f e at l (xt) d tJ - Eb [e a m.+ u (x (ma+))]

11)

o

= lim Eb [e O m.+, ma+ = ma < + oo] u (a) b+a

= Jim Eb(e a m.) u (a)

_ [1 - k_ (a)] u (a). 8,0

Because mo+ = m,,, if mo+ < + oo and 0 f Q, 7b) is a special case of 6b); the proofs of 4a, 5 a, 6a, 7a) are similar (see D. B. RAY [2] for the original of this method). A special case of 3, 4, 5, 6, 7) is the fact that the GREEN operators map B (Q) into the space D of /unctions / E B (Q) such that

lim/(b) = 1(a) i/ Pa[ma- = 0] = I

12a)

lim/(b) _ /(a) i/ Pa[ma+ = 0] = 1; bia this will be important for 3.7. Problem 1. A simple MARKovian motion D, defined as in 3.1, is .-.

12b)

:ti

Strict MARKOV if

? 01= 1

...

P. (li m (G. /) (xe) = (G. /) (xe) , t `[Consider

/ E C (Q) ,

/ (oo) - 0.

r__,

a MARKOV time in, let in,, = ([m 2"] + 1) 2-" (n z 1), let e = e (w) (0 S e S 1) be measurable Bm+, and select / E C [0, 1] such that / (oo) = 0. Because (m" = k2-n) E Bk2-.. (k z 0) and a (wmj = e (see 3.2.10c) ), it is found, as in 1.7, that e-at /[x (t + m)1 d tl

Ilk

L

ntmo E. [e(wt.) f e'"`e/[x (t +

0

...

+f00

E.I e

d1

* 00

= lim

Ilk

J

nt+oo

E.(e Ga /[x (m")])

= E. (e G. / [x (m)]).

a = x (m) .] ICI

P.[x (t + m) E d b B,,,+] = Pa[x (t) E d b]

17'

.ti

i.e., inverting the LAPLACE transform, and expressing the result in terms of conditional probabilities,

96

3. The general 1-dimensional diffusion

Problem 2. A simple MARKovian motion D defined as in 3.1, starting afresh at the special MARKOV times ma = min (t: x (t) = a) aEQ ma_ = lbim mb a > inf Q

a < supQ,

nta.r = blimmb

also starts afresh at the general MARKOV time, i. e., it is strict MARKOV.

[3.3, 3.4, 3.6 can be copied unchanged. Given a dense countable subset I of K. including all the left isolated points of K,, P. [x, < xx E K. for some s > t Z 0]

S ZP.[m!<+oo,x(s+mj) <1 for some s>0] IEI

s E P. [mi < + oo] Pi [m1. < + oo] 1EI

= 0,

i.e., the path cannot cross down over a right singular point; the proof that it cannot cross up over a left singular point is similar. Because

G.B(Q)C D, it follows that P.(lim(G.f)(x,) = (GaJ)(xg), t? 0 = 1, f E B (Q), and an application of the result of problem 1 completes 11

the solution.) Problem 8. Given a diffusion D with state interval (0, 1], if 0 is entrance in the sense

that lim E0f (e-mf) = i fjo 1

uP.(ma<+oo)=0,

then it is possible to define a new diffusion D' with state interval [0, 1], Pb = Pb (b > 0), and PP[mo+ = 0,limmo(1t) = +ool = 1. left.

to

[Define P = P112 X P118 X P114 x etc., and using w1122, u,1tg,

denote sample paths starting at J, }, etc., let

mu _

n? 2,

mui (Tut «1)

tZr and let w' be the new sample path

t=0

x'(t)=0 Because

=x(t- mu,.,Wu,)

mu,St<muS-1,

=x(t- M6, W112)

m;,2S<+00 .

P (m;rn < + oo) Z E [e-m',M] = E [e

n

(w 11

m11,,

1

n t + oo,

3.6. GREEN operators and the space D

97

0.'

P[.Iim m*,/,, = 0] = 1, or, what interests us, P f li m x' (t) = 0] = 1, and j ` It +oo it follows that Po (B) = P [w' E B] is a non-negative BOREL measure on B of total mass + 1 such that PO [mo+ = 0, ±i+m mo (ws) = + ool = 1. 1 Choose / E B [0, 1 ] ; then ll

+ 00

(G,; /) (0) = Eo

t/ (xi) d t

i 0f e

+ao

= lim Eo I f e-a: /(x (t + in. (wm ) + m)) dl] `40

0

m=mljn,

t

Te-AtI(x(t +M.))di

=1imE11ai .40

n Se< x1

L0

= lip (Ga /) (e)

_ (Gaff (0+), which implies P;(lim (G. /) (x,) = (Ga /) (x1) ,

rIt

i z 01 =1, and to com-

a°,

plete the solution, it suffices to show that [x (t) : t _> 0, Po] is simple MARKOV (see problem 1). But, if 0 S e1 S 1 is measurable B., if limo e1(w;) = e1, if e2 E C (0, 1]. and if e2 (oo) =_ 0, then, computing as above, + o0

Eo l e1 llll

f e_at o

= of+oo lim

et(xt+,) dt] +m E' l el (w+,) f e-°'' e2(wt+,+m) dt l L

m = ml/n

0

_ lim, Ell. [e1 (Ga et) (x,)]

=

i

+oO

E;[el (w+,) (Ga eo (x,+m)]

= E; [el (Ga e2) (x.)],

and, inverting the LAPLACE transform, one finds

a = x (s).] Problem 4. Give an explicit isomorphism of G1 B (Q) into C [O. 1] in case PO (ml < + oo) > 0. u' = P. (ml < + oo) -1 u. [Consider the mapping u E G1 B (Q) Because K_ n [0, 1) = 0 , u (a+) = u (a) and bl jm Pb (ml < + oo) = blijra P. (mb < + oo) Pb (ml < + oo) = P. (m1 < + oo) (a S 1) , i. e., k+ (a)] u(a) and u' (a+) = to (a) (a < 1); as to u' (a-), u(a-) oo) Pb (ma < + oo) Pa (ml < + oo) k+ (a)] x bfm Pb (m1 < + = lim bla x Pa (m1 < + oo) (a z 0), i. e., u' (a-) = u' (a) (a > 0), so u --o. u' is a possible isomorphism.] °'v

...

VII

Ana

4.B

all

'ti

P o [x (t + s) E d b I B.] = P a [x (t) E db)

Itb/McKesa, Diffusion processes

7

3. The general 1-dimensional diffusion

98

3.7. Generators

Consider the GREEN operators G. (a > 0) acting on the space D as defined in 3.6. Given a, P > 0,

Ga-Gp+(a-fl)GaGp=0,

i) because

+ CO

Gp Ga I = E. [ f e4' (Gal) (x,) ds

2)

0 +00

+00

=E. f e-p' ds f e-' /(xs,,) 10

0

]

T

=E.[ +'e-' /(x,)dr f

e.-(P-a)ado

T=t+S. =s

o

t00

=E. I (a-ps-` a _ aT, 0

/ (xe) dT

=(oc-fl)-1(Gp-Ga)/, and it follows that the range D ((5S) - G,D C D and the null-space G;'(0) - D fl v: Ga/ = 0) are independent of a > 0. Because of the definition of the space D, if a E Q and if / E D, then P. [li m / (xj) = f (a) ] = 1, permitting us to deduce from / E G. 1(0)

that + CO

3)

0 me

t

Gp /

E. [pt n fi f e-p`l(xr) 0

dt]

= /.

G. : D - D ((a) is therefore invertible, and, defining 4)

(55 =1-GI1:D((3)-+D

and letting I - (53 act on both sides of 9) with a = 1, it is found that

Gp1=(fl-0)

5)

fl>0.

is the so-called generator o t D.

Given a MARitov time m, a > 0, and u E D (0), if f = (a - (55) u then, as in 3.6.8), 6) u = Ga / = E. [ e-ae / (xl) dt + E. [e-am (Ga fl (xm)]

f

=E. [ f e-01(a 0M

- (55) u(xg) dt + E.[e-'m

3.7. Generators

99

and, for E,(m) < -3-.oo and a j 0*, 6) goes over into E. B. DYNKIN'S formula [4] : r m

u E D (@), E.(m) < + oo.

E. [u (xm)] - u = E. I f (0 u) (x1) dt

7)

0

(53 is computed from 7) following E. B. DYNKIN [4, 5).

Given a trap a E Q, Pa [ma, > t] = e" (0 S x < + oo) (see problem 3.2.7), and, if x = 0, then P. [m = + oo] =1 and (( u) (a) = 0, x-1 < + oo, and using DYNKIN'S while, if x > 0, then 0 < Es formula,

((i u) (a) E.

8)

n1im E. [

,

f

((53 u) (xi) dt

lim P. [m. < n] u (a)

t

_ -u(a), u

(t u) (a)

(a)

E.(m_)

u E D (CS) .

o44

Given a E Q such that (0 u) (a) = 0 for each u E D (0), a (G./) (a) = / (a) for each / E D and a > 0, and taking / E C (Q) C D so as to have 0 S / (a) < / (b) (a + b), it is found that P. [x (t) - a, t z 0] = 1, i. e.,

a is a trap. Because of this, if a E Q is not a trap, then ((3u) (a) > 1 for some u E D (05), and using the fact that (5S u E D to select a neighborhood

B of a: b < a if a is a le/t shunt b > a i/ a is a right shunt c < a < b it a is non-singular

B = (b, a] = [a, b) = (c, b) such

that @u > 1 on B and introducing the exit time m =

min (1: x (t) ( B), DYNKIN'S formula applied to u and m /5 is shows that mAn 10)

E. (m) S nt+oo Jim Ea[ f (0 u)(x1)dt S211 u11 <+oo. Lo

.t.

..,

But then DYNKIN'S formula can be applied to each u E D (6) and

exit time in, provided B is small enough, and it follows that 11)

((53 u) (a)

_

E. Cu (xm)) - u (a) E. (m) Boa

u E D((53),

* / depends upon a but it does not. 7*

400

3. The genera( 1-dimensional diffusion

i. e., making cases, 12a) ((3 u) (a) = li m

u(b) P.[m, < mom] - u(a)

12b)

(t $ u) (a) = Jim bja

12 c)

(0 u(a) = lim 0a

if a i s a l eft sh unt

E.(m&Am 0)

bfa

u(b) P.[mb < m°°] - u(a)

if a is a right shunt

E.(mb Amm)

u (c) P. [m. < ma] + u(b) P. [m, < m.] - u(a)

ba

/11

i f a is non-singular.

(B is then a local operator, i.e., i f a E Q and if ul E D ((S)) coincides with as E D ((s) on a neighborhood B of a:

B=a

if a is a trap

= (b, a] = [a, b) = (c, b)

ba c
if a is a left shunt if a is a right shunt if a is non-singular,

-,.-.

then (($ u,) (a) = (($ u=) (a). Problem 1. Given u E D (CS) and a neighborhood B of a E Q as de-

scribed above, if u (b) S u (a) on B and if u (a) ? 0, then ((B a) (a) S 0. [Use 3.7.12).) W. FELLER [9] proved that if

"+5g

is local and if it is also negative (. in the sense of problem 1, then it has to be a differential operator of degree S 2 (see 4.1 for more information on this point). 3.8. Generators continued

Here is another method of computing Q. Consider

r' (E. [u (x.)] - u)

E>0

and define 2)

CAS' = limo 45.,

III

fin'

where & is to be applied to the class D (W) of functions u E D such u that (W u = Jim ($. u exists pointwise and belongs to D and sup 40 &>O < + oo; it is to be proved that 3)

i.e., 4a) 4b)

'_

D(W) = D((s)) E [u (x.)] - u W u =1im .;0

4b) is to be compared to 3.7.11); the use of ( W. FELLER [5].

u E D(O).

was suggested by

3.8. Generators continued

101

I-.

DYNKIN's formula 3.7.7) applied to u E D((1) and m = e > 0 implies that (SS' = CSS on D (0) C D ((SS') because

1 ±m 03, u = im e r E. I f (0 u) (xr) dt] = O u

5 a)

8

8

0

and 5 b)

110. U11 s 110 U11,

and, to complete the proof, it is enough to show that D (t?) C D (0). But, if u E D (O') , then 6) Ga(a - Oi)u

=aGau - limGa (0,u a G. u - lim e 'E. I f Cal [u (x:+.) - u (xe)) dt) 0+00

f

`40

= u,

41

.

u (xr) dt - f er u(x) dt

'o$

=aGau -lime' E. {(e1 - 1) +-

]

.,C

and, since (a - (33') u E D, u = G. (a - ($Y) u E D (0) as desired. (SS' = CS can be used to give a new proof of the fact, noted at the end of 3.7, that 03 is a local operator. Because the path cannot cross up over a left singular point or down over a right singular point, it suffices to show that

i o t-I P. (mb < t) = 0

7)

a < b,

as will now be done using the method of D. B. RAY [2].

Given a
8)

Pa (mb < t) = Pa [m + mb (wm) < t]

M. = m.

VII

SPa[m
= E-(Me < t, Pa[mb(w) < tI B,a+])

=P.(me
P [ma A m A mb > s] [slt] Pa(mb < t) Pc [ma A m. A mb > n t] Pa (mb < t)

2

S t)

VIII

nS(,n) (a. b)

Z PC [ mb E (n 1, (n + 1) t] *5feltl

S1

s>0.

3. The general 1-dimensional diffusion

102

But, since P, [ma A m A mb > 0] = 1, iii up t-1 Pa (mb <0 < +00, 840 and so [use 8)] him t-' P. (me < t) < + oo, completing the proof of 7) ; as a matter of fact, 8) implies P. (me < t) < constant x 0, t8, etc. 14 0), proving

lirnr"PQ(mb
10)

a
Problem 1. Give a new proof of 10), using DYNKIN'S formula to check

Es(e-"E") S [a Ee (me A m A mb)]-'

a
e

and then making use of Pa (mb S t) S e f e-'1ePa (mb E d s) S e E. (e'e-' "") 0

(see problem 4.7.4 for still another proof).

[Given a < b such that Pa (mb+ < + 00) > 0, if / E D is 0 to the left of b and positive elsewhere and if a > 0, then y = (G`f) (b) > 0,

0 < u - y-' G,/ = E.(e-"m') E t and solves (Ju = au on (a, b), and using DYNKIN'S formula with in = n A M. A M. A Mb, it is found

that, if a < c < b, then 1 z EQ[u (xm)] - u (c) = E, f ((t3u) (xe) dt)

au(a) E, (m) tau(a)

?

as n t +oo.] 3.9. Stopped diffusion

We will prove that i/ Q' is a subinterval of Q. closed in Q. and if e is the exit time inf (t : x (t) ( Q' ), then the stopped motion D' = [x (t /\ e) :

t Z 0, Pa : a E Q'] is itself a diffusion; this fact was used in 3.5. Consider the standard description of the stopped motion with sample paths w' : t - Q', universal BOREL algebra B', MARKOV times m' = m' (w*), subalgebras Bm. + , and probabilities 1) Pa(B')=PP(B) aEQ', Bi-- (w: w*, EB'), B'EB';

the problem is to show that, if B' E Bm.+, then 2)

P: [B', x (1 + m') E d b] = E: [B', P. (m.) (x (t) E d b)] .

But, if m' = m' (w') is a MARKOV time for the stopped motion (in standard description), then ignorning paths not starting from Q', m (w) - m' (w;) is a MARKOV time for the original motion because 3) (W: in < t) = (W: m' (w,*) < t) B'=(w':m'
=(w:w'AgEB') EBe Use 3.2.1) and eA t = e(w)At (see problem 3.2.6).

t?0;*

3.9. Stopped diffusion

103

III

in addition, if B' E Bm.,. , then B - (w : w4 E B') E B(mnr)+ as is clear from

B fl (w : m A e < t)

4)

=(w-, EB')n(m'(w)
((w,*)e E B') n(m < t) fl(m < e) *

+((w;)iEB')fl(e
I00

All

tZ0.

E Bt

Using

a=mA e+e(wmnr),

5)

x (t + m, w*,) = x ((t + m A e) A (e (wmAt) + m A e))

6)

= x (t A e (wmnr) , wmnr) ,

and 1), it is found that

P:[B';x(t+m')Edb] =P.[B,x(t + m,w,)Edbj

7)

= P.[B, x(t A e(wmnr), wmnr) E db] = E.[B, Pz(mA<)(x(tne) E db)] = E.[B, P. (m. (ro ), ;)(x (t) E db)]

= E: [B-,

E db)]

as desired.

Consider the generators 0 and 4W of D and D'. 0 is a contraction o l C55' in the sense that, if u E D (0), then 8a) and 8b)

the restriction u' of u to Q' belongs to D((i')

®' u' = (i u on int Q' and at those endpoints of Q' at which

P:[e>0]=1, as will now be proved. Given u E D (@), if f = (1 - G3) u, then

fED

9a) r

u = G, l = E.

9b)

fe-II(x,)dt + E. [e-e u (xe)]

,

and, if 10)

f' (a) = l (a) = (GI /) (a)

=0

a E Q' ,

Pa [e > 0) = I

a E Q%

Pa[e = 0] = I

a= oo,

* Use the fact that w E B' if and only if (w'); E B' for each I > W(e); see 3.2.10c).

3. The general 1-dimensional diffusion

104 then

t'ED',

11 a)

D' being the space D attached to D', and on Q', G' to = E: I T e-4 to (z) dtj

11 b)

= E. [Tie'/* [x (t /\ e)] d t l o

J

+m

=E. f e4/'(z1)di +E.I f eldt/'(xe)J l e

= E. f e-t t (zt) dt + E. [e-e (G1 /) (xe)] 0

=G11 = u.

8a) is clear from 11) ; as to 8b), 11 b) implies that on Q', 12)

CS'u'=(Gi-1)/'=G1/-/'=(b3u+/-/'=c5u P.[e>0]=1,

and 8b) follows. G is now seen to be local in the sense that, if A is the closure of a neighborhood B of a E Q:

it a is a left singular point ba c < a < b it a is non-singular, then 05 agrees on B with the generator (l' of the stopped motion PE: EA] e=inf(t:x(t)(A); this usage of local is to be compared with the usage in 3.7 and 3.8. Problem 1. What step of the proof that D' is a diffusion fails if e .ti

,,d

B = (b, a) = [a, b) = (c, b)

is a general MARKOV time? Give a concrete example.

[5) is false; indeed, if D is the standard Bxowlaian motion and t

..'

if f E C (R') is positive, then e = min (9: f / (x,) d s = 1) is a MARKOV 0

time, and if 5) held and m = I were < e, we should have e

t

e(wt)

t

1 = f t(x.) ds = f t(x.) ds + f ;(x (s. wi))ds = f /(x.)ds + 1, 0

which is absurd.]

0

0

0

105

4.1. A general view +.+

Problem 2. Given e = inf (1: x (t) ( Q'), show that, if P,(e < + oo) > 0 on Q', then the stopped conditional motion D' = (x (t A e) :I L 0, Pa(B I e <+ oo) : a E Q'] is a diffusion (see 4.3 for an application). [Given m', in, B', B as above, if P: (B') = P.(B I e < + oo) on Q' and if db C Q', then, as in 7),

P: [B*, x(t + m') E db]P.(e < + oo) = P.[B, x(tAa(w++Ae), WrAe) E db, mAe + e(wMAe) < +00]

=E.[B,Pz(mne)(x(t/\e)Edb,e <+00),m/1e< +oo] = E.[B, P%(mAc) (x (t) E d b) Pz(mAe) (e < + oo)]

= E. [ B, m A e + e (wmAC) < + 00, P; (-A0 (x (t) E d b)] = E.[B, e < + 00, P=(mAe) (x (t) E db)]

= E: [B', P;(m.) (x(t) E db)]P.(e < +c ), completing the solution.]

4. Generators 4.1. A general view

A particle starts at time t = 0 at -1 S 1 < 0, moving at speed + i until it hits l = 0; at that moment, it begins a reflecting Bnowxian motion on [0, + oo), stopping at the passage time ml to 1 = 1, wait-

ing at that place for an exponential holding time e with mean and jumping at time ml + e to the point oo. E. B. DYNKIN'S formula 3.7.11) can be used to check that (3ju coincides with 1)

-1 S1<0

((3Yu)(1)=u+(1)

i

U (O)

u (E) 62

1= 0

0<1<1

=}u'(l)

.r.

_ -3u(1)

1=1

on its domain D ((13) = D ((Y) comprising all functions u E C [-1, 1] fl

Cl[-1, 0) fl C2[0, 1) such that 2a)

u (0) = i u"(0+)

2b)

u+ (0) = 0*

2c)

iu"(1-) = -3u(1),

* u+(0) = 0 is automatic from the second line of I because e-l[u(e) - u(0)] is bounded, as e . 0.

4. Generators

1 06

especially, u -+ ('3u) (1) is a differential operator of degree 2,1, or 0 according as I is a non-singular point, a shunt, or a trap. Given a reflecting BxowNian motion x+ as described in 2.1 with passage time m1 = min (1: x + = 1) and an independent exponential holding time e with law P.(e > 1) = e-39, then

0 St<-1

3)

-lSt<-1+nt1

=x+(t+1) =1

-1+m15t<-1+m1+e t? -1+m1+e

= 00

;01

a.°

4n.

is a (non-standard) description of the motion starting at -1 S 1 < 0. 0, D (0), and the associated sample paths admit a similar description in the general case as will be explained below. W. FELLER discovered an expression for G$ as a differential operator i in the case that the GREEN operators map C (Q) into itself and discussed it in ,,,

a series of basic papers [5, 7, 9, 10, 11). E. B. DYNKIN [4, 5, 8] found an elegant probabilistic method based on 3.7.12) of deriving FELLER's

0 in the conservative case, at the same time giving it a more probabilistic expression. The expression for OP on the shunts, the properties

of killing measures, and the concrete expression of the sample path in terms of W and a standard BROWNian motion, as described in 5.1, etc., are published here for the first time. Begin with the non-singular case : 4 a)

Q=[0,1]

4b)

Pa[ma_=ma+=0]=1

5 a)

Ea (e-A'.+) = lim Eo+ (e-"4) , Eo+

5 b)

E1 (e_ml-) = lb m

b4O

0
(e-'n+), E,_ (e-tmi) = m Eb (e-m'-). E1_

btl

4b) states that (0, 1) contains no singular points; see 4.5 for a dis.L'

cussion of 5). Given 0 < a < b < 1, the hitting probabilities and the mean exit time 6a)

Pab(E) = PE[ma < mb]

6b)

Pba()=PE[mb<ma]

6c)

a<
eab (s) = EE (ma A Mb A In

can be used to introduce a (continuous, increasing) scale s = sab, a (non-negative) killing measure k = kab, and a (positive) speed mea-

4.1. A general view

107

sure m = mab according to the formulas 7a)

S ( d ) =Pat, ( ) Pb, (d ) - Pba (4) Pab (d

7b)

k(d ) =

P+'(dE)

-

)>0

z0

m(dst) _ -[eob(dE) - eab(sr) kab(dE)] > 0

7c)

a < < b,s

in terms of which 0 can be expressed as a differential operator

a
(0.u)()=u+(dt)

8a)

m(d)

the precise meaning of 8a) being

V

'!s

(d)-u+ (c) - f u(f)k(di) a
8b) f ((l

[e, d)

[e, d)

Given 0 < a' < a < b < b' < 1, the new to the old according to the rules 9a)

are related

Sa b (d) = B (aa b ) S. b (d)

a < < b, v

(d) = B (a. b.)-1 kab (d)

9b)

)_1

9c)

ma. b. (d

) = B (a b

mab(d4)

where a b. /I

_

Pb a (b) Pa b (b)

0.'

B( a- b-)

pb a (a)

> 0,

Pa b (a) +1r

and, using this rule, it is a simple matter to introduce universal scale, killing measure, and speed measure such that 8) holds for all 0
k (0) = lim p0(m' - +o) < -F 00 s(b) - s(0) boo

10a)

lob)

m(0) =1im s(b) - s(0) bt0

E.(m'nm.)

<

- -00 1

112

f k [0, ) s (d) < + oo

11 a)

0 1/2

f m Co, E) s (d) < + oo,

+

11 b)

0

and

12 a)

' e+(

(0 u) (0) m (0) = u+ (0) - u (0) k (0)

_ m [s

u E D (05),

)]-1 [e (ri) - e (E)J (e = P.a P... e.&) and e+ (d.) is the

BOREL measure induced by the interval function a+(E. t1= a+(rt) - e+(f)

(f < ii).

4. Generators

1 08

where - u (o) u' (0+) = u+ (0) = lim us (b) (b) - s (0) a+0 with the understanding that k (0) = m (0) = u+ (0) - 0 in case s (0) - s (0+) = - oo; similar results hold at I in case El (e-m=-) = 1. D (0) can now be identified as the class D ((W) of functions is E D 12b)

.ti

111

with

u (0+) = u (0) is (i -) = is (1)

13 a)

13 b)

if

EO.,

if

Et_ (emi) = 1

1

such that, for some u' E D, 14)

u'

0 < < 1,

k(d)

m (d E) = u+ (de) - u

15a) u' (0) m (O) = u+ (0) - u (0) k (0)

if

15b) u'(1) m(1) = -u (1) - u(1) k(1)

if El(e ml-) = 1,

16 a) u' (0) = - x (0) u (0)

Eo (emei) = 0,

if

16b) u' (1) = -x(1) u(1) tf El(e mi-) = 0,

E. (e- m--) = 1

x (O) = Ee (m )'t

x(1) =

and 05 can be identified as the global differential operator W mapping is E D ((3') --o. u' in agreement with 9). (3' is unambiguous because

WED and 0<m[a,b) (0
0 at 0 depends in part upon s, in, and k near 0 as epitomised in the table below; in this table, 0 is said to be

i

} an exit point if + oo > f k (f , 4] s (d s) + J m

I k (0

,

] s (d ) + Jim (o, t] s (d ¢) ,

f4'

an entrance point if + oo >

i] s (d )

0

0

0

and (tJu) (0) = in (0) -t [u+ (0) - is (0) k (0)] stands for u+ (0) = is (0) k (0) in case m (O) = 0 this classification is due to W. FELLER [4]. exit and entrance

Pe[Tne+ =0)=

0

Eo+

(05 u) (0) =

or

1

1

u+ (0) - u (0) k (0)

exit not entrance

entrance not exit 1

0

I

0

I

0

0

-xu(0)

-xu(0)

M (0)

N P,[mo+=0]=1 - x u (0)

ii

P,[me+ = 0] = 0

neither exit

nor entrance

u+ (0) = 0

4.1. A general view

109

(, at I depends upon s, m, and k near 1 in the same manner. Consider, as a second example, the singular case: 17a)

Q=[0,1]

17b) 17c)

Po [m, < +001 > 0 P. (mi- A m- = + 001 = 1

17d)

Eo(m,Ant) < +°°: '
the content of 17b) is that [0, 1) contains no left singular point. Q breaks up into the shunts 18) K. fl[0,1) = [0, 1) fl (a: P. (m.- = +ooJ = 1), the non-singular intervals QA = [In, rK)

19)

nZ1,

E,a(e ma.+) = 1

and

a single isolated trap at 1, and on U [l,a, re) , it is possible to express (J as 20)

r..

nal

( u) () m (d E) = u+ (d) - u (E) k (d E)

21 b)

( u) (la) m (In) = u+ (IA) - u (I+,) k (In) ,

1. < < r,a A

21 a)

where, for In < a < b < rn and some positive constant B depending upon a and b, r"9

B[pab( ) Pba(dE) - Pba( ) /ab(d )J > 0

22 a)

k(dl) n(d) =

22b) 22c)

p;.(dE) p., (E)

z0

A. (E)

- [ea b (d E) - ea b (E) k (d )J > 0*

as in the non-singular case described above. At the left end, 23 a)

u+

u+

k(In) = lim

23b)

u (b) - u (la) =1im b+t. s(b) -s (I.) P,S(b)

-s(l)

,

m A

23c)

'n (l")

h ml,. Ea (b)

S

(l,.) )

with the understanding that u+ (ln) = k (l,a) = m (la) .= 0 in case

s (l.) = s (l,a-(-) _ - oo, and the additional condition X f k [la, ) s (d ) + f m [In, E) s (d i) < + 00

24)

nZl Q.

QpI

holds. p:,, p;,, etc. are computed using the universal scale s of 22a). not the local scale s.& (de) = pr (E) p,. (dd) - p,. (E)

4. Generators

110

Besides this, the hitting probabilities and mean exit time 25 a)

Pb(a) = P.(mb < +oo)

25b)

eb(a) = Ea(mb /\ mao)

aSb

can be used to introduce a shunt killing measure k+ and a shunt scale s+ :

0 5 k+ (d E) = p

26a)

k+(0) = 0, s+ (d

26b)

(E < b)

= pl (d E)

(E < 1)

0 S k+(1) = P1-[m1 = +oo]

E) _ - [eb (d) - eb (E) k+ (d t)]

(E

[e1(d ) - el (E) k+ (d$)]

< b)

(E < 1)

such that b ill

s, (b) - f s+ (d E) is continuous 111

27 a)

0

(a < b)

27b)

s+ (a) < s+ (b)

27c)

s+(1) + k+[0, 1) < +00

28a)

f

(a.b)na+

(0) u) (E) s+(d E) =

f

[u(dE) - u(E) k+(d E)]

(a,b)nK+

a
and, for l < E < re, s (d E) k+ (d E) = p

29 a)

(,)

29b)

(' P (n) k (drl) ll.-E)

f P(n) m(dry),

s+ (d E) =

[ix. 4)

P being the solution of P+(d E) = P(E) k(dE)

30a)

30b)

P+

l<E
p (l.) k (la)

=1.

30c)

28a) can be put in the alternative form 28b)

(6 (3u)u ()a = lim e

n

u(b)-u(a)-f[

aEK+n [0, 1),

ns+I1


[1 - k+ (dE)] being meant as a suggestive notation for e ,-j+ (a b) x

a
4.2. ( as local differential operator: conservative non-singular case

111

..r

D (0) can now be described as the class D ((.Y) of functions u E D such that, for some u' E D, 31 a)

f u' ()m(d)=

31 b)

32)

J

[a. b)

[a, b)

u' (ln) m (1) = u+ (ln) - u (1) k (1) ,

u' () s+(d) =

J

J

[u(d) - u() k+ (d

(a, b) n H+

(a, b) n K+

)]

05 a
U, (1) = 0,

33)

and i can be identified as the global differential operator 0' mapping u E D ((SS') --> u'. 0' is unambiguous because u* E D, 0 < m [a, b) (l S a < b < rr,) , and s+ (a) < s+ (b) (a < b) . A similar description of (55 holds in the general case; the expressions

r"'

become a bit complicated but nothing new happens and the proofs can be carried through using the special case explained above as a model and appealing to the splitting of 3.5 and the stopped motions of 3.9.

The reader may wish to skip the singular case at a first reading ; this means that he should read 4.2-4.7 and 5.1-5.9 and then come back and read 4.8-4.10, 5.10, and 5.11. 4.2. 0 as local differential operator: conservative non-singular case Consider the non-singular conservative case:

OSa51,

Pa(m,o=-{-00)=1 Pa(ma+=0)=1 Pa(ma_=0)=1

1)

2a) 2b)

0
0
suppose, in addition, that 3a)

Pa(mo<+oo)>0

3b)

Pa(m1<+00)>0

4a)

PO (mo+ = -}- oo) = 1

4b)

Pl(ml-=+oo)=1,

5)

Ea(moA m,)<+oo

a<1 a>0,

05a51,

Vas

and let us construct a scale (i. e. , a continuous function with s (a) < s (b) (a < b)) and a speed measure m (i.e., a non-negative BOREL measure

with m [a, b) > 0 (a < b)) such that 6) (S u) () = U_ (d_) -_ m(d6)

m(de)

u E D(ffl,

* 32) contains the matching condition u (b -) = [1 - h (b)] u (b)

0 < E < 1, (0 S b < 1).

4. Generators

112

where u- and u+ are the one-sided scale derivatives of u: u- (a) = bta lim s(a) u (a) -- s(b) u (b)

,

u (b) - u (a)

u+ (a) = lim b4a s(b) - s(a)

and 6) is short for

u- (b) - u- (a) = f 0 u d m

7a)

[a. b)

u+(b) - u+(a) = f (SS u dm

7b)

0
(a. bl

Define

s()=Pe(mi<+oo)=PC (ml<ma)

8)

let s (0) = 0, s (1) = 1, and let el be the indicator of the point 1; then CIO 9) u = E.(e-` m,) = e G. ei t s

and making e 10 in

ex [G.el-G.el+(a-e)G.G.ej = 0

10)

proves

aG.s=s,

11)

or, what interests us, s E D(O) MS = 0.

12a) 12b)

Because s (a) S s (b) (a < b), if s (a) = s (b) for some a < b, then either s (b) = 0 contradicting 3 b), or s (a)/s (b) = P. (mb < + oo) = 1, and 13) 0 < E. (e-m.) = E. (e-me, mb < ma) = E. (e-me) Eb (e-m.) < E. (e---e) ,

which is absurd; in brief, s is a scale.

Because of 5), if a < < b, then 14)

1 = Pi (ma A Mb < + oo) = Pt (ma < mb) + PC (Mb < ma) and, solving 14) and s () = Pe (m < + oo, mi (im) < + oo)

in = M. A Mb

r-.

15)

= Pe (ma < mb) s (a) + P1 (Mb < ma) s (b)

for PE(ma < mb) and P1(mb < ma) in terms of the scale, Pe (ma < mb) = s(b) - s(6) s(b) - s(a)

.ten

16)

a
P1(mb < ma) = Sss(b) -- s(a) s(a)

4.2. Ib as local differential operator: conservative non-singular case

113

Defining eab to be the mean exit time 17)

a < j < b,

) = Et (ma A Mb)

Cab (

implies that, if a < $ < b and if m = M. A mb, then

16)

18)

e(

EE [m + mo (wm) A mI (em)]

) = eo I

s

= eab (b) +

s (b)

s

- s (a) a (a) +

s (b)

- s (a) a (b)

i.e., e is a concave function of the scale. e is also a member of D (0); for, if f is the indicator of (0, 1), then rm. Am,

G,f =E.I

j o

e

j 0 in G.+(a-e)G.Gj f=0,

CG

19)

it is found that e = G . [ f + a e] , or what interests us, 20a)

eE D((5S)

-(.3e=f=1 (0<E<1)

20b)

=0

especially, e is continuous on [0, 1] because of 2) and 3) (see 3.6). Now, in general, if e is concave and if e(0) = e(0 +) = e(1) = e(1 -) = 0, then I

0
e(a)=- f G (a, b) ei (db)

21)

0

where G is the Green function G(a, b) = G(b, a) =

22)

[s (a) - s (o)] [s (1) - s (b)]

aSb

s(1)-s(0)

and e- (d b) = e+ (d b) is the non-positive BOREL measure induced by the interval functions e- [a , b) = e- ( b)

23 b)

e + (a , b ]

-

e - (a)

b - e+ = e' O

a

e-- (a)

e

a

e(a) - s (b)

= lim bta

s(a) - s(b)

=

e(b) - e(a)

0tl

2 3 a)

-

'

b 4a s(b)

- s(a)

10,

Here is the proof. Because e is concave, 24 a)

24 b)

a (b) - e (a) t s(b) - s(a)

e (a)

-

e (b)

e+ (a) (b j a) ,

e+ (a) < -(- o o (a > 0) ,

e- E j

j e - ( a) ( b t a) ,

e- (a) > - oo ( a < 1) ,

e- E J ,

It6JMcKean, Diffusion processes

s

4. Generators

114 and

a
e` (a-) = e+(a-) = e- (a) e-(a+) = e+(a+) = e+ (a)

25 a) 'N+

25b)

a > 0,

and it follows that the interval functions 23) induce the same nonpositive BOREL measure e - (d b) = e+ (d b) . Because of e3 E 1 and

e(0) = e(0+) = e(1) = e(1-) = 0, 26 a)

a (b) - e (0) ? et (b) [s (b) - s (0)] ? et () [s (b) - s (0)]

26b)

e(1) - e(b) S ei (b) [s(1) - s(b)] 5 of (}) [s(1) - s(b)]

bS b?

and using this, it develops that 1

27)

0
f G(a, b) e-(db) 0

s (t) - s (a)

(' [s(b) - s (0)] of (d b) +

J $M-S(O) 0
+ s (a) - s (O)

s(t) - s(o)

f J a
[s (1)

- s (b)] et (db)

`- s(1) - s(a) [[s(a) - s(0)] e+ (a) - e:1 (b) s(db) + - s(t) - s(o)

f

+ s(a) - s(o) I s(1) - s(0)

as stated in 21).

1

-e(a)

-[s(4) - s(a)) a+(a) + fe± (b) s(db)

L

a

J 1

21) is now applied toe = E .(m0 A m1) , with the result that e = f G d m,

where m is the speed measure 0 28) m (d b) _ -et (d b) ? 0. Because the eab in 18) is positive, m[a, b) > 0 (a < b). 6) can now be proved in a few lines. Given u ED (95), DYNKIN'S formula 3.7.7) implies S (F) - s(0) s(1) - s( ) 29) s(1) - s(o) u(1) + s(1) - s(0) u(0) - u( ) m = m0 A m1 = EF [u (xm)] - u (s) 1

=Efl f (0a)(xt)dtI

0<
0

and to establish 6) it is enough to prove 1

30)

E.I

dt =f G, dm

/E

4.2. Q$ as local differential operator: conservative non-singular case

415

and to differentiate the resulting formula 31)

s(l)

- s(o) u(1) + S(1) _ S(oj u(o) - u( ) 1

= f G( ,,1) (0 u) (r!) m(dr7) 0

Given B C [0, 1] with indicator /, uB = E. [

because

m,nmb

uB () = EE f / (x:) d t +

32)

fl (xi) d t] is concave

0

uB (a) + s (E) - s(a) uB(b)

s (b) - s

s(b) - s(a)

s(b) - s(a)

0

UB(0+) = uB(1-) = 0 because 0 S uB S e01, and, choosing B = (b, 4] (0 < b < 1) , an application of 21) coupled with 0) - s(E) u(b) u() = eb1() + s(7) - s(b)

u= u'(b,l], 801

b:5' e,

i-+

33)

implies

u(b,l] Z J Gdm

34a)

(b,1]

u[b,l] = rlatb u[a,l] i>= f Gdm.

34b)

[b, 11

But now 34c)

u[0,b) Z

fGdm CO. b)

for the same reasons, and adding 34b) and 34c), it appears that

e=u[o.b)+u[b,1)? f Gdm+ f Gdm= fGdm=e, [b,1]

[0, b)

0

i. e.,

u(b,1]= f Gdm

35)

0SbS1,

(b, 1]

leading at once to 30). Problem 1. Consider the splitting of the standard BROwxian motion

II'-

into independent . = (t: x (t) = 0), excursions en : n Z 1, and signs en : n z 1 (see 2-9); the problem is to compute the scale s (di) and the speed measure m (d i) for the skew BRowNian motion made up from

standard BRowxian 8 and en : n ? 4 and skew signs e, = ± 1, g«

4. Generators

116

P.[e,= + 1] = a, n z 1, 0 < a < 1. (a = 4 for the standard BRowxian motion .)

-a)-1 =a-1

-a) de

m(de) = 2(1

m(de) = 2a de

for for

S 0, z 0;

in fact, if 8.: n z I is a (BOREL) numbering of the open intervals of the complement of 8 and if m*1, m+1 are the skew BROwNian pas-

sage times to ± 1, then s(+1) -s(O) = Po[m*1 < m;r] =

s(+1)-s(-1) -

na1

Po[e = -1, m-1Am+1E 8n]

_ (1 -a) I Po[m-1Am+1E8.] = 1 -a, Sat

s ) - $(s (o) = 1

a and for the rest, it is sufficient to point out that the skew and standard BROwNian motions

or, what is the same, tun

agree up to the passage time to 0.] Problem 2. Consider a particle moving among the integers Z1 accordi.+

ing to the rule : wait at x (0) =I E Z1 for an exponential holding time e1 > 0, at time e1- make a unit step (± 1) with probabilities depending upon 1:

0
exponential holding time e2 > 0, etc. W. FELLER [12) pointed out that the generator (J of such a birth and death process can be expressed in terms of a distance (scale) and positive weights (speed measure) :

°.'

(Su=

u+ - u-

m

where u - (1) =

u (1) - u (1- 1) s (1) - s (1- 1)

aye

-f-

1+1)-u(1) u'(1) _ Us(1+1)-s(l)

a<1
A

(b) -- s (1) Pt [ M. < mb] = sS(b) - s (a)

0 < m (1) -=

s(1+ 1)-s(1- 1)

Is (1) - s (1 - 1)] [s (1 + 1) - T(1-)] E1(el) .

Give a complete proof using the method presented above. !ML

4.3. (3 as local differential operator: general non-singular case Now consider the general non-singular diffusion D with state intera.+

cow

val = [0, 1] and let us prove the existence of a scale s, a non-negative killing measure k and a positive speed measure m defined on (0, 1)

4.3. 05 as local differential operator: general non-singular case

117

such that, for each it E D (@) ,

m (d ) = u- (d ) - u

(O u)

k (d ) = u* (d ) - u

k (d )

0 < a < b < 1, it is clear that, up to the obvious change of scale, the conditional diffusion (see problem 3.9.2) 2)

D*=[W,B,PE(B)=PC(w,;,EBI m<+oo),aS Sb)

m = man mb satisfies all the conditions of the preceding section, and we conclude that, for the scale

s* () = Pf (Mb < + 00)

3)

the speed measure m * (d ) = -e* * (d t) *

4)

e* (E) = Ee (mb)

and the appropriate Green function G*, m

5)

E* [

ff

b (x:)

dt]

=

J G*

t)) f (77) m* (dr1)

a S S b,

f E C (a, b)

Define f* = //P.(m < +oo) on [a, b]. Given u E D (05), it is clear from DYNKIN'S formula that, for

aS$Sb, 6)

[EE [u* (xm)]

- u* (E)] Pf (nl < + oo)

= EE [u (xm) , m < + oo] - u (s) mAnOO

=Ef f

((

u)(xr)dt

0 1.00

+

_ dtEf[((iu)*(x1)P?,(m<+oo),t<mAn'.) 0

+'o

+00

= f dtEE [((iu)*(xe),t < m]PE(m <+oo) 0

m

r

1

= EE I f ((i u)* (xi) dtJ Pt(m < +oo), 0

and recalling 5), it results from 6) that 7)

-u* () +

s* (b)

- s* (a) u* (a) + so (6

6

=

f G*

17) (() u)* m* (d77)

s* (a) u* (b)

a S S b,

b

* - and + applied to functions e* (with a star) indicate left and right differentiation with respect to s*.

4. Generators

118

or, what interests us, that (( u)* (t)

8)

a< f < b,

u*+(dt)

u E D(('$).

Consider the new scale

s(dt) =p(E)'s*(dt) p(s) = PE(m < +oo), a

9)

e S b.

Given at > 0, Ga I E D (0), ((bSGe 1) * = (aG,, 1-1)* S 0, and using 7), it results that (Ga 1) * is a concave function of s* ; thus, l lim a (G. 1) at+oo

is a concave function of s*, and computing -1 + in terms of s, we find that

-1+(ir)=p+(t)E T

10)

10) permits the introduction of the non-negative killing measure

k(dt) =p(t)-l p+(dt) =p(t)-'p-(dt)

11)

and defining also the positive speed measure 12)

m (d ) = p (t)-' m* (d t), it results from 8) that, for u E D (tl ), (05 u) (t) m (d t) = p (s) -1 (45 u)* (>t) m* (d t) = p (t) -1 u* + (d

13)

p-1)+) (d f) =p(f)-'(u+p - up+) (d t) = u+ (d e) - u p -1 p+ (d e) = u+ (d 4) - u k (d

)

a
as in 1).

13) is a local formula; the global formula 1) will now be proved. Given some new s', k, m' such that 13) holds, and defining 14a)

g1

14b)

EE (e-a me)

EE(e-amo), 1".

if u($) = Ga [(a - E) V 0] , then u (a) >0, gs = u (a) -1 u (a < < b) ,

and using 13) with s', k', m', one finds 1 5 a)

a gs

m'

gz+

ga

a < t < b **

k*

and likewise 15b)

a g1

m' (d ) = g; + (d ) - gl ($) k ' (d )

Because g1 E f, g2 E 1, and 0 = gi+ (d ) g2 g1 (E) g2+ (d'), the Wxoxsxian gi+gs - glg2+ is constant (> 0). s' (di) is therefore deter+4n

mined up to a constant factor (> 0) and, fixing s' (di), 15) deter* - and + applied to functions e (without the') indicate left and right differentiation with respect to s. ** + applied to function gindicates right differ. nt,ation with respect to so.

4.4. A second proof

119

mines k' (di) and m' (di) : 16)

s' (d ) =Is (d E) ,

m' (d E) = l-1 m (d )

k' (d l:) = l-1 k (d ) ,

1= s Qa, ) > 0. Now each point 0 < < 1 belongs to some interval (a, b) on which are defined

s, k, and m such that

13)

holds. Given two such

...

intervals the corresponding s, k, m match on the overlap as in 16), and t'.

using this, it is a simple matter to piece together these local scales, speed measures, and killing measures so as to obtain global ones for which 1) holds. 4.4. A second proof

nab

how

4.3.1) can be proved using an elaboration of the method of 4.2; the conditional diffusion of 4.3 is thus avoided.

Given 0 < a < b < 1,

t..

Pba()=Pe[mb<ma] Pa..

) < Pba(7l)

<17

pb6EC[a.b] Pba(a) = 0,

Pba(b) = 1.

2 c) is automatic ; as to 2 a), if pb a () = 0 for some a < rl < b, then 0 < F.,, [e-m,] = En [e-m., ma < mb] < Ea

which is impossible, and since p, a () < I ( < rl) for similar reasons,

f
=p,,a(')Pba(17)
'0.

as desired. Because p7 a (f) is continuous in q (rl >) , pb a (rj) is conti-

nuous (, > a), and the estimate p,,a (a+) S P7 [mb < s] 0

(e f 0, a < sj < b)

completes the proof of 2b). Changing the roles of a and b,

aSf5b VII

pab(f)=Pe[ma<mb]

3)

satisfies the analogous 4a)

4b) 4c)

Pab(s) > Pab('l) pab E C[a, b] Pab(a) = 1, Pab(b) = 0.

< q

4. Generators

120

convex Junction of t = pba; for, if a < a' < E < b' < b, Pb- a (') pab (b*) pab (') = Pa b ('9 pab pba (b ) pba (E) = pa b (E) Pb, (a) +

and, solving 5 b) and VII

P (9 = Pa* be (') + Pb- a (E)

6)

in terms of p and t = pba for t(E)-P(E)t(ai = t(E)-t(a*) 7a) Pb- a () = t(b') t(a') + p1(bt) - i(a )E) 7b) P.- b- (E) = t(bb) t(a) and substituting them back into 5 a), one finds ,.,

-

8)

pab(E) =

t(E) - t(a')

t(b') - t(E)

t(a') pab(a)

I

t(b -

b-)p(t(E)a.)

(1

t(b') - t(a-) [1

(bp) -]pea) )

Pb a(b )

X t(E) -

t(E)

pab (a) +

t(b') _ t(a')

pab(b )

as desired.

But then - p.&(E) 0 ? p' = Jim p.07) t(i)-1(E) 7141

9)


satisfies

p'Et

10a)

limp'(e),

10b)

and, defining the scale e

11)

s(') = sab(E)

it is found that

= f [pab dpba - pba dpab] = f [Alb a

a

t+ =pea() = oliE

'.1

12)

e

13)

pab(E) = , ,

t((n) s 17)

--t (E)

p. P.07)

s (E)

-t

= [pab - t p*]-' E t

S(t) - s(E)

=ptEf

and (see 11)]

1. pbapab - pbapab = t+[pab permitting the introduction of the killing measure; 14)

15)

0Skab(dt)=p+(de)

PM

p=pab,pba

di,

4.4. A second proof

121

Given a < a' < be < b, the same method applied to and pb.1 leads to a scale sa.b. and a killing measure ka.b. for as < ; < b', and solving the identities 5) for P.iO

ja b () _

16a)

n

o (6 b'/

p0.(b')

p.e(a')

B

pb.(a*) pb.(b')

p.,(0)

16b)

pb a

p..(a*)

B

I

Ip.,(a') p.a(b') I

a

be)

1a b'1 a

b

_- (a

++r

0.4.0.

(

a b1

B

p..(a*) po.(b

)

be))

(a.

ab

a simple calculation leads to the rules (d ) = B-1 Sab

17a) -pt

17b)

b

B=B(a

a

a'

ka be (d ) = B kab(d )

be

Coming to the speed measure, introduce the mean exit time

aSesb VII

e(e)=eab()=EE[meAmbA m-]

18)

and choose (a, b) so small that (%u > 0 (a S e S b) for some u E D (0) (such intervals (a, b) cover (0, 1)).

Given a < e < b, 19)

EE[u(x(ma A mb A m.))]

- u() = Ee

m1 meA moo QO u dt

f

0

Zme 0uXe(s), showing that e is bounded and that

e(a+) = e(b-) = 0;

20)

in addition, 21)

)a .r+

and, using 19) with (a', b') in place of (a, b) to check that sup ab is small for small (a', b'), it follows from 21) that eal,E C[a, b]. Now substitute 16a) and 16b) into 21), finding 22) B i a'

)C(E)=B(aa

a (a') +

+B

a

a'

)e(b.).

4. Generators

122

i.e., collecting the determinants, 23)

:

0ZB

a

e(b.)

e(a.)

b)e0.0.() = pba(be) Pba(:) pba(a*) pab(b') pab(E) pab(a')

a

21) states that a

t
a

lies above the linear interpolation

(a' < t < b'),

I () = Pa* be (E) a (a') + Pb- a (t) a (b')

and, thinking of I as a straight line because of i3.

1t (dE) - kab(dE) I(s) = 0,*

it is natural to describe the situation of 21, 22, 23) with the phrase; e is concave relative to the linear form

u*(dt) - kab(dF,) u(),** and one expects b

r

24)

a < e < b,

e(E) = tab(E) =,1 Gab(E,11)mab(d,?) a

where mab (d e) is the (positive) measure - [e* (d ) - k (di) e (i)] and Gab is the GREEN function 25)

S rl

Gabs. 17) = Gab(i, S) = Pba(S) pab(?1)

Consider, for the proof of 24), a subdivision d of [a, b] and, between each pair of successive division points a S a' < be b, define the interpolation 26)

) = P. be (E) a (a') +

ee (

+Pb..()

e(b')

a'

E S be

.v.

'ht

CAD

as in the diagram. e4E C(a,b], ea Se, e4 teas the modulus of d I. 0, and m4 (d i)

(a < < f < b) is concentrated at the - [e a (d 4) - kab (d E) ee (t)]

Dia`ramm t

division points of A.

is non-negative; for, if a < < b is a division point of d and a< a' < < be < b, then [use ea S e, ea(e) = e((), and 23)] ''AA

if

e(b')

ee(b') ee(e) ee(a') 27)

fib. (b') Pb a ('E) Pb a(a' )

Pab(b') Pab(s) Pab(a')

Iv`

ma

S

e(a')

Pb. (b') pba (') pba(a*) pab (b') Pb a( ) pab (a1

(.) =1im 101) - 1M l' [ ) = lim 1M - 100 +tte Sab(f) - SabW n+ S.07) - Sab(s) ** See M. HEINS (1) for a special case of this idea. 1

< 0,

4.4. A second proof

123

and transforming the determinant on the left, it appears that e,6 (a,) ed(b') s(E) -

s(b') 0;5 lim ..It

28)

A. R)

s(b') -

P.s(b')

s(s)E)

-

ea ()

s(.) - s(a )

pab() Ps ($) -

s

++^

b

Po.() - Pe.(a')

Pe.(b') - Pe.( )

ed

- es () - kab

a*)')

ed (e)

ed

pba (

pba (E) pba (E) = pba (S) -pba() -kab (S)pba (s)

ee (fi)tt

ed (E)

ee ( )

'(Z 'A,

pba () pba () Pab( ) -pab(E)-kab(s)Pab(S) pab() pab($)

t) pab( ) Pab(') pab( )

-md (E) ed (f) ee (f) 0

fib. (f)pba() = and() X[pbaPab - pbapab] = and( )

0

pab ( ) pab ( )

i.e., and Z 0 at each division point. ed = f Gab dmd because e' = ed - f Gab dmd is a solution of

a
29a)

e' (a+) = e' (b-) = 0,

29b)

as a simple computation proves, and, as such, it has to be = 0; for example, if 0 < d = max e' and if a < n < b is the least root of e' = d, ti.

(a, b]

then [use 29 a)] 0 Z e'- on a left neighbourhood of 71, contradicting

d > e' e Z ed = f Gab dmd now shows that, if a < c < b, then the total mass nd [a, b] of

a<Zc

nd (d f) = pba (e) and (d f)

30)

= pab ( ) and (d )

is bounded, and selecting subdivisions d with moduli 10 such that WA

converges to a non-negative BoIEL measure it on [a, b], one finds b

b

31)

ed = f Gab dmd to = n(a) pab() + n(b) pba() + f Gabdmab, a

a

where 32)

inab(dS)

a <
n (d i)

b>E?c

pab and

33)

n(a) = e(a+) = n(b) = e(b-) = 0,

completing the proof of 24).

4t!

fibs( )n(d)

4. Generators

124

mab is the speed measure. 21) and 17) are now used to show that mab

transforms according to the rule

(d) = B (a. b ) mab(d )

34)

and the proof of 4.3.1) is completed as in 4.2. Problem 1. Prove that, for a < b, pa a () -

(b)

s

s

- s (a) . Pb. (6) - ss(b) - s (a) s where = holds if and only if k (a, b) = 0. [p (e) = pab (e) satisfies p + (d ) = p k (d E) , p (a) =1, p (b) = 0, (b)

and therefore

p() S

s(b) - s(E) s (b) - s (a)

s(b) - s(E) - s(a) p(a) + s(E) s (b) - s (a) p(b) = s (b) - s (a)

- j( )

p cannot touch l inside (a, b) unless it is a linear function of s, i. e. unless p+ (d E) = p k (de) = 0.] Problem 2. Prove that P. (mb S t) is a convex function of s = s (a)

for a S b and for a ? b. [Pa (mb 5 t) 5 Pa (mb (wmEnm,I) 5 t) = pe, (a) PE (mb s t) + Al I (a) P,, (Mb S t) for

< a < , < b or e > a > r) > b; now use the result of problem 1.)

Problem 3. Consider the class D (0') of continuous functions u such that u+ (b) - u+ (a) = f k(dE) + u' (.) m(dt)] 0 < a < b < 1 (a.bl

for some continuous u' and let W be the differential operator ((&' u) (E) M U* (E) =

u+(d E) - u (E) k (d E) m(d f)

0 < < 1;

the problem is to show that W u S 0 at each non-negative local maximum of u (s) (0 < e < i) (see problem 3.7.1). (I>- 0) and [Given 0 < a < E < b < I such that u (,J) S u (49' u) (rj) > 0 for a < t) < b, it is found that b

0>Pab()u(a)

u(b) -u(E) = f Gab(e,r1) (GY u) (ii) m(drj) > 0,

which is absurd.] Problem 4. P. (fl b < + oo) Pb (ma < + oo) is either - I on (0, 1) X 'LS

(0, 1) or < 1 on (0, 1) x (0, 1). D is said to be persistent in the first case and transient in the second.

4.5 (l at an isolated singular point

125

[Given 0 < a < b < 1 with Pa (ntb < + oo) Pb (ma < + oo) = 1, Pb ( ) = PE (Mb < + o) is a solution of

p()=1=a,b

p+ (d ) = p

)

0<
`r"

k (d

and, using 0 5 Pb E t , it is found that Pb (E) = 1 (0 < < A). But then k (di) = 0 (0 < < b) and since k (di) = 0 (a < < 1) for similar reasons, p,, () = Pe (m,, < + co) (1 > > b) is a solution of

p+(d) = 0 p(,1-)=1

i.e.,

1

p+()=0 (0 < < ,) for each j< 1.]

0<
4.5. CAS at an isolated singular point ,.,

Given a non-singular diffusion D on an interval Q with endpoints 0 and 1, if 0 ( Q, let it be added and let PO [mo+ = m W = + oo] = 1 . D will be modified so as to have 'L7

I a)

e1 = Eo (e-m.+) = e2 = bifm Eo+ (e 14')

I b)

e3 =

Eo+(e-m.) = e4

= bl of Eb (e-ms+) .

Because of 2a)

e1,e3,e,=0or1

2b)

el S es ,

2c)

e1 e3 = e2 es

e3 5 ei

[see 3), 4) and 5) of 3.3], if el < e=, then 3a)

e1=0,

e3=e4=0,

e8=1,

1 b) is automatic, and (see problem 3.6.3) PO can be modified so as to have PO [mo+ = 0] = 1, i. e., e1 = 1, while i/ el = ez and if e3 < e., then 3 b)

e3 < 1 = es,

e1 = es = 0,

I a) is automatic, nto+ = mo A m in case x (0) > 0 and mo+ < + oo, and the desired modification is [x' (1); t 0, P.], where x' (t) = x (t)

=0

t < mo A mo+ t?moAmo,

4. Generators

126

D is now modified at I also, so as to have I E Q and E,(e-m,-) = limE,-(e

4a)

btl

4b)

E,-(e

m,) =

omEb(e-m,-);

this modification will be understood below.

Given u E D (0). if PO (mo. < + oo) = 0, then (63 u) (0) = - x (0) u (0)

5 a)

x (O) = Eo (m.)-1, *

while, if PO (mo. = 0) = i and if a dependence c, u (0) + c$ (Ou) (0) = 0 prevails for each choice of u E D (0), then [c1 + c2 a) (Gs f) (0) = C2/(O) for each /ED and a > 0, and choosing f (0) = 0 < f (b) (b> o), (G` f) (0) > 0 implies cl = cs = 0; in brief, if PO (mo, = 0) = 1, then u (0) and (()u) (0) are independent for u E D ((s).

But if Po [mo+ = 0] = 1, then, as will be proved below, u+ (0) = am u(b)-u(0) s (b) - s (o)

6)

=u =

(0

+) **

exists for each u E D ((si) , and a dependence holds between u (0), u+ (0), and ((33u)(0):

(0 u) (0) m (0) = u+ (0) - u (0) k (0)

5 b)

u E D (0),

where 7a)

k (O) -

of

= + oo) po (m s (b) - s (o)

b

M (O) = boo E.(m A 7b) s(b) - s (o) complementing 4.3.1) :

***

(($ u) (e) m (d ) = u+ (d i) - u (E) k (d i)

0 < < 1.

Beginning with the proof of 6), if f E D is such that /(a) = 0 (a > j) then u = G, / E D (0) satisfies u (a) = E. (e-m}) u (j) (a S 1). Because u (J) > 0, 0 S u (a) E t (0 < a S J) ; because P0(mo+ = 0) = 1, u (0 +) = u (0) > 0; and, using 0 u (a) = u (a) (a < 1-) to deduce from 0 S u E T that O S U+ E T to the left of 1, it follows that }

k(0,.*]+m(0.f)Su(0)-1fuk(dE)+

8)

10+

e

EO(mo,) > o;

o+

S u(0)-1[u+(1) - u+(0+)] a-l - o if e - +00.

u+(0) - 0 if s(0) = -00. k(o) = m(0) = 0 if s(0) = -00.

1

1

< +oo.

4.5. 0 at an isolated singular point

127

8) implies the convergence of the integral

f

u+(j)

>j;

f 0+

o+

- u+(0+)

for each u E D (Gi) , proving the existence of u+ (0 +) ; as to the rest of 6), either s (0) > - oo, in which case u+

b

u

(0) = io s (b) - s (o) =

lizn blo [s (b) - s (o) ]-10f u+ s (d

)

= u` (0+),# or s (0)

oo,

in which case u+ (0) = 0, u + (0 +) =0 owing to

u () - u(0) = f u+ d s , and k (O) = m (0) = 0 in agreement with S b). 0+

Given s (O) > - oo, DYNKIN'S formula,

10)

P. (m,. .--

(0 u) (0) = lim b4o

9)

.

Fo ( A moo))- u (0)

(Gi u) (0) Eo (m, A moo) P9 (m& < + s (b) - s (0)

- u(b)-u(0) - PO (m1<+°°)-1-1 u(0) +0(1) b 0, s (b) - s (0) s (b) - s (0) and an application of 6) coupled with the independence of u(0) and (G3u) (0) implies the separate convergence of the coefficients 11 a)

11 b)

Po(m,< +-)-1- 1 = P0(m,= +°°) S (b) - s (o)

s (b) - s (O)

E.(m,Amao)Po(m,<+oo1-` s (b) - s (0)

=

-1- 0(1)

E0(m,Amoo) +0(1) s (b) - s (0)

to non-negative limits k (0) and m (0) < + co. 5 b) is immediate from this. Chi satisfies similar conditions at 1; if u E D(@) and if Pl (ml- = 0) = 0 then

(Gi U) (1) _ -x(1) u(1) while, if P1 [m1_ = 0] = 1, then 12 a)

12b)

(05 u) (1) m(1) _ -u'(1) - u(1) k(1),

where

u (1)

u(b) bti s(1)-s(b)

fly

13)

x(1) = E1(tn.)-1,

* Po[mo+ = 0] = 1 implies u(0+) = u(0).

4. Generators

128

and

(m, = + oo) k(1)=lbfl P,s(i)-s(b) w!-

14a)

A m.) m(1) =1bm EI(m, s(i)-s(b)

14b)

Problem 1. Give an alternative proof of ((SS u) (0) m (0) = u* (0) - u (0) k (0)

u E D (O)

in the case

PO (m, < + oo) > 0,

Eo(m1 A mao) < + oo.

Use

A ( ) = P?

(m, < + °O)

poi ( ) = PE (mo < m1) e1

EE (m, A mom)

as in 4.4 to define the killing and speed measures Pi(d_)

0<$<1

P, ( )

k(0) =

PL(o) Pi(O)

_

P; (0+) P,(o)

m (d e) _ - [ei (d E) - e, (!`) k (d ))

0<'e<1

[ei (0) - e1(0) k(0)]

m (0)

and express

._w

p,( ) u(i) - u() = Ee

miAm ,

f

(6 u) (xs) dt

u E D(05)

0

by means of the GREEN function

)=

E S ,,

B

P, (E) Po1(')

0sE<1. V

B = p (E) pa,

P,(E)P6,07)

1A1

G( , rl) = G(7,

4.6. Solving ( u = a u '.7

Consider the modified non-singular diffusion of 4.5; it is obvious from the composition rule 1)

Ea (e-a "")

= E a (e-a "') E e (r "q)

a
oc > 0 ,

that 2 a)

g, (a) = E. (e-0'0*) = E} (e- a4) -I

0SaS

j < a:<

1

4.6. Solving 0 ,u = a u

129

t9'

satisfies

Eae-a m+ ) (

g' (a) gl(b) g1 (a) < g1 (b)

3 a)

4a) 5 a)

a
a
gl (a-(-) = g1 (a) < + oo

0Sa<1

g1(a-)=g1(a)>0

0
Now as in the argument before 4.5.8), if 0 < b < 1 and if / E D satisfies / (a) = 0 (a S b) and / (a) > 0 (a > b), then u = G`/E D (CAS) satisfies u (a) = E. (e-a m+) u (b) (a S b). Because (CSu) (a) = x u (a) (a S b) and u (b) > 0, g1 is now seen to be a solution of 6a)

((&' gI) ( ) = x gI (

O<<1,

)

coupled with a gl (0) m (0) = gi (0) - g1 (0) k (0) *

91 (0) = gi (0+) = 0

Eo

I

1111

7a)

Eo (e-m.+) = 0,

being the differential operator

(

( u) () =

u+(d E) - u(f) k m(d f)

A similar argument shows that (e a m})

2 b)

gs (a) = E a = E} (e-a me)-'

j< a S 1 0SaS}

satisfies loo

gala)

4b)

a>b a>b V

A

E.

3 b)

92(b)

ga (a) < gs (b)

92 (a+) = g: (a) > 0 g$ (a-) = gs (a) < + o0

5 b)

and 6b)

0Sa<1

0
(0' 92) () =ags( )

coupled with

ags(1) m(1) = -gz(1) -g2(1) k(1)** Sao

7b)

gs(1) =g2(1-) =0

E1(e-m'-) = I E1(e-m,-) = 0.

000

g1 and g, span the solutions o l

NY u)()=au(.)

8)

as will now be proved. ' gi(0) = gi(0+). M(O) = k(0) = 0 if s(0) = -00. if s(1) _ +oo.

' gq (1) = g;(1-). m(1) = k(t) = 0 Itb/McRean, Diftusion processes

9

130

4. Generators

0vxw

Given 0 < a < b < 1 and a solution u of 8), the fact that the

+.o

rail

determinant g, (a) g, (b) - g1 (b) g, (a) is < 0 permits the choice of constants cab and cba such that u' = u - [cab gt + cba g,] = 0 at a and at b.

Because u' is a solution of 8), if a < t < b is a positive local maxi-

of,

mum of u' (a), then 0 < a u' (t) = (CY u') (E) 5 0 (see problem 4.4.3), which is absurd. u' (t) is then S 0 (a < E < b), and, using -u' in place of u', it appears that u' (e) = 0 (a < E < b). But then u (J) = cab g1( ) + + cba gs () (a S t S b), and as the reader will check, cab and cba do not depend upon a and b; this completes the proof. Because b gig:-gtg2lab

9)

= f [g2 050 g, -g1ClYgJm(dt) 0
the WRONSHIan

B = B [g1, go = gx (E) gs (E) - gt

10)

gs (E)

=gi()gsW - gt( ) g+( ) is constant (> 0); this will be useful below.

Given a > 0, let j(db) = k(db) +am(db) (0 < b < 1), let 0 be called '0.

an exit point it

/s (d a) J j (d b)

+00 >

a

0+

an entrance point if + 00 > Jj (d a) /s (d b) , a

0+

note that this classification is independent of a, and let us check the statements of tables I and 2 below. Table I

>0 _+00 =0

=0 =+00 =0

<+00 =+0o < + 00

_ +00

< + 00 < + 00

8

<+00

< + 00

= + 00

< + 00

= +00

entrance

=0

g1(0)

z

g2(o)

< + 00 < + 00

0

gi (0)

>0

+

g2 (0)

0

++v

neither exit not entrance

+

exit not

entrance not exit

exit and entrance

fg,dl

+

112

trs

V

fg2dl

C+

< + 00

+

0

4.6. Solving ( u = a u

131

Table 2 exit and entrance

PO [mO+ = 0] =

0

Eo+ (e-m,+) =

i

or

exit not

entrance

ndther exit

entrance

not exit

nor entrance

0

I

0

1

0

0

I

u+ (0) - U(0) k (0) (Cl u) (0) =

-x u(0)

-(0)

-x u (0)

i/ PO [mO+ = 01 -1

-x u (0) if P0[mO+=01=0

u+ (0) = 0

A glance at line i of table 2 will explain the usage of exit and entrance; in table 2, (0u) (0) = m (O) -1 [u+ (0) - k (O) u(0)] means u+ (0) = k (O) u(0)

if m (0) = 0; in table 1, g2 (0) = g+ (0 +) .

Endpoint 1 is said to be 1-

a

+00>

an entrance point if

+ oo > f j (d a)

A.,

an exit point it

s (d a) f j (d b) 1-

a s (d b) ;

f

=gi(1-), P1[m1-=0], limEb(e-m=), btl

etc.

can again be described as in tables 1 and 2. We label the statements of table i like the entries 11, ... , 64 of a 6 x 4 matrix; the reader is advised to take a pencil and to check off each statement as it is proved. Because gi (0) ? 0, row 5 is evident from gi (J) - gi (0) = 19, dj, 0+

and, using g2 ( ) - gQ (0) = gt d j, row 6 is seen to follow from 000

0+

W4+

row 4. Because 0 > g+2 E t, g+2 (0) = g 2 (0 +) < 0, and so gt (0) < + oo implies that s (0) > - oo and

+00 > g2(0) - gt(f) - [s(4) -s(0)]g:( )

/j (e, f] s (d

_ 0/s(d)gt dj z g2(?))0 E

i.e., that 0 is an exit point, proving 23 and 24, from which 33 and 34 follow using the fact that gigtis smaller than the constant B =g;g2 -g1g+.

12 and 14 follow from 42, 44, and -glg4 S B; as to 42 and 44 them9*

132

4. Generators

selves, gQ (0) > - oo implies that

+oo>g+()-ga(0)= f gtdj?gi() 0+

o+

(=

i. e., that 0 is an entrance point as it be. Given that 0 is entrance but not exit, gi (0) = 0 (see 33), e

0 < gi ()/g1( ) = g, ( ) -' f gi d j s 9 (0, 4], 0+

and so

0

ds S fi (0, fl ds = J [s( tai

ff

0

0+

-

s (J)]

d1 < +oo

proving 13. 43 follows from -g1g2 5 B. Given that 0 is an exit point,

0 s gsW -'{(j) - ga (i)] = g:( )-'.g9 dj, f

VII

gs+(f)

!In

and so r}

Jigs' ds - lggs to S,l 1(E, J) s(dE) < +°o, 0

0

t

V

:.o

proving 21 and 22. 32 is now evident from gig= - g1gQ = B > 0 and E

'AI

- gs W g1()

-g+2 () f gi ds 0 i 21 and g= (}) - gs+ (0) = J g, d i

-gi (¢) f g+ ds, and 41 follows from 0

g2 (0) j (0, 1] (see problem I for addi-

o+ p.1

tional information about 11 and 31). Table I is now established. Table 2 follows at once.

Problem 1. Given that 0 is both an exit and an entrance point, construct the solutions g. and g' of

ag(e)=(t?g)()

11)

0<

}

(')

such that

g.(0) = 0, g.+(0) > 0, g.() = I 12b) g' (0) > 0, 90+(0) = 0, go (}) = I and prove that g.(g') is the smallest (greatest) positive increasing solution of 11) subject to g (J) = 1. 12a)

[g. = B.IB. (J) and

g' =

B. 1B.

(1)

B. = g1 gs (0) - g1 (0) ga

B' = g, g' (0) - gi (0) g2

4.6. Solving 09 u = a u

133

,..

solve 11) and 12), 0 S g, S g*, and if 0 S g E f is another solution of 11) with g(]) = 1, then the obvious independence of go and g' per°.R

mits the choice of c. and c' such that g = cog. + c'g'. go S g S g' is now clear from g, S g' and g(0)Ig (0) = c' ! 0, g+(0)lg. (0) = c. ? 0, g(]) = 1 = c. + Co.] Problem 2. Compute the increasing and decreasing solutions g, and

2IbI-''D' (-1
gsand the WRONSKIangig,-g,g+for@'= (see 6.8 for an application).

[g, (q, a) = g: (a, -a); setting

_ (y +

2) -1, g2 (a. a) m"`

_ IcK,6 (2# 2a a'1' ) and

I Ip(2 f j all'P) + g.

gI (a, a) = 1'(1

for a

0;

..a

the WRONSKIan is 1'(1 - fl) r(f)/2p.] Problem 3. Determine the scale and speed measure for the BESSEL (-d + d , 1 d and check that motion with generator 05' = dr r2 2

'.r

P. J,lim0 r (t) = + oo] = 0 or I according as d=2 or d Z 3 ;

prove also that P.[r > 0, t > 0) = 1 for d z 2 (see 2.7). [s(dr) = r'-d dr,

m(dr) = 2rd-' dr,

s(+oo) Z + oo (d $ 2), +00

+00

f sdm= f mds=+oo.] I

1

Problem 4. Compute the increasing and decreasing solutions gl and g. and the WRONSKIan for the BESSEL operator 6' = 2 (--am + y a )

(r > 0), and use gt to prove that lim

E,

(e-..r m.)

=

Y+ 2

Y> O,

What is the meaning of the disappearance of the & ?I

gs(a, )=K0(f i), gigs - gigs =1, EI (e

m.)

= Ko(i)/Ko(E 2a

,

K0()

* F. B. HILDEBRAND [1; 161-167) is helpful. 1 See problem 6.8.4 for a related result.

-lg $ (" f O).)

4. Generators

134

Problem 5. Give a new proof of D. RAY'S estimate glo

t-"Pa(mb St)=0

nZ1

a
;17

l

for the general non-singular diffusion, using to check VII

V11

e-1 P. (Mb S t) S g1 (a)Ig1(b) S [1 + a p= + a' p4 + a' p. +

.

]-1

a = t-1, where 424-!

f1

ps" = f s (d 1) f m (d t2) ... f a+

a

Wad

b

124-1

s (d a"-.)

f m(d : ")

nZ

1

a+

a

boo

(see 3.8.8) and problem 3.8.1). Problem 6. A non singular diffusion D is persistent if

k=o

13)

((l u) (o) m (o) = u+ (0) (CS u) (1) m(1) = u+(1)

in case s (0) > - 00 in case s(1) < +00 -}-

14a) m (o , 4] < + oo, 14b) m[4, 1) < +oo,

and transient otherwise (see problem 4.4.4 for the definition of persistence).

[Given a persistent diffusion, if - s (0) < + oo = m (0, J1 and if 0 < a < b, then Pa (mb < + oo) = lim 91(a) Slim g' (a) [s (a) - s (0)] a f o gt(b) - to g1(b)

t0 S $(a)s(0) s (b) - s (a)

a l0,

which is impossible; in brief, s (0) > - oo implies m (0, 4] < + 00. But then Eo+ (e-'"-) = 1, because of persistence Eo (e-mo+) is likewise 1,

and now the fact that Pe (mb < + oo) is a solution of o = P+(0) - P (O) k (O)

p (O) = 1,

Eo

1

In'

implies 13) and 14a). The proof of the converse is similar: using 13) and 14 a), p() = PE(mb < +00) is found to be the solution of l01

p+ (0) = 0

Eo (e-

+) = 1

p(b)=1; as such, p ( ) = I + constant x [s (b) - s (¢)], the constant has to be 0

either because s (0) and p+ (0) = 0, etc.]

oo and 0 S p S 1 or because s (0) > - co

4.7. m as global differential operator: non-singular case

135

Problem 7 after J. L. DOOB [3]. Given a conservative diffusion with

s (0) > -oo and m(0, 4] _.+oo, limPb[lim x(t) = 0] = I even if 0 tt+0° btO is not an exit point. [As in the solution of problem 5, s (0) > - oo and m (0, }] _ + o0 implies Po+ (mb < + oo) = 0 (b > 0).

(a S S 4)

P,, [ma A mi < + oo]

because it is a solution of

(03'u)()=0

(a S5

u(a+) = u(4-) = 1,

and so lim P. I lim x (t) = 0, ? limlim limP. [ma < + oo, mb (wM.- + 001 e40

.40 boo 410

81+00

=lml lfo P.[m<+oo]Pa[mb= + oo]=hm ajmo P.[ma<+00] o ?limlimP.[maAmi<+00,m}=+oo]=1.] .1o a4o

Problem 8. Prove that a non-singular diffusion in the natural scale

on Q = R1 with P. (x (t) S a) = P. (x (t) > a) = 1/2 for each t > 0 and a E RI is a standard BROwNian motion up to a constant scale factor. [Derive first the D. ANDRE reflection principles Pa (mb < t) = 2 Pa(x(t) > b) (a < b) and Pa(mb < t) = 2 P4(x(t) b) as

in 1.7, then take their LAPLACE transforms and deduce that -g1g2 = 0'O

gig2 = 1/2 (gigs - g1g2 = 1) and hence grg, = constant; the rest is plain sailing using Qi g = a g.] 4.7. 0 as global differential operator: non-singular case

0 can now be identified as a global differential operator 0'. D (CS') shall be the class of functions u ED such that, for some

u'ED, u+(dE) - u(s)

1)

2a)

u' (0) m (0) = u+ (0) - u (0) k (0)

u+ (0) = u+(0+) 2b)

in case 0 < E < 1

u' (1) m(1) _ -u-(1) - u(1) k(1)

"-(I) = u- (1 -) u' (0) _ -x (0) u (0)

3b)

u'(1) _ -x(1) u(1)

e+1

3 a)

4a)

U (O) = u (0 +)

in case

Eo (e-m.+) = I

in case El (e-'"i-) = 1

in case Eo 0 in case El (e-m+-) = 0 in case

j(b, 4]s(db) <+co 0 0

4 b)

and (il

u(1) = u(1-)

in case f j[1,b)s(db) <+co

shall be the map u E D ((SY) --+ W.

4. Generators

136

Because m (a, b) > 0 (a < b) and D (W) C D, W u = u' is unambiguous. (53' ) 0 as is clear on using 4.5 (see 3.6.6)) and entries 21 and 22

of table 4.6.2 in connection with 4)), and, for the identification of 05 and 0', it suffices to show that D ((') C D (0). But, if u E D ((53') , then (1 - (53') u E D, u' -u- G, (1 -@')u E D ((5}') is a solution of ( u' = u', and, as such, it has to be = 0;

for example, if Po[ma, = 0] = I and if I is neither an exit nor an entrance point, then u' (1) = 0 because ((53' u') (1) _ - x (1) u' (1) = u' (1),

u' (0 +) = u' (0), to (E) = constant x g, (E) (0 S < 1) because g, is unbounded near 1, and the constant has to be 0 because 0 < (53' g, (o) m (0) - [g= (0) - g2 (0) k (0)] ;

the proof in the other cases is similar. (W' stands for a local or a global dillerential operator below, the letter (53 being kept for the global operator; for example, ((53'u) (f) = u' (E)

(0 S E < 4) is short for the statement that both u (E) and u' (() = [u+ (dE) - u (E) k (dE)]/m (di) are continuous (0 < E < }), coupled with 5 a) u' (0) = u' (0+), u' (0) = u+ (0+) U (O) = u(0.4-), u' (0) m(o) = u+ (0) - u(0) k (O) in case Eo (e-'".-) = 1, 5 b) u (b) and u' (b) bounded near b = 0, u' (0) = -x (0) u (0) in case Eo(e-"'o+) = 0.

and 4a). As an illustration, 4.6.6a) and 4.6.7a) can be summed up as (0 5 E < 1), ((550 gl) (E) = a g, (E) and 4.6.6b) and 4.6.7b) as (Ygs) () = a g2 (E) (0 < f S 1). 4.8. 0 on the shunts Given a diffusion with state interval Q = [0, 11, if 1 a) PO (ml < + oo) > 0

P,(m,_=ma,=+o0)=1

1b)

Eo (m, A m ) < + 00 , then it is possible to define a shunt scale s+ (E) and a shunt killing measure k+ (d $) with k, (0) = 0 such that, for u E D (0), 2a) u (E) k+ (d f) (0 u) (E) s+ (d E) = 1 c)

f u(d) - f

f

[a, b) f1 R,

[a, b) f1 R+

2b)

u(1-) = [1 - k,(1)] u(1)

26

((M ul (a) = lim

[a. b) f1 R+

aSb<1,

k+(1) = P,- [m, = +00] u (b) - u(a) - [(.
bra

s+(b) - s. (a)

aEK+n[0, 1),

4.8.0 on the shunts

137

.'S

X where n [1 - k+ (d f)] is meant as a suggestive notation for a
((53 u) () in (d) = u(d) - u

3 a)

k (d)

In < < rn

(J u) (la) m (In) = u+ (In) - u (In) k (ln) u+ ( u (b) - u (I.) ln) = b+m s (b) - s (I.)

3 b)

as in 4.3.1) and 4.5.5b).

4b)

s + (d

)=

s (d ')

p(0

r p (71) k (d,7)

A

) = sP ( dM)

In < Se < rn

14. 4)

f

p (17 )

m (d ri )

In

< x, < rn, A

k+ (d

A

4a)

A

Given n ? 1,

[to. I0

p being the solution of 5 a)

p+ (d b) = p (b) k (d b)

5 b)

p+ (In) = p (In) k (In) ,

I,, < b < rn p (In +) = P (In) = 1

a special case of 4) is

6a)

k (l n )

(I., b) = lim 4s(b)-s(I.)

6 b)

m (In )

s+ (b) - s+ (I.) = lira b 4l. s(b) - s(I.)

the global condition

r

7a)

k+ [0, 1) S z k[In, ) s(d ) + k+ (K+) < +00 "al Q.

7b)

s+ (1) - s+ (0) 5

nil Q..1 m

[In, e) s (d ) + s+ (K+) < + 00

has also to be met. Here is the proof.

Given a 5 b, if 8a) 8b)

pb (a) = P. (Mb < + oo) eb (a) = Ea (mb A mao) ,

then [see 1)] 9 a)

9 b)

pb (a) ? pl (0) > 0 eb (a) 5 el (0) < + 00 ,

4. Generators

138

and using 10a)

al

lob)

b

pb(a) = Pb- (mb < + oo) = 1 - k+(b)

limPb(a) = Pa(ma+ < +°O) = 1

b>0 a < 1*

and the composition rules pb (a) = p, (a) pb(c)

11 b)

eb (a) = e. (a) + pe (a) eb (c)

aScSb, VII

11 a)

it is a simple matter to show that pb (a+) = pb (a)

12b)

Pb(a-) = Pb (a) [1 - k+(a)]

a
13 a)

eb (a+) = eb (a)

a
13b)

eb(a-) = eb(a) [1 - k+(a)]

a S b.

VII

12a)

and

as to

c { a in i 1 b) and use lob) and Jim e. (a) = Ea(ma+A mo) = 0; as to 13b), let c t a in 12) is clear ;

13 a), let

ova

eb (c) = ea (c) + P. (') eb (a)

and use

(c < a S b)

li a ea (c) = In P. (0)-1 Eo (ma A m W - me A me*, in < + °O)

=

mA(0)-'Eo(ma A m,, - me /1 mm) = 0.

Because of the composition rule 11 a), the shunt killing measure k+ (d a) as defined in 14)

k + (d a) = p' (d a)

a < b**

does not depend upon b. 10 a) is consistent with the present definition. k+ (0) = 0, and k+ (0, 1) < + °o because pl (0) > 0. Coming to the definition of the shunt scale,

aSb, IV'

ve(a)=P.()

15)

is continuous [see 12) and 13)] and, using 11), if E < i S b, then 16)

Vb (n) - Vb (#E) = Pb (')'1 [po (') eb ('7) - eb ( )]

-See 3.3.1 oc) and 3.6.6a).

q(q)

<0

** k+ (d a) = d )g p, (a) if p, is continuous.

4.8. (J on the shunts

139

so that, taking differentials and using 11 b) and 14), the (continuous) shunt scale s+ as defined in n

0<

17)

f Pb (a) vb (d a) E

n

f [eb (d a) - vb (a) Pb (d a)] E

n

f [eb (d a) - eb (a) k+ (d a)] E

III

tj < b

S. M - S.

does not depend upon

b.

Because Pl (0) > 0 and el (0) < + oo,

s+ (1) -s+ (0) < + oo ; in addition (use of vb (a) = 0), b

18)

eb (a) = Pb (a) vb (a) = Pb (a)

s+ (d

f

a

PA (S)

b

aSb.

= f PE (a) s+(de) a

2 c) can now be proved as follows. Given u E D (9), if aE K+ n [0, 1) , then DYNKIN'S formula implies b ja

Pb(a) u(b) - u(a) 4th

(0 u) (a) = lim

19)

E.(m, A moo)

- [Pe(a)-L -1l u(a) = lim u(b) - u(a)e,(a)/p+(a) b ja

as in 4.5. Because of 18), eb (a) /Pb (a) = s+ (b) + s. (a) + o (1) (b l a) , and it

is a simple matter to conclude from 14) that n [1 - k+ (de)) as descria <E$b

bed below 2b) is identical to Pb (a) ; this completes the proof.

Consider the indicator / of the point 1; f E D [see 1 b)], -G. f = E,(e-11 i) e (G. f) (1) increases to P1 E D as c + 0, and so, letting et 0 in

EX[G.-G.-}-(a-E)G.G.] (I,

20)

it is found that 21)

Got

PI =Ply

i.e., 22a)

P1ED(G)

22b)

OP, = 0

0,

4. Generators

140

as in 4.2.12). An application of the composition rule i1 a) coupled with 3) now implies 1 < a < r., a S b Pb (da) = Pb (a) k (d a) 23 a) 23 b)

In S b,

Pb* (1) = pb (1n) k (1,)

from which 4 a) and 5) are immediate. b'4

1moon m,

As to 4b), if / is the indicator of (0, 1), then G,/ = E.

increases to e1 E D as e 4 0, and so, letting e t 0 in

f e .1/(x1)di 0

[G.-G.+(a-e) G.G.)0,

24)

it is found that e1=G.[/+a e1]

25)

i. e.,

26a)

el E D(M)

26b)

(05el)(b)=-1

b<1

as in 4.2.20). An application of the composition rule 11 b) now implies

e+(da) - eb(a) k(da) = -m(da)

27a) 27b)

1, < a < r,,, IN S b,

eb(IN) k(ln) =

from which 4b) is immediate on using 17). 7a) and 7b) are clear from 4a) and 4b). Coming to the proof of 2a), if J+ = K+ - U IN , then nZ1

EE [measure (1: x(t) E [a, i) n j.. t S m1 A M-41

28)

f

pn () s. (d n),

[aVe.l)nK+

as will now be proved.

Because a sample path starting at t cannot enter [0, ) n j+ without crossing down over a point of U IN, nZ1

kin

El[measure (I: x(t) E [0,1) n j+. t < m1 A mom)]

29)

Eb [measure (t : x (t) E J+ , t s ml A M.41

= Pb

= pb (E) ei (b) - Pb (f)

' Eb[measure (1: 1 S x (f)
nil

= Pb (E) e1 (b) - Pb (s)

I pb v 1. (b) er. (b V

r,,> b

r.

I

I Pb v r. (b) = f pn () s+ (dn) - Pb ()r.>b e

1

= f pn

s+

as stated.

f pn (b V 1.) s+ (dn) bV1.

r. (dn) - ' f pn () s+ (dn) r.>b bV1.

f p,,(E) s+(dn), [e.1) n R+

b=va

4.8. 0 on the shunts

141

2a) can now be proved in a few lines. Given u E D (05), if a (dt) - - [u (dry) - u (17) k+ (d,1)] , then

a S b,

Pb (a) u (b) - u (a) = f pe (a) a (dry)

30)

[a. b)

the proof being identical to that of 18), and now an application of z

DYNKIN'S formula coupled with 28) implies f p,1 (9) a(dii)

31)

IM)

= A () u (1) - u (s) m1Am,

f

= Ee

((55 u) (xe) dt

0

m, A m

f

= Ee I

e5m,An'

r.>e

J

x(1)EJ+ 00

f

(0) u) (n) S. (d r1) + X PI.Vt() rw>e

[;. 1) fl E+

f

+

f pn () (0 u) (rl)

s+ (d n) +

if, l) f1 E+

f

Eve

(5 u) (xe) d i + N prv e

(l3 u) (xe) d t

0

I P,,(1 V e) a(d71) [4Ve.r ) a (dry),

po

U [liV e. r,.) ,w>1

i. e.,

f pn () (e u) (1l) S+ (d r1) =

32)

if. l) n x+

f p,

a (d 17).

[e, i) n K+

.ti

2a) is immediate from this.

Problem 1. If K+ = [0, 1), then the motion starting at a < 1 is just translation at speed 1 in the scale s+: x(t) = s+1 (t + s+(0, a))

t < m A s+ (a, 1]

= 00

tZ m'. <s+(a,i]

=1

IZs+(a,t) <

moo ,

where s+1 is the inverse function of s+ (0, a]. [x-1(b) = mb if mb < + oo, and using the obvious '!3

a p s+ (d EE) = p (d E) - p one finds p(a) P(b)

=

k+ (d i)

e-of+(a,b)

E
fl [1 - k. (d )]

o<e5b Ea (e- a me)

=

Ea[e_az-1(b),

now the result is evident.]

mb < +00]

a
4. Generators

142

4.9. 0 as global differential operator: singular case Continuing with the case 1a)

PO (ml<+oo)>0

1 b)

Pl (mi- A m,o = + oo) = 1 Eo (m, A

1 c)

+00,

CS can now be identified as a global differential operator W.

D (4) shall be the class of functions u E D such that, for some

u'ED,

f u'

2 a)

(a,b)

u'

2 b)

[a,b)nK+

[a,b)

l,
m (l,) = u+ (ln) - u

k

u (d,) - f

f u' (4) s+ (d i) = f

3 a)

k(d)

m(d) = f u' (d ) - f u

[a. b)

[a.b)nK+

n n

u($) k_ (di) a< b < 1

[a.b)f1X+

u(1-) = [i - k+(i)] u(1)

3b)

u' (1) = 0,

alb

4)

and CS' shall be the map u E D ((3S') --> u'

.

Because in (a, b) > 0 (a < b) on Q. and s+ (a) < s+ (b) (a < b) , (3S : u u' is unambiguous, and using 4.8, it is clear that for the identification of (3S and C3S', it suffices to prove D ((3S') C D ((5S) .

But, if u E D (09), then (1 - (3S') u E D,

u'Mu-GI(1uED((3S')

5a)

is a solution of 050u' =u'

5 b)

and, as will be proved below, 0 < d = sup u'

leads to a contra

og; E:5 1

diction. u' (f) is then S 0 (0 S 1), and using - u' in place of u', it develops that u' (E) = 0 (0 S S 1), i. e., u = G, (1 - (3S') u E D (0). Given d > 0, if lim b = a and if is t d (n t + ao), then nt+OD

0< d = Jim u' of +co = u' (a) = Pa_ [ma < + - oo] u' (a)

in case b ? a in case b < a

i. o.

i. o.,

and, using 0 < u' (a) S d in the third line, d = u' (a) is seen to hold in that case also. .d.

But, i/ a is non-singular, then (see problem 4.4.3), u' (a) = (&a) (a) S 0, contradicting d > 0; i/ a = 1 , then 0 < d = u' (1) = (0' u) (t) = 0;

4.9. 0 as global differential operator: singular case

143

and, if a E K+ f l [0, 1) , then choosing b > a such that u'

d/2> 0

(a S t S b) and noting

u'+() = u'+(la) + f [(i' W) (r!) m(dn) + u' (r1) k(dn)] (lw.el

> u'+ (l,) (C3' u') (la) m (ln) + u' (l,) k (1k)

aS1. S:
0

it develops that u' (b)

= u' (a) + a$

f u+ d s +

f

[(@ u') (17) s* (drl) + u' (rj) k+ (drl)]

K+f1(a.b)

> U. (a) = d, which is also absurd. Problem 1. Consider the differential operator

b E R - R1 - o,

(0' u) (b) = m ab)

f.:

cap

bpi

where u+ (a) = u a [b - a] -1 [u (b) - u (a)], m (db) is a speed measure on R', and u E D (C)') = C (R') fl (u : (53' u E C (R')); the problem is to determine all generators CS of conservative diffusions on R1 such that CS' D 0.

[One possible choice is to declare ((5S u) (0) = 0 for each u E D (0). D (C)) is the class of functions u E D (CS'), left continuous at 0 0

1

if f m [-1, ) d < + oo, and right continuous at 0 if f m (, 1] d e <

g^.?

-1

0

< + oo. If in (0, + 1) < + oo, it is possible to introduce a right shunt at 0, i.e., to define u(E) - u(o)

((3j u) (0) =1im

81O s+ (E) - s+ (o) -G1

with s+ (a) = f m [0, b) d b and m (0)

0. D (OS)

0

functions it E D (C)')

is now the class of

0 ,

left continuous at 0 if f m (- 1., ) d < + oo, -1

act

with u (0 +) =u(0) and (@u) (0+) _ (0u) (0). If M[-1, 0) < +00, a left shunt can be introduced at 0, while if m [-1, 0) + m (0, + 1 ] < < + oo, it is possible to define u+(b) - u+(a) m(a, b] o-.

b,& 0

1..

((SS u) (0) = lim

at0 with m (0) 0 and D (03) the class of functions u E D (C)') with u (0+) = u (0) and (OS u) (0+) _ (CS u) (0).]

4. Generators

144

4.10. Passage times

Consider, as in 4.8, the case 1 a) Po [m1 < +oo] > 0 I b)

Eo[m1 A m00] < +oo, introduce the letter Q' = U [Is, r,J, and let us show that [mb: 0 S b S 1, n21 Po] is identical in law to 2) s+([0, b) fl K+) + f l p ([0, b) x dl) -h oo x p((0, b] x + oo) (0,+00) O S b 51, Vii

VII

where + oo x 0= 0 and p (d b x dl) (0 S b g; 1, 0< 1:5 + oo) is ,y.

the PolssoN measure with mean n (db x dl) : e-n((a.b) x+00) = Pa[mb

'ft

<

n (K. x dl) = 0

3 b) 3

a
< -boo] =a<1:,b fl [1 - k+(dE)]# `"'

3a)

n (db f1Q ' x dl)

c)

[ mb: 0 S b S i , P (B

l > 0.

= s (db)lim atb s(b) - s(a)

.=+

'ti

< + oo) ] is differential, PO (l m M. = Inb,

Pojalumma=mbj=1 (OSbS1).

so ms, can be expressed as

i

05 b 5 1

f l p ([0, b) x dl)

VII

1(b) -}-

4)

+00

(0'.00) .ti

relative to the conditional law PO (B I m, < + co), where t E C [0, 1] is independent of the sample path and p (d b X dl) is a POISSON mea-

sure.' Taking p ((a, b] x + oo) to be independent of 4), differential, e-"

Iii

and PoissoN distributed with mean n ((a, b) x + oo) - -1g Pa [mb < + oo] VII

(0 S a < b S 1) as in 3 a), mb can now be expressed as 5) t (b) + f 1 p ([0, b) x dl) + oo x p ((O, b] x+ co) 0;5 b S 1 (e. +oo)

.ova...

relative to the unconditional law P0. t and n have now to be identified with the expressions in 2) and 3). Given a S b inside some interval of Q', Pa(mb z 1) is a concave function of s (a) (see problem 4.4.2), permitting the definition of

n' (b x dl) as in

a t b, 1 > 0 0 S [s (b) - s (a)]-' Pa (mb ? 1) t n' (b x [l, + co]) if l > 0 is not a jump of n' (b x dl). Using the solution g, = E.(e-° m') of 0 S b < 1, 7) (0* g,) (b) = a g, (b) 6)

VII

See the explanation below 4.8.2).

' See the note placed at the end of this section.

4.10. Passage times

145

it is immediate from +Co

a f a-al n' (b x [1, + oo]) dl

8)

0

+00

=a1 is f e°I[s(b) - s(a)]"1 P0(mb 0

f [1

= lim [s (b) - s (a)]-' atb

=

(0.+ol gi (b) -1

gs

1) dl

- e-°I] Pa(mb E dl)

(b - s (a )

g (b)

that

gi(b)

- e.--I] n' (b x dl) _ g, (b)

f [1

9)

ga(b)

(0. +ao1

and

f s (d ,) n' (f x + oo) = l +m lg gi (a) = -1gPa [mb < -f - 00] .

ti.

10)

[a. b)

v^,

But now, using 5) to check G_a[I(b)-l(a)]fo f '[1-e °']n([a,b)xdl)-n((a. blx+co)

11

05a
Cpl

=

=Ea(eame)gi(a)

gi(b)

.,,

it appears from 7). 9), and 10) that 12) a [1(b) - 1(a)] + f [1 - e-111] n ([a, b) x dl) + n ((a, b] x + co) (0.+CO)

_ 1 ga(b)

-

g gi(a)

=1g u(a)

- 1gPa[mb < +oo]

u = p-'gf E C[0, 1],

= f u(d)) - lgP4[mb < +0o]

P = P.[m, < +oo]

[a. b)

p(d))

IgP0[mb < +00]

[a. b)

gi (E)-i [(d' gI)

k+ [a, b)

s+ (d 4) + gI (.) k+ (d I

+ [a, b)J fi Q

gi (

f

[a.b)fQ

)

s(d ) --- lgPa[mb < +00]

f [1-e

jam

=as+([a,b)f1K+)-k+([a, b)f1Q')+

.r-

f

[a. b) f1 S+

(0,+001

-IgPe[mb < +oo]

f

(a, b) f1 Q

It6/McKean, Diffusion processes

s (d )

f [1 (0, +oo)

- e-°1] n ( x d1) '^*

=a s+([a, b) fl K+) +

-IgPa[mb < -boo] . 10

4. Generators

146

t (b) - t (a) = s+ ([a, b) fl K+) and 3) follow, and the proof is complete.

Note 1. Differential processes with increasing paths. Given a differential process p (t) (t z O) such that p (0) = 0 , p (t) E t , p (t) = p (t +) <+ oo

(t ? 0), and E [e -n (I)] is continuous (t z 0), P. LEvY [1: 173-1801 proved that

p(t) = c(t) + f+00p([o, t]xd1)

1)

t ;>-. 0,

0+

where c (t) is continuous and independent of the sample path, and 2)

(dt x dl) = the number of jumps of p of magnitude l E dl in time dt

is a POISSON measure with mean n(dt x dl) such that Co

f (1

t>0

((0, t] x dl) < + oo

o+

I~>

(see the note placed at the end of 1.8 for the definition of PoissoN measure).

P. LEvy's formula [1: 173 -1801: 4) E[e-a(P(':)-P((i))] +00

= exp[-a[c(ts) - c (t,)) - f (1 o+

- e-°t) n((t,, t=] x dl)] a>0, tl St=

is immediate from 1), and 3) is immediate from the fact that the left

side of 4) is > 0. The following proof of 1) is adapted from K. Iro [1]. Given p as in 2), 5) c (1) - p (1) - f l p ([0, t] x dl) t? 0 o+

is continuous, non-negative, increasing, and differential, and, as such,

it has to be independent of the sample path as will now be proved in the case I = 1. Because

rma E(lRt)

I.L. = [c(.)_c('')] Al, k S n, n? 1

S man E[1 - el-+]

S akZnx E [1

-

P(k p(!n1 )1] n)

e-L

f_P(!)

-e

D(ri), E

ale

=ma E x°)

S E[e-P(1)]-l max [E(e'()) - E(e , ale

6)

_

'(k"1)1-1

(7k

))1

4.10. Passage times

147

tends to 0 as n t + oo, it follows that if P < a < y, then 7)

- y Lm Inf

e

Z E(fk)

a4 400,5

= Jim sup H e-YE(I.k)

n1+oo ksn lim sup II [1 - a E (lnk)] nt+oo ksn lim sup 17 E (e-afwk)

nt+oo ksn

a £1Kt

=1im supE(e ":So +

nt+00

Lit,

-a £

lim infE(e k, nt+oo = lim inf jj E (e-ah*k) ..a

nt+oo ksn S lim inf II [1 - P E (lnk)] nt+oo ksn

: =e wt+ooksw -plimeup

and, letting

t a and y f a, one finds E[e ac(i)] = e

8)

-a llm

£ E(ink)

a+ +00 ksw

0,

III

which is impossible unless c(1) is independent of the sample path. Coming to the proof that P ([t1, t2) X [1 ,1s)) (t1 < ts, 0 < 11 < 12) is PolssoN distributed, if k 9 lnk = 0 or I according as (f n , n) x [ll, 14)) = 0 or z 1, 10)

'Z1

then, much as in 6), [1 - e-a] msax P (lnk = 1)

SmaxE[1 - e-aInk] ksn

k

S E [e-ap(')]-' max (E le-2

,- E t n) k

'.S

ksn tends to 0 as n t + oo, and as in 7), one finds -a Sl f 11) E le ks.

[

nt+oo

J

- urn H 5 n

= lim 17 [1 - (1 - e-a) P(lnk = 1)] nt+oo ksn

..r

=expt-(1-e-a`)n'imakYP(lnk=1)] a>0, 0

0

completing the proof in case tl = 0 and tq = 1. 1o*

4. Generators

148

+-'

Because of the meaning of ,p, it is clear that it is differential in t, and, to complete the proof that , is POISSON measure, it suffices to check the independence of m+ = ,p((O, 1) x (11,1=]) and I 1 p ([0,1) x di)

(0 < li < 10-

(0.111

Consider for this purpose,

l_k= jl,p([kn 1

12a)

,

kk) xdl)

(0.11]

12b)

1

,) x(11,10,

let f denote a (non-stochastic) subset of 1, 2, ... , n, and let f denote the (stochastic) set of integers k S n such that lnk > 0.

Because the event t = m can be expressed as the sum

UU (lnk=tk, kin), tEt I

where T_ is the class of all sets t = (t1, ..., tk,

of non-negative inte-

gers such that tk > 0 or = 0 according as k E f or not 13 a)

tt + is + ... + to = m,

13b)

it follows that

E[1+=m,e

14)

kEi

tk,kSn,e

E1

tEZ

kEf

I7P(l1'9*k=1k)17E[lnk=0,e kEi

tESC &Ef

_II1P i

(114k

tESC &Ef

I7 E

W

x

= tk) H P (ink = 0)

0. e

lift

aik]

r

17 E [i

kEf

=o

ear"k

II P (twk = 0) kEi

II P(t.+t= 0)HP(lwk=0) JI E[1..

I

P (1+,L. = I't' k S n) X

kEi

17r.0-° >a-1 S'i .

tESC

0.e = O] ,re

=r

kV

s`,

kEf

kEf

But the sets f and f figuring in 14) contain at most m members; thus, 15)

lim HE [a-a1k I Ink = 01 = 1,

nt40D kEf

4.11. Eigen-differential expansions

149

and as n t + oo, 14) goes over into E[D+ = m, e-aQ-] t X P (ln. = tx, k S n) E [e-` 0- I m« = 0]

16)

=n

= P[Z1+ = m] E(e-0- I n., = 0],

completing the proof. 4.11. Eigen-differential expansions for Green functions and transition densities

Given a non-singular diffusion modified as in 4.5. let Q' be the 1, closed at I if E0+ unit interval closed at 0 if E0 E, (e-'"'-) = El_ (e-m') = 1, and open otherwise, introduce the stopping time m = min (t : x (t) ( Q'), and let us use the idea of eigendifferential expansion to construct transition densities p(t, a, b) such that P6[x(t) E db, t < m] = p(t, a, b) m(db) i)

(t, a,b)E(0,+oo)xQ'xQ', where 2)

0 5 p (t, a, b) = p (t, b, a) is continuous on (0, + oo) X Q' X Q'

3)

p(t,a,b)=fp(s,a,c)p(t-s,c,b)m(dc) t>s>0, Q.

4a)

4b)

ap(t,a,b) _ e

a,bEQ'

p(i,a,b)*

if E0(e-mo+) = 0, b E Q p(t,1-,b)=0 if EI(e-m'-)=0, p+s (t, 0+, b) = 0 it E0 (e-mo+) = 1, E0+ (e-m,) = 0, b E Q *s ps (t, 1-, b) = 0 if El (e-m'-) = 1, Ei_ (e-ml) = 0, bEQ', p (t, 0+, b) = 0

bEQ.

4c)

W being the local differential operator described at the end of 4.7. The method is outlined only; see H. P. McKEAN, JR. [2] for the proof in a special case, S. KARLIN and J. McGREGOR [1] for a beautiful

method which can be adapted to the present needs, and H. WEYL [1]

for the classical eigen-differential expansion for STURM-LIOUVILLE operators. As a general source of information about HILBERT space, B. Sz-NAGY [1] is suggested.

* 09 is applied to a or to b. ps(t, a, b)

1im 1+4

p(t. , b) - p(t, a, b) s (s) - s (a)

;_,

p (t, a. b) - p(t, a, Pi (t, a, b) = 1im s(b) - s(e) ?+&

etc.

4. Generators

150

Given a > 0, let g, and g2 be the solutions of 5 a)

(0Y g,) (b) = a g, (b)

0Sb<1

5 b)

(W g2) (b) = a gg (b)

0
described in 4.6, let B be the WRONSKIan g11g2 - g,g (= constant > d) , introduce the GREEN function G (a, b) = G (b, a) = B-1 g, (a) g$ (b)

6)

a S b, a, b E Q-,

and let us prove that, for / E D, f G(a, b) 1(b) m(db)

7)

Q

a E Q.

= Ea[f e-°`I/(xi) dt]

is the solution of 8a) 8b)

u E C (Q*)

(a - W) u (b)

subject to 9a) u(0+) = 0 it Eb(e-m.+) = 0, 9b) u (1-) = 0 it E, (e-ml-) = 0,

1

a,.

10a) 10b)

u(0+) = 0 it u - (1-) = 0 it

b E Q-

(b)

E0 (e-m.+) = 1.

E1- (e-"`') = 1 0 E0.

E, (e-m'-) = 1,

Et- (e-m') = 0

(see 2.6 for a special case). As in 2.6.16), VII

fG(a, b) m(db) Sa-1;

11)

Q.

for example, if 0 E Q', if E,_ (e-mi) = 0, and if 0 < a < 1, then a f G (a, b) m (d b)

12)

Q.

= B-1 192 (a)

f

0$b
g, dm + g, (a)

f ' g: dml

asb<1

1

S B-1 [g,(a) f 9(d b) `}- g, (a) 9(d b)l asb<1 0
= 1 + B-1 gs (a) (a g, (0) m (0) - e (0)] + B-1 g, (a) g2 (1)

51 Both 13a)

u

fG/dm Q.

4.11. Eigen-differential expansions

151

and M

Ga = E. ( f e-a`1(xl) dt

13 b)

/

o

r..

are solutions of 8), 9), and 10) as is clear in the case of 13 a) on diffe-

rentiating and using table 4.6.1, and in the case of 13b), on noting 14)

G. 1 = Ga 1 + E. [e a (G. 1) (xm)] g2 g2 (0)

+ g1 g1(1)

m

-I-

- Ga 1 +'

PO [m0+ = +00] 1(0) + X (O) +a

P1[m1_ = +°`) X(1) -f- a

/(1)

But then u' = u - G;/ E C (Q') is a solution of ((SS' u') (¢) = au' (6) ( E Q') subject to 9) and 10), and, as such, u' = 0 on Q' in agree-

II:

ment with 7) ; for example, if OE Q' , if E1 (e-m'-) =0, and if E1_ (e-m=) = I then, as in the identification of f as a global differential operator in 4.7,

u' (b) = constant x gE (0 S b < 1) because g1(1 -) > 0, and the constant has to be 0 because

0 < k(0)92 (0) - gY (0) + m (0) ($ g:) (0+) Now think of G. as acting on the (real) HILBERT space H of (speed) measurable functions At defined on Q' with 11 h1s = h$dm < +00, V modulo null functions. Y Q'

f

Because f G d m S a-1 and G(a, b) = G(b, a), 15)

JIG; hIl2

= fdm(fGfGfhdm)2 S fm(da) fG(a, b) m(db) fG(a, b) h(b)sm(db) S a-1 f f G (a, b) n: (da) h (b)2 m (d b)

Sa-=fh(b)2m(db)

= a -$ ll hlla , 'N"'

i.e., GQ is bounded (11GQJ] 2 Sa-1) and symmetric (G.* = Ga).

G: is also positive-definite; indeed, using

>0

A

Ga-Ga-}-(a-P)G.G;=0

16)

to check that lims-1 [G,*, CdO

- Ga] = - Gas

in the operator norm and noting 0 GQ na 4 0 (a t + oo), it is found +00

that G. = f Gp dA is non-negative definite. A second application of 4

4. Gegerators

152

16) now implies that the null space H fl (h : G; h = 0) is independent of a, and it follows that if (Ga h, h) = 0, then 0 = lim fGp h = h Pt+°° as desired. G; is now invertible, and using 16) once more, one finds that the range G; H - H (Q) is independent of a and that

G.-' =a - Q

17b)

£1 a 1 - GI*-i ' : H (Q) - H. :.,

17a)

Q is symmetric (Z* = Q) and non-positive definite (Q = lum (e - G,1) S 0).

H (Q) is the class of functions h E H such that, for some u E D

and h'EH,

u-h

18a)

18b)

a. e. (m)

0 < < 1.

m(de) = u+(d f) - u(e) k(d f)

hh

subject to

h' (1)m(1)=-u-(1)-u(1)k(1)

if IEQ'

lit

it

u (0+) = 0 i/ Eo (e-mo+) = 0,

Eo+

u (I-) = 0 if E1(e-'"i-) = 0,

El_ (e-"'1) = 1 Ea, 0

ply

20a) 20b)

k (O) m (O) = u+ (0) - u(0) k (O)

u+(0+) = 0

21 b)

E1-(e-m,) = u- (1-) = 0 if E1(e-"'3-) = 1, 0 u(0+) = 0 if Eo(e-1°°+) = Eo+(e-II-) = 0,

22 a)

Eo (e-1°o+) = 1.

0 E Q'

1

21 a)

if

b.,

19a) 19b)

112

+ oo > f [s () - s (b)]2 m (d b) o+

22b)

u (1-) = 0 if E1(e-'"1-) = E,-(e-m-) = 0 1-

.'t

-{- oo > f [s (b) - s (J)]2 m (d b) ,* 112

and Q is the map h E H (Q) - V. Now introduce the spectral representations o+

Q = f y p (dy)

23 a)

-00

o+

23

b)

Ga = f Ip (d y) a - y -CO

where 4 (dy) is the appropriate spectral measure from (- co, 01 to

projections (4(-oo, 01 = 1). 22) is an instance of H. WEYS.'s limit circle case.

4.11. Eigen-differential expansions

Given I' = (y1, y=] (- oo < y1 < y$ operator; in detail, if e = 24a)

153

P (1') is a CARLEMAN

0)

e2(y,.)) is the solution of

0
(®

= (1, 0),

e' (y, 4) _ (0, 1)

..a

24b)

a (y,

'L7

and if OEQ' 1 EQ',

e(y,0) = e(y,0+) e(y, 1) = e(y, 1-)

25 a)

25b)

then

p (I') h = f e (I', a. b) h (b) m (d b)

26 a)

0

a (1', a, b) = f e (y, a) f (d y) a (y, b)

26b)

r

f e (I', a, b) 2 m (d b) < + oo

26 c)

o

a. e.

(m),

where f (dy) is a BOREL measure from (- oo, 0] to 2 X 2 symmetric non-

negative definite matrices 27a)

f (dy)

f11(dy)

=

11/21(dy)

27b)

/11(dy) Z 0,

122(dy) Z 0, /12(dy)a

/12(dy)

/(y) /12 (dy) =/s1(dy),

/11(dy) f::(dy)

and e f e is the inner product of e and f e in 2-space. = f e h dm is the so-called FouRIER transform; it satisfies the Q,

PLANCHEREL theorem o+

f

28)

(dy)

hps

-00

23 b) implies the eigen-differential expansion o+

29 a)

G (a, b) = f (a - y) -1 e (y, a) f (dy) e (y b)

a>0,

(a,b)EQ'XQ',

and inverting 29 a) as a LAPLACE transform, 1), 2), 3), and 4) are found

to hold for the kernel o+

29b)

p(t, a, b) = f ev' e(y, a) f(dy) e(y, b) -00

(t, a, b) E (0, + oo) x Q' x Q' ;

4. Generators

154 °».

the proof of 3) is based upon the estimates 0+ _w.

f (a-y)-'f11(dy)=B-1g1(j)g2(j)<+°°

30a)

o+

f (a -Y)"sffI(dy) 5 (a

30b)

B)-1gi(+j)gs(j)

< +oo

-00

and

31)

I ez (Y

) I + I e9 (Y, ) I < eoonat.ntYT7l (y y

- oo) on closed subintervals of Q',

permitting the differentiation of 29b) under the integration sign. Given 0 < b < 1, if p (t, 71) ( < rl < b) is the kernel 29b) for the stopped diffusion

[x(t A e):IZ0,Pa:aSb]

.a.

a=mb,

1-t>0, a
then P (t, a, b -) =0 (a < b) and `1r)

32)

III

.^.

leading to the identification of u (t, a) = Pa [Mb < t] as the solution of a E [0, b) n Q'

a 0, t > 0,

E0(e-'".+) = 0

34b)

u (0+, a) = 0 u(t, 0+) = 0 uq (t, 0+) = 0

34c)

li mu(t,a)=1

t>0.

33 b)

34a)

u (t, a)

.r^.

at u (t, a) =

.Vi

t > 0,

33 a)

1,

Eo+ (e-m-) = 0

If 0 and I are exit or entrance, then (see table 4.6.1), the trace ...

i r (GQ) - f G (, ) m (d ) is < + oo, with the result that G. is come. pact and f (dy) is concentrated on a series of simple eigenvalues 35)

0 ? Y1 Z Ys >

etc.

1 > - oo. j - oo,nagI yw

If s (1) - s (0) and m (0, 1) are both < + oo, then -2 36)

lim -Yw nt+ao X2112

= [!}1s(db) m(db) 0

,

4.11. Eigen-differential expansions

155

I

being the HELLINGER integral inf E V s (Qn) m (Q,J

J V s (d b) m (d b)

Q, )Q) (see H. P. McKEAN, JR. and D. B. RAY (1], and, for the L

nZI (U

classical case, COURANT and HILBERT [1]). S. KARLIN and J. McGREGOR [1, 2, 4]

have developed similar

eigendifferential expansions in connection with the birth and death processes described in problem 4.2.2.

Problem 1. Compute the FOURIER transform for the standard BROwxian motion. [e, (y, ) = cos }I21 y I

sin 21y1

e2 (y,

,

21yI

l I I (d Y) =

dy 27r y2 I -Y1

/12 = /21 - - 0,

,

is : (d Y) =

21y1dy .I 2n

Problem 2. Express the FOURIER transform for the BESSEL motion

attached to (3' = 2 ( b + d b 1 b) (d Z 2) in the special form +00 o+ h= r ei(Y,')/II(dy)f .1

-00

o

-°=

choosing for e, the solution of (CS' el) (b) = y e, (b) (0 < b < + oo) ,

eI (0) = 1, e1 (0) = 0. Evaluate p (t, a, b) using 29b) and A. ERDELYI [1: 186 (39)], and check it against 2.7.4). 2-

b) = 2

I r (Z) (V2 y I b) 12 J d

z

J

111(dy)

= 2 2 P(±)-2 l 1-2lYl2 -I dy,

2

01+61

p(t, a, b) =

s, e

2t(a

b)

'

Id

'

...

.

I (f 5 J

b),

and

ja b )

Problem 3. Given a conservative non-singular diffusion, define b.

pI = s, III

po

rf

P. (b) = f s(di)1.1 p"-s (n) m(d71)

n z 2;

the problem is to show that p (1, t, ,q) is an analytic function of 71 stn

(0 < 7) < 1) in the sense that for each t > 0 and each 0 < < 1 , with coeffinZo cients

p (1, t, 77) can be expanded as a power series Z c

/

t

C2n = @on ph(t, S , +00

c2n+1 ='i '-'

tt4)

P)3 (t, s, 113)

4. Generators

156

[Because el

= naoy"pzn, e2 = nZ0Ynpzn-i (p-i = 0), and ,

0

n

j ereIYI' Ie(y, E) f(dy) (pzn(rl),psn-i(rl)) -c G+

J eveI yIIn e°Ylrl 2[lii(dy)

na0 -,0

+ lzz(dy))

-2

<+00 for some constant c, an application of 29b) implies 0+

f e l e(y,t)f(dy) e(Y,rl)

_00

o+

= f er'e(Y, ) f(dy) -00 a ereyne(y, )f(dV)(f,0) 0+

+ 2y p:n-2 (17) ,I er`Y' e(y, 4) f(dy) (0, 1), U

-00

0

and to complete the proof, it suffices to note that 0+

c:n = f eve yn e (Y, 4) f (dy) (1. 0) -00

o+

= f ele(y,4)f(dy)Wne(Y,1) -00

= @** p (t, 4,

)

and III

0+

2

4i1=

f er' yn e (y. 4) f (d y) (0, 1) - 00 0+

=

f eve e(y, 4) (d y) (05'n e)+(j) - 00

Problem 4. If p(t, a, b) S p(t, a, a) for each (t, a, b) E (0, + oo) x x Q' x Q', then, after a change of scale, x (t) = b (t) = 00

t<e t ? e,

4.11. Eigen-differential expansions

157

C'.

where b is a standard Bxowxian motion and e is an independent exponential holding time with conditional distribution P.(e > tj B) = e-" (x Z 0). o+

f ev a (y, a) f (dy) a+(y, a) is continuous from the right (0
and so 0+

fe't e (ya) f (dy) e+ (y a ) ...

urnP(t.b'(b)_P(a)a'a) = 2

-00

=2limP(t,a,b)-p(t,a,a) =0 s(b) - s(a) bja III

proving that p (1, a, a) = c1 (t)

0
does not depend upon 0 < a < 1.

+00

o».

Because of this c2(a) = f e-`t c1 dt = B -1 g1(a) gs(a) is also constant 0

00°

(0 < a < 1), and using this to solve B = gigs - gl g i for g1 = constant x ecs and g, = constant x e-`d [c3 = (2c2)-1], an application of (Y gl = ag1 implies k (d b) = c4 s (d b) and m (d b) = c5 s (d b) , where 0:5C4 and 0
because s (0) > - oo would entail either 0 = x (0) g1 (0) + (0'g1) (0) = [x (0) + a] g1 (0) or

0 = k (0) g1(0) - gi (0) + m (0) (3'g1) (0)

_ [k (0)

- Vc4 -+ ca + a m (0) ] g1(0)

,

contradicting g1(0) > 0; the proof of s (1) _ +oo is the same, and

now a change of scale brings 0' into the form (' = 2 D2 - ca (Q = R1, ce = c4/c5

0) ]

Problem 5. Give a proof that p (t, a, b) > 0

(t, a, b) E (0, +oo) xQ' xQ

using the eigen-differential expansion 29b) and the CHAPMAN-KOLMOGOROV identity 3). o+

[p (t, a, a)

f eVte(y,a)f(dy)e(y,a)>0

(1>0,0
-00

because e (y, a) f (dy) e (y, a) Z0 and O,

f (a - y) -1 e(y, a) f(dy) e(y, a) = G(a, a) > 0.

4. Generators

158

Given s > 0 and a, b E Q' such that p (s, a, b) = 0, the CHAPMANKOLMOGOROV identity

(t<s)

p(s,a,b)= f p(s - t, a,c)p(t,c,b)m(dc) Q. III

coupled with p (s - t, a, a) > 0 implies p (t, a, b) = 0 (t < s). But

moo'

p (t) = p (t, a, b) is a LAPLACE transform, and, as such, has to be = 0 +00

(t > 0), contradicting f

p (t) dt = G(a, b) > 0.]

0

Problem 6 after KARLIN and McGREGOR [5]. Consider the space

of points c' E (0, 1)" (n ? 2) such that c1 < c, < etc. and, for t > 0, define e(t, a', b') to be the determinant p(t,a,,b") p(t,a,,b1) p(t,a1,b,) p (t, aa, b1)

p (t, a,, b,)

. .

.

P (t, a,, b0)

I p (t, a., b1)

p (t, a,., b,)

. .

.

p (t, a,.,

I

Consider also the joint sample path x (t) = (x1 (t), x, (t), etc.) E [0, 110 of n independent particles starting at a', let Pa. be the corresponding law, and let m be the crossing time sup (t : x1 (s) < x, (s) < etc., s < t). .41

Given Be = B1 x B. x etc. C (0, 1)" such that B1 lies to the left of B,, B. to the left of B,, etc.,

P0.[m>t,x(t)EBe] = f e(t,a',b')m(db'), Be

where m (d b') = m (d b1) m (d b,) etc. Give a direct probabilistic proof of

this using

Pa. [m>t,x(t)EB'] = Z (o) P.. [In > 1, x(t) E o Be] 0

= S' (o) (Pa. [x (t) E o Be] - Ps. [m < t, x(t) E o Be]) 0

=

f

0

Be

where o is a permutation of 1, 2, ..., n, (o) = f 1 according as o is even

or odd, and oB' = Bo1 x B., x etc., and deduce that e(t, a', b') > 0 for each t > 0 and each choice of a' and be (see KARLIN and McGREGOR [5]

for additional information and G. POLYA [1] for the original result in this direction). [Because Pa. [m > t, x (t) E oB'] = 0 unless o = 1, Pa. [m > t,

x(t) E Be] = f ti m(db') - f' (o) PQ[m < t, x(t) E oB'] is clear, and it Be

0

0

is also clear that the joint motion starts afresh at time in. But a per-

4.11. Eigen-differential expansions

159

r-,

mutation that interchanges two of the coinciding particles is odd and has no effect on the future motion, so (o) Pa. [m < t, x (t) E o B'] = 0, 0

Pa.[m > t, x(t) E B'] = f em(db')

leaving

Bte)

't7

as desired. e (t, a', b') z 0 is now clear, and an application of the CHAPMAN-KOLMOGOROV identity e (t, a', b') = f e (t - s, a', c') e (s, c',b') x

m(dc') to the case a' = b' plus the fact that e(t, a', b') = e(t, b', a') implies e(t, a', a') > 0 for each t > 0 (if not, then e(1/2, a', ...) - 0 and Pa. [m > t12] = 0 which is not possible). Given (1, a', b') such t=1

now

that e(t, a', b') = 0, a second application of the CHAPMAN-KOLMOGOROV identity implies e(t - s, a', a') e(s, a', b') = 0 (s < t), i.e., e (s, a*, b') = 0 (s < t), and so a (s, a', b') = 0 (s > 0) as the LAPLACE transform of a (signed) measure. But for t > 0, 0 < e (t , . , .) near the diagonal a' = b' , and using the obvious HEINE-BOREL covering to select a chain ci, c2, etc. of l < + oo points leading from a' to b' such that 0 < e (t, ci , c2) , e (t , c2, c;), etc., a final application of the C,,

CHAPMAN-KOLMOGOROV identity implies e(lt, a', b') > 0, contradictIOU

ing e(., a', b') = 0 and completing the proof.] Problem 7. Check that

t-' G (1-1,) [p (t, E,

)

1>0.

p (t.

is convex in t, and so

+Cc

f. OD

af a-I" p(t,

dt>p(a f e'°rtdt, ,e)=p(a-'. it.

0

)]

0 C.'

Problem 8. D is transient if and only if +00

.--.

limG(a,b)= f p(t,a,b)dt<+oo

a,bEQ'

0 ..+

(see problem 4.4.4 for the definition of transience). [Because the scale distance between 2 points a < b of Q' is < + oo, {in

flimG(e, E)-'s(d ) 80 b

= lim I aj0 a

B

9192

s (d E)

b +

a

= lim lg a10

g, (b) g, (a)

f -)

'Z!

= lit Jrlrg,

s (d E)

g: (a) g: (b)

_ - 19Pa(nib < +oo) Pb(ma < +oo)

4. Generators

160

now use the solution of problem 4.4.4 and the fact that the magnitude of the slope of G is S B-1 x [gig, - glg4] S 1.] Problem 9. Compute lim G (a, b) in the transient case. 010

aSb,

(lim G (a, b) = l ato

o G (b, a) = h1(a) h,, (b)

0
0
hlh2-hl hy = 1

((53'h1)()=0

(0S <1),

(t'ht)( )=0

(0< 51).]

Problem 10. Compute

m=maA Mb

EE(m")

in the conservative case. Use the solution of problem 9. [Consider the (transient) stopped diffusion

(x(IA m):iZO,PE(B):aSb] `10

with generator 05 and GREEN operators G;/ = ESL f e-

(x) dt

u = 1 - E. (e a m) is a solution of

(a-()u(E)=a (a<E 0), and using this, it develops that

E. (m") = -

la

-)" a u

= limn![-aGQ"«11 + G "1] aIO = n] Go+ 1,

Gp, being the GREEN operator based on the GREEN function [s(s) - s(a)][s(b) - s(n)] s (b) - s (a)

a < S,j < b.]

Problem 11. Given a persistent non-singular diffusion, if e is a non-negative BOREL measure defined on Q' such that f e (d a) Pa (x (t) E B) = e (B)

(t > 0, BC Q') ,

then e is a constant multiple of m on Q' (see G. MARUYAMA and H. TANAKA [1] for a different solution and additional information; see also 8.2).

4.12. KOLMOGOROV'S test

161

[Because f e (d a) p (t , a, b) m (d b) = e (d b) (1> 0, d bC Q') , /=- d e/d m

satisfies GQ / = floc. But then (l f = 0 on Q', showing that /+ = conv.-

stant, and f = constant follows from / Z 0 in case s (0) = - oo and s (1) = + oo, from /+ (0) = (® /) (0) m (0) in case s (0) > --- oo, and z:.

from -/- (1) = ((3Y /) (1) ni(1) in case s(1) < +oo (see problem 4.6.6).]

Problem 12. Given a conservative non-singular diffusion with Po [mo+ = 0] = 1, if p (t , b) is the passage time density at Po [mb < 11 (see 32), 33), 34)], if 0 < h (t) E t (1> 0),

and

if

lim Po [mh(f) 5 t] = 0, too

then

f p (t, h) dt < + oo

implies

o+

Po [x (1) < h (t), 14 0] =1 (see 1.8 for a special case). f p (t, h) d t = + oo o+

should entail Po [x (t) z h (t) i. o., t 4 0] = 1, but the proof eludes us. 4.12. Kolmogorov's test KOLMOGOROV'S test for the 1-dimensional BROwxian motion (see 1.9)

.iv

will now be established using the elegant method of M. MoToo [1]. Consider a persistent non-singular diffusion on Q = [0, + oo) with scale - oo < s (0) < s (+ oo) = + oo, speed measure m (Q) < + oo, sample paths w : t -+ x (t), and probabilities P. (B), and let e, < es < etc.

be the successive passage times to a > 0 via b > a. m (Q) < + oo implies

y = Ea(e1) < +oo,

1)

indeed, using 4.2.28) and problem 4.5.1 E. (e2) = E. (Mb) + Eb (ma)

2)

b

= J s(a V , b) m(dE) + f s(a, b A t] a

_ [s(b) - s(a)] m(Q) < +00. Iv'

Because the excursions x (t) _1 : e,S t < e,, are independent and identical in law, the same is true of the differences ex - ep_, (n Z 2), and an application of the strong law of large numbers implies

Pa[ nt+co lim

3a)

1,

or, what interests us, 3b)

Pa[2y<e,t-1<e,,<2ny,nt+oo]=1.

1t6/McKean, Diffusion processes

11

4. Generators

162

4)

Given h = h (t) increasing to + oo with t, it follows that St< (w: > h(2n y) i.o.) g. = max(x(t) : t t + oo) C (w : j, > h (eA-1) i.o.) C (w: x(t) > h (t) i.o.,

i.o.), and using the probabilistic significance of the scale s to check

P1[I.> h(ny)]

5)

h(n y) > b

= Pb [mh($,,) < ma]

_

s(b) - s(a) s (h (n y)) - s (a)

~constant x s [h (n y)] -1

to t +00.

the independence of the 1. coupled with the BOREL-CANTELLI lemmas

implies the alternative 6)

P. [x(t) > h (t)

i. o.,

t t + 00] = 0 or 1

+00

according as f s (h) -1 d t

converges or diverges.

Or.

.-'IN

ray

Consider now the radial (BESSEL) part yx1(t)= + - + x(t)1: Ca 0 of the standard d -dimensional BROwxian motion of 2.7, let R be its standard description with sample paths w : t -. r (t) E [0, + oo), BOREL algebra B, and probabilities P . (B) , and let us show that 6) can be applied to the motion R' =_ [r' (t) = e-i rs(e` - 1) : t .'a: 0, P.J. Given a MARKOV time m' for R', m = e'"' - i is a MARKOV time for R, and introducing the BOREL algebra

7) Bm.+ = (B' : B' fl (m'< s) E B[r' (1), t S s] for each s ? 0)

=Bm+=(B:Bf (m<s)EB[r(1),ISs] for each sZ: 0) and using the fact that the scaling r (t) -. jr r (tls) (s > 0) maps [r, Pa] into [r, Py-a], it is found that 8)

P. (r' (t + m') S

b I B;,,.+)

= P.(e-(1+'"') r2 (e"' (e`-1)+m) S b Bm+) = Pale' ' s r2 ((e,-1)/s) S b) s=

= Ps =P

(e-4 r2 (e'- 1) !-, b) (1) _< b)

in brief, R' is a conservative non-singular di//usion.

a = r (m)

4.12. KOLMOGOROV'S teSt

163

+''

Computing its generator (J', if p (t, a, b) is the BESSEL transition density relative to the speed measure 20-1 db [see 2.7.3) and 4)], then

'ti

P. (r* (t) E db) = p(et -

9)

1,

Va, e,`/t2 j) edt/2 btd12)- 1 db

= p (1 - e-t, e-t12

1' a ,

Y

O) b(d12)

-1 d b

(t, a, b) E (0, + oo) x [0, +O°)',

and since a

2aa

(d - a) a , P (I - e , e-1/2

Its

[

10)

e-t

pi - 2

d- 1

P22 _

its

=

2 e-112 y a

P2]

.

lib)

0

'O.

(t, a, b) E (0, +oo)s

and P0[r' (t) > 0] = I for t > 0, it appears that 05' is the differential operator based on the scale b

a
-d!2 et/2 d

A

U..

s (b) - s (a) = J.

11 a)

a

and the speed measure

0
m [a, b) = 2 fd12) -1 e-412 d E

11 b)

a

m (0) = 0

with an entrance barrier at 0 and u+ (0) = 0

12)

u E D(@*).

,p.0

Because s (+ oo) _ + oo and m (Q) < + oo, 6) can be applied to R and translating 6) into BESSEL language and noting the estimate s (b) .' 2b -(dll) eb12 (b t + oo), it develops that if h (t) t + oo with t and

if a > 0, then 13)

P. [r (t) > t -+I h (t + 1)

=

P. [r' (t) > h (et)2

1. o.,

i. 0.,

t t + oo]

t t + oo] = 0 or I

+00 OHO

according as f h (t)d a-h (9)'J2 d t converges or diverges,

or, what interests us, that 14)

Po[r(t) > JFI h(t)

i. o.,

t t +oo]

+m

_

>Z.

= r P. (r (1) E d ) PE [r (t) >'t + 1 h (t + 1)

i. o.

t t + oo]

0 00

= 0 or I according as f hd e-0/2-L' converges or diverges. NIA

When d = 1, this is KOLMOGOROV's test, and when d Z 2, it is the DVORETSKY-ERDOS test (see A. DVORETSKY and P. ERDOS [1]). 11*

5. Time changes and killing

164

The same method shows that if 0 < h(t)1.0 as t t + oo, then 15) P0[r(t) < l/i h (t) i. o., t t + oo] = 0 or I according as +00

r

ilg hl-, d:

(d

_ 2) converges or diverges.

+00

f

kd-sdi

(d >_

Because PO is unchanged by the transformation x (t) --1. t x (1 /1), it .Z'

follo.,., from 15) that if 0 < h (t) + 0 as t 4 0, then 16) PO [r (t) < ft h (t) i. o. I f 01 = 0 or I according as

f Jlghj-l a

(d = 2) converges or diverges

.Z"

0+

f hd-a

dt (d

3)

acv

(see A. DVORETSKY and P. ERDOS [1] for the case d Z 3 and F. SPIT-

ZER [1] for the case d = 2; another method leading to a cruder result will be found in problem 7.11.2). Problem 1. Give a complete proof of 15).

5. Time changes and killing 6.1 Construction of sample paths: a general view

Given a 1-dimensional diffusion, singular or not, a complete description of its generator 05 is now before us. 0 coincides on its domain

i..

with a differential operator ' of degree S 2 expressed in terms of invariants (scale, speed measure, etc.) via the formulas 4.1.8) [or 4.1.31,32,33)] and each invariant has a simple probabilistic meaning embodied in the formulas 4.1.7) [or 4.1.22, 23b, 23c, and 26)]. But the actual situation is even better, to wit, each choice of invariants specifies a differential operator CS' via the formulas 4.1.8) (or 4.1.31, 32, and 33)], and (aS' generates a non-singular (or singular) diffusion, the original invariants being expressible in terms of this

hop

end

,,!

diffusion via the formulas 4.1.7) [or 4.1.22, 23 b, 23 c, and 26)); in brief, a per/ect correspondence is obtained between differential operators (53' on the one hand and non-singular or singular diffusions on the other. The proof occupies the rest of the present chapter, but first a rough explanation of the leading ideas (time substitution, annihilation, shunt.CS

ing) in series of special but typical examples. Beginning with the simplest example, if D is a standard BROwxian motion with sample paths t - > x (t) and if a > 0 is constant, then a

$.1. Construction of sample paths: a general view

165

non-standard description of the diffusion associated with W IN = J a2 u" "''

is the scaled standard BRowNian motion x' = a x, or, what is the same in distribution, x' (t) = x (a' t) : t a 0. .fl

Now suppose 0 < a is continuous but not constant. Using the first prescription, the suggestion is that x' will be like x while it is near x' e., x' (t) = a + f a [x' (s)] d x

1)

a = x' (0) ;

0

this is the idea of K. ITO {2J. But to deal with the general non-singular U)'

diffusion, the second prescription is better; this suggests that x' is a standard BROwNian motion run with a new clock t' = t' (t) that grows like a' (6) t while x' is near 6, i. e., dt' = a2 (z') dt, 2a) or, what is the same, 2b) dt = o-'-(z') dt = a-2[x(t')] dt'. 2b) states that to is the inverse function f-' of the additive functional t

r

f (t) = f o-2 [x (s)] d s, and noting that f 9-2 (x) d s = f t (t, 6) in (d

),

0

0

°.a

where t is the standard BxowNian local time and m (d ) = 2a'2 (E) d is the speed measure of 03', it is now just a small jump to conjecture that a (non-standard) description of the diffusion on Q = R' associated with 3)

m(d)

is X* (t) = x (f-1)

4 a)

with

f = f t dm;

4b) see 5.2 for the proof.

Because the time substitution t - > f -1 depends upon the BROWNian

era.

path, it is often called a stochastic clock; the idea occurs in G. HUNT [2(2)] and a hint of it is contained in P. LEVY [3: 276]. H. TROTTER suggested the possibility to us. VOLKONSKII [1] obtained such time substitutions but did not find the local time integral for f ; see also 8.3. A second instructive example is that of the non-singular diffusion on (0, + oo) with generator 5a) 5 b)

u=

m(dl;)

m (0) (CJ' u) (0) = u* (0) ;

5. Time changes and killing

166

the method of time substitution still applies with

f = f t (t, ) m (d ) ,

6)

the time substitution of 2.11 for the reflecting Bxowxian motion being a special case.

.

0

Now let us explain how to construct the non-singular diffusion with killing associated with 7)

= u`(dE) -

k(dE)

III

on Q = R'. Suppose to begin with that k has a constant density x g 0 relative to m so that 4 = 0 - x with Q$ as in 3) ; then x* is just the motion x associated with 0, killed (i. e., sent off to the extra state co) after an exponential holding time tno, with distribution 8) P. [mao > t jx] = e-"e; .L'

this suggests that if x is continuous but not constant, then the chance that the dot particle, moving along the undotted path should not be annihilated before time t ought to be .N.

(l [1 - x[x(s)] ds] = e .Sr The time substitution 4) permits us to conclude that the local

9)

times

t (t, a) = lim

10)

boa

measure (s: a x(s) < b, s m(a, b)

s)

exist (5.4), so 9) can be expressed by means of a local time integral: i

f x[x(s)] ds

11)

= f t(t, )

0

and using this formula, the suggestion is that for a general killing measure k(db), the dot motion is identical in law to the undotted motion, annihilated (sent off to oo) at time m with conditional law

x]=e ftdm;

12)

the proof occupies 5.6. As the reader will easily believe, the relation between the motion on [0, -}- oo) with generator 5) and the motion associated with 13 a)

13 b)

". u =

u' (d ) - u (F) k (d F) m (d t)

m (0) (00 u) (0) = u* (0) - u (0) k (0)

* n is meant to suggest a continuous product.

5.2 Time changes : Q = R1

167

is just the same, except that the local time integral f t dk is used in 0-

(see 5.6; the elastic BROWNian motion of 2.3 is the simplest example of this). As an example exhibiting all the essential features of the conservative case, let (Y be the differential operator 14)

(®* u) (a) = J

't7

v,N

12)

u (a)

u(b)-u(a) =lim bja s(b) - s(a)

=0

"al

being the intervals of the complement of the standard '41

Q = U (ln)

CANTOR set K and s+ the shunt scale s+ (b) =

15)

(r,, A b - ln) + s+ ([0, b) fl K),

50 with the understanding that

(nZ1)

m^.

16a)

( 'uEC[0,11

16b)

uED(C?).

x' can be built up out of BROWNian motions on [ln, rn) with reflection at ln, hitched end to end, with a lag during the passage of the CANTOR

set so that for paths starting at 0 S a S 1, 17)

s+ ([a , b) f l K) = measure (s : x' (s) E K, s

t)

b = x' (I), t

M* .

The simplest method of doing this is to begin with a reflecting BROWNian

motion x on (0, + oo) stopped at I and to perform the time substitution t --> f -1 with

measure (s: l < x (s) < r'!

a<sRsb

s S m,,,) + s+([a, b) n K) VII

18) f (t) _

a.)

a = x(0), b = a V x(t), m,. = min (t: x = rn); the proof will be found in 5.10 and problems 5.10.1 and 2; see also 5.11 for a discussion of annihilation on the shunts. Creation instead of annihilation is introduced to obtain a sample path model of (3S' _ (1/2) D2 + x (x > 0) ; this model occupies 5.12 his 15.

I+.`

^A,

6.2. Time changes: Q = R1 Given a standard BROWNian motion on Q = R1 with scale s (db) = db and local times t = t (1, b) , let m (d b) be a speed measure on R1 and

let us prove that if f -1 is the inverse function of

i)

f(t) =f(t,w) = ft(t,b)m(db) Rl

t?0,

168

5. Time changes and killing

then [x (f -'), P.] is a (non-standard) description of the conservative diffusion with the BROwxian scale and speed measure m, i.e., with generator C`1

2)

'u = u+(db) m (d b)

Because t (t, b) E C ([0, + oo) x R') and vanishes outside minx (s) < b < max x(s) 3a)

f(tf)=f(t)<+oo

(t > 0),

1 ?0,

esp., 3b) f(0+)=0. Because t (ti, b) > t (t1, b) (t= > t1) on a set of positive speed measure

and t (+ oo, .) = + oo (see problem 2.8.7), 4 a)

f (t:) > f (t1)

and

4b)

f (+ 00) an +oo,

and it follows that f -I has the same properties, esp., x (f -') is continuous.

Now let w' : t -- x' (t), Bi , B', and Pa (B') = P. [x (f -1) E B'] be the sample paths, BOREL algebras, and probabilities for the standard description of the motion D' = [x (f -1), P.], let w-1 : t -. x-' (1) denote the sample path x (f -1) , and let us prove that D' is simple MARKOV. Given B' E B, with indicator function e (w), e (w) = e (w-') is measurable

Because of GALMARINO'S lemma (see 3.2), it is enough to show

that if

f-'(t,u) <s

5b)

x(0, u) = x(0, v)

0

VII

5a)

S.

then 1^D

6)

a (u) = e (v).

But, if 0 S t, then 5 a) implies f -' (8, u) S s, 5 b) then implies f -1 (8, u) = f -' (6, v) because f -' (0) is a MARKOV time, and the resulting VII

x(6, u-1) = x(f-'(8, u), u) = x(f-1(8, v), s) = x(9, v-') 0 S t, 7) coupled with the fact that e' (w') is the indicator of a member of Bj implies 8)

as desired.

a (u) = e' (u-') = e' (v-1) = e (v)

5.2. Time changes: Q = R1

169

But now it follows that, with B', e', and e as above,

P:[B',x'(t+s)Edb]

9)

= E. [e* (w-1), x-1 (t + s) E db]

= E.[e(w), x(f-1(t + s)) E db] = E.[e(w), x(m + f-1 (s, wm)) E db] = E.[e (w), P' (x [f -1(s , wm), wm] E d b I B..)]

m = f-10)

'-'

= E.[e(w), Px(m)(x[f-1(s)] E db)] = E.[e' (w-1), PZ (t) (x' (s) E d b)] = E: [B*, PZ.(j)(x' (s) E db)],

1'1

and that finishes the proof of the simple MARKOV character. As to the strict MARxovian character, it is now enough (see 3.6) to show that the GREEN operators

Gal = E;

10)

e-ag /(x') dt]

= E. [°'e_h/(x_1) dt] 0

10+

map C (RI) into itself (see problems 2 and 3 below for an alternative proof).

Given a < b, if u = G;/ for / E C (R1) and if in = ms, then f-'(t + f(m)) = in + f-'(1, wm),

11)

l

r1 (m)

u(a) = EaI f e al /(x-I)dtl

12)

0

+ Ea [C-m)

f e at l[x(f-1 (tw), w)] dt

((m)

ii.

= Ea [ f e-a1/(x-1) dt + Ea[e-af(m)] u(b), 0

and letting b f a in 12), P.1 iln f (m) = f (0 +) = 0J = 1 implies u (a +) = u (a) ; the proof of u (a -) = u (a) is similar.

D' _ [x(1-'), P.] is now seen to be a conservative di//usion, and using 13 a)

ma (w-1) = min (t: x (f-1) = b) = f (nib)

together with the evaluation 13b)

EE[t(manmb,rl)]=G(E,r]) VI!

5. Time changes and killing

170

(see problem 1) to compute its hitting probabilities and mean exit times 14a)

pba() = P; (M; < ma) = Pe[f(mb) < f(ma)] a < < b

= Pe (Mb < ma) = b - a 14b)

eabR) = EE(maA mb) = Ee[f(ma) Af(mb)] b

Ee [f (ma A mb)] = Ee [

t (ma A mb,

(d rl)a

b

=fG

I

t,) in (d 21)

a

it appears that its scale is a positive multiple of the Brownian scale and its speed measure the reciprocal multiple of m, as desired. Problem 1. Use the formula for the standard BROwxian local time t tao

Ee f e

t (d t, rl) =

0

a>0

2

to give a direct proof of 13 b). (Et It (ma A mb, fl)] Ill

lim mEfl f e a't(dt,17)]

m=maAmb

0

+ao

e-a't(dt,71) - e-* In f e-a't(dt, j, w;,)J

= 1imEe

e-y2ale-of-Cat e- aalz(m)-.7I 1im EE ado

J

2 2a

IL

= a Ee[jx(m) -'7i = 2 [la - qI G

- 1 -riI]

b-a +I b - nj b-a

n).]

Problem 2. Prove that if m' (w') is a MARKOV time for the standard description of x (J-1), then m (w) = f -' (m' (w-1), w) is a MARKOV time '^"

...

for the standard BrowNian motion, and that if B' E Bm.4 , i.e., if B' n (m' < t) E B' (t > 0), then B =- (w: w-1 E B') E B, i.e., B n (m< t) E B1 (t Z 0) (use GALMARINO'S theorem; see 3.2). C)'

Problem 3. Give a direct proof of the strict MARxovian character of x (f -1) using the standard description, the result of problem 2, and the method used in 9).

S.3. Time changes: Q = (0, + °°)

171

5.3. Time changes: Q = [0, +oo) Given a speed measure m (d b) on Q C RI and standard BxowNian local times t, the statement of 5.1 was that if f -1 is the inverse function

f = f t dm, then the time change t

of

J-1 sends the standard

BROWNian motion into a (non-standard) description of the diffusion D'

with the BROWNian scale and speed measure in, i.e., with generator

_ u+(db)

1)

y-0

m(db)

The case Q = RI was justified; the cases 1

f e dm = -}-oo

.+.

2a)

Q= :" 0 '+0 0 )

0+ 1

2b)

Q=[0,+oo),

m(0,1) =+oo,

fEdm<+0 Or

Q = [0, -1- oo)

2 c)

,

M (O' 1) < +00

will be treated here; the case Q = [0, 1] is left to the reader. In case 2a), 0 is neither exit nor entrance. In case 2b), 0 is exit but not entrance.

In case 2c), 0 is exit and entrance, and two possibilities have to be distinguished according as ((Y u) (0) = 0 111

3a)

or m (0) (03, u) (0) = u+ (0)

3 b)

0 < M (O) < -1- oo.

The precise statement of our result is as follows: i/ m (0) = 4- oo

4)

in case 3 a),

'a.

if

f (t) = f t (t , ) m (d ) in case 2a) and 2b)

5 a)

0+

f (t) = f t (t, ) m (d ) in case 2c) = 3)

5 b)

0-

with the convention 0 x +oo = 0, and if f -1(t) = max (s : f (s) = t)

6)

then the motion [x (f -1) , PQ : a > 01 is a (non-standard) description of

the diffusion D' with generator W. Given a BROWNian path starting in [0, -1-00), let us have a good +CO

look at f= f t dm. 0

5. Time changes and killing

172

Granting 7a)

P, [f (m0) < + 00] = 0 or 1 according as 1

Y> 0)

J' d m diverges or converges 0

'ti

and 7b)

Pr [f (m1) < + oo] = 0 or I according as

m[0,1)=+o0 or m[0,1)<+oo

(1>1 0),

the principal features off and f -1 (t) are as follows:

in case 2 a),

is continuous (1 < m0) , f (l1) < f (4) (t1 < is < m0) , f -' is continuous (t L> 0), f (f -1) = t (t ? 0), f -' < m0 f (m0) = + oo, (t z 0), and f (+ oo) = m0 (see diagram 1) ; f

in case 2 b) and 3 a) the same, except that f (m0) < + oo, f (m0+) _ + oo,

f(f-') = tnf(me), f -I(t) < mo (t < f(mo)), and f-'(t) = mo (t>f(mo)) (see diagram 2) ; coq

in case 3 b), f is continuous (1 ? 0), f (tl) < f (1$) (11 < t=) on each Bxowxian excursion x (1) : t E 8n (U ,3n = (t: xt > 0)) leading into

naI

r.'

(0, + oo), f is flat on each excursion x (t) : t E n (lJ 3n - (t : xi < 0)) 0Zl

leading into (-oo, 0), f-'(t) = f-'(t+) < +c (t (i Z 0), and x (f -1) is continuous (see diagram 3).

;? 0), f(f-') = t

7a) is proved as follows.

Given a Bxowxian path starting at l > > 0, m' = f (m,) is identical in law to the passage time to e of a diffusion on R' with the f-Yt

yf Diagram I

Diagram 2

Diagram 3

Bxowxian scale and speed measure m' = m on [e, + oo) (see 5.2), so E, [e-°`m; ] = g2 (e) where gq is the decreasing solution of m (d ) = ag (t; )

5.3. Time changes: Q = [0, + oo)

173

(e > 0), and since 1

g2(0+)

8)

according as

+ oo

f d m z +oo, 0

1

Pill Ono) =+00]=1 if f edm=+oo,

while P, [f (ino) < + oo] = I

0

l

1

if El[ f t(m0, )dm]= f Edm<+oo. 0

0 .°o

7b) is immediate from the fact that t(1, ) > 0 in the neighborhood of 0 for l > m0. Given t Z 0, s-'> 0 All

(w:f-1(t)<s)=(w:t
9)

implies that in = f -1(t) is a Markov time; also, f -1(t + s) = in + f -1(s, w 1)

10)

because, in cases 2b) and 3 a), t > f (m0) (< + oo) implies

f-1(t+s)=MO =f-1(t)=in

11 a)

(see diagram 2) and 11 b)

f -1(s, w 1) = f -1 (S, w 0) = 0,

while, in the other cases, W -1(t)) -1(t)) = t

12a)

12b)

f (s + m) = f (m) + f (S, w;0

=t+f(s,w+11) (see diagrams I and 3), and therefore 13)

f-1(t + s)

=max(0:f(0) St+s)

=m+max(0:f(0+in)
as desired.

[x(f-1), P.] can now be identified as a conservative non-singular diffusion on [0, + oo) and its (local) generator 0' can be evaluated on (0, + oo) just as in 5.2; the only novel point is the evaluation of the

5. Time changes and killing

174

generator at 0, and for this, it is enough to note that PO [f -1 = x (f -1) = 0] = 1 unless 3 b) holds, in which case DYNKIN'S formula

(@* u) (0) = l;m E.(m')-1[u(e) - u(0)]

14)

m' = max(t: x(f-1) = e)

coupled with 15)

E0(m') = E0((f(m.))

=

f o-

r-.'em(O) in case m (O) >0 = O (E)

in case m (0) = 0

implies the expected 16a)

m (0) > 0

u) (0) _ . o E0 (m') -1 [u (e) - u (0)] = m

16b)

(o)

u * (0) = 0

M (O) = 0.

6.4. Local times

Consider the non-standard description

D' = [x' =xP.]

f = f tdm Q

the general conservative non-singular diffusion on Q = R1 or [0, + oo) or [0, 1 ] with scale s (b) - s (a) = b - a, based on the standard Bxowhian motion with sample paths w : t -- x (t), local times t, and probabilities PQ(B). D' has a local time of

measure (s: z* (s) E db, s

1 a)

V (t, b) =

1 b)

t' (t, b) = t(f -1(t), b)

t)

m (d b)

at each of the non-trap points of Q; indeed, if / E C (Q) and if t is smaller than the dot passage time to the traps, i.e., if f-1(1) is smaller

5.4. Local times

195

than the standard BROWNian passage time to the traps f -' (+ oo), then 2)

JI(x.(s))ds 0

I

= f /(x (f -')) ds 0

1-1 M

=

f 1((x (s)) f (d s) 0

= lim nt+oo

/(x(k2-")) Q (k - 1) 2-"-, k2-")

Y k2-" < j-'(f)

= lim f nt+ooQ

I

/ (x (k 2-")) t ([(k - 1) 2-", k 2-'), b) m (d b)

k2 '
1-40

=f f /(x(s)) t(ds, b) m(db) Q0

= f t(f-'(t), b)1(b)m(db) Q too

because t(ds, b) is flat off the visiting set (s: x(s) = b), and thinking of the original integral as

I

1) follows.

t)1(b) ,

measure (s : x' (s) E d b, s

V can be used to express the transition densities p' (l, a, b) of 4.11: 3)

a E. [t' (t, b)] = p' (t, a, b)

(t, a, b) E (0, +co) x Q- xQ',

as will now be proved. Given points a < b of Q',

rf

+00

III

4a)

E.

1

e-,"t'(dt,a)]

0

m = E.

f e-a e t' (d t, a) + Es (e-o'm;) Eb 0

nib

E.I f e-a'

y=

4b)

a)

In

1 -E.(e a'I)E.

(e" m:)

and introducing the GREEN function +00

G'= f ec"p'dt=B-'g,g2, 0

B = gi ga - gi gs

5. Time changes and killing

176

and letting b I a, it follows that pla,

g1(b) g0(a) (b - a)-1 E, f e-91 bla

°

pro

y = 1im

5)

gl(b) - g, (a)

b-a

g:(b)

_ gl(b)

gs(b) - gs(a)

b-a

f

= B-'91 (a) g, (a) bi{ (b - a) -1 E. I B-1

=

a)

f 0

to (dt, a)J

g1 gs

=Go(a,a) 1^p

because (see problem 2.8.3) 6a)

Ea-e'a1)to(dt,a) [1(1 o

I JJ

S Ea[(1 - e

ami) to (m;, a)]

'AI

and 6b) But now

S E.[(1 - e-ams)2] VEa[t* (me, a)=J a)91I2) = o (b - a) = o (Ea [t (Tub,

Ea[t'(mb,a))=Ea[t(mb,a))=b-a. [+ 00

f e--*' to (III, b)

7)

0 o-1

= Ea (e-a m`) Eb [Y0' e t' (d t, 0

=

b)J

Ea(e-amj)

G' (b, b)

= Go (4, b),

and, 3) is immediate on inverting the LAPLACE transform and differentiating.

.>, a0)

6.6. Subordination and chain rule Given a conservative non-singular diffusion D with state interval Q, sample paths w : t -+ x (1), and probabilities P . (B) , a second conservative non-singular diffusion D' is said to be subordinate to D provided 1 a)

1 b)

2a)

its state interval Q' is part of Q, it has the same (natural) scale, i f l' = inf Q' is a shunt o l Q', then either l = inf Q is a shunt of

Q or - oo =I< 10 , 2b)

i f r' = sup Q' is a shunt o l Q', then either r = sup Q is a shun t of Q or -(- 00 = r > r*.

5.5. Subordination and chain rule

177

D' is subordinate to D if there is a time substitution t -+ J-1 map-

ping D upon D'; note that if l > - oo is neither exit nor entrance, then

l
0 < Pa[jlim x (t) = 11

3)

+00

(see problem 4.6.7), and no time substitution t -> f -1 can alter this fact or make x (f -1) behave as if l' (- 1) were a shunt. On the other hand, i f D' is subordinate

e

to D, if m' is the speed measure of D', and if e is the passage time of the sample path w:t ->x(t) to the traps of Q', then the time substitution t -i f -1 based upon f

f

t

t

t

e

e

maps D upon D', i. e., P.* (B)

5)

= P.[x(f-') E B'] aEQ', B'EB

Re)-tom

T

Diagram t b

N

G3.

as will be proved below in the special case:

Q=[l,+oo) Q'

l-> -oo l' z - 00

+ oo)

s[a,b)=s'[a,b)=b-a

b
a..

Given 1 > - oo and a sample path

Diagram 2 a

'L'

,..

w : t -+ x (t) starting at a point x (0) ? 1, t i f 1' is a trap of Q', then e < + oo and f +o. has the aspect of diagram I a or of diagram 1 b according as l' is exit or not, while, it l' is a shunt of Q' , then either f

p".

l= - ooorl> --- ooisashuntofQand has the aspect of diagram 2 a or of

diagram 2b according as l = l' or 1 > P; in both cases the identification of x (f -1) proceeds as in 5.3.

+ Diagram 2b .C3

As to the case C = - oo , if x (o) > - oo, then f has the aspect of diagram 2 a, and the same is true if x (0) =1' itA/McKean, Diffusion processes

oo is a shunt of Q' 12

5. Time changes and killing

178

0-

because then P

[mo < + oo] =1 and E

[f (mo)] = J. I j dm < + oo;

oo is a trap of Q' then f (0 +) _ + oo,

on the other hand, if x (0) = l'

x (f -1) _ - oo, and the proof is complete. Given conservative non-singular Df subordinate to D, if

tSet

f}(t)= ftdm}

6a)

at = +00

t > et "C3

as in 4), then x (f *') is identical in law to D, and using its local times it appears that if t+ =

r.+

t:e

't!

f(tj= ft+dm_

6b)

t > e,

where e is the obvious passage time of x(f ,1) , then x (f =') and x [f

.,5

f

= +00

r-+

I-..

Q_

are both identical in law to D_; this suggests the chain rule:

3

f;' (f-') aft

7)

x(0) E Q-

established below.

°a'

t

Because 0 < f-1(t) is the same as f (0) S t, 8)

f;' [f-' (t)] = max(s : f+ (s) S f _1(t))

= max (S: M. (s)] S 0, gyp y--

T Diagram 3

and so it is enough to prove that f (f +) = f _ . Given t z 0, f +1 [ f + (t)] = t unless f + is constant on a half-closed interval (tl S t < tE)

as in diagram 3; such an interval 8 corresponds to an excursion of the sample path into R' - Q+ or to its arrival at time t = tl at a trap of Q+, and subordination implies that f- is likewise constant on 8 with the result that f- [f+1 (f+ (t))] = f- (t)

9)

Because of 9), 6b) implies 10)

f [f+ (t)]

= f- [f;' (f+ (t))] = f- (t) +OU

f+ (t) s e

f+(t)>e,

and it remains to prove f- (t) = + 00

.,±

11)

in case f+ (t) > e

5.6. Killing times

179

CD-

.ti

But, since J-1 (e) z e-, it appears that e J. (c-), and hence f+(t) > e implies t > e-, which, in turn, implies f_(t) = + oo as desired.

5.6. Killing times

Consider a non-singular differential operator

u+ (a)

u+(db) - u(b)k(db) m (d b)

Jim

u (bb

0
- au (a)

no-

((33' u) (b) =

1)

with k and m finite on closed subintervals of [0, + oo) and either 2 a)

to (0) < + oo,

0 S k (0) ,

(03'u) (0) m (0) = u + (0) - u (0) k(0)

or

((33'u)(0)=xu(0)

2b)

05x<+00,

and let us show how to construct the corresponding diffusion; it will be apparent that the method used below can also be applied to the most general non-singular differential operator of 4.7. The idea is to introduce the non-singular conservative diffusion D with generator ((33u)(b)= m(db)b)

4a)

((33 u) (0) m (0) = u+ (0)

0
(d

3)

in case 2a),

in case 2b), 4b) ((53 u) (0) = 0 sample paths w : t --> x (t) E [0, + oo), BOREL algebras B, probabili-

ties P. (B), and local times measure(s: x(s)Edb, s St) m(db)

5a)

t(t ' b) =

5b)

t(t ' 0) = lim

measure(s: x(s) <e, s < m[o,

E)

b>0

in case 2a) ,

+oo, to

to adjoin to its path space an extra coordinate 0 < m extend P. to this new coordinate according to the rule

6)

tz0

P.(m,>IIB)=e-cu)

with +00

7a)

10) = f t(t, b) k(db)

in case 2a)

0-

+Go

7 b)

1(t) = f t (t, b) k (d b) + x measure (s : x (s) = 0, s S t) 0+ in case 2b), 120

5. Time changes and killing

180

and, introducing the new sample paths

<m

w°:t-sx°(t)=x(t)

8)

t

= 00

m,,,

to prove that D' = [x°, P.] is a (non-standard) description o/ the di/"-'

/usion with generator W. The construction of the elastic BROWNian motion explained in 2.3 was a special case of this prescription. Consider for the proof that D' is simple Markov, the sample paths w' : t - x' (t), killing time ma, = min (1: x' = oo) BOREL algebras B, and probabilities Pa (B') - Pa(w°E B') of its standard description. Given s 0 and d b C [0, + oo), one finds 9) Pa(B', x' (t + s) E db) ti.

0.?

=Pa(B',x'(t+s)Edb,m:,>t+s) E6 [B , x (t + s) E d b, e- I ('+')]

= Ea[B, a-t'), x(s, wi) E db, at(,.W.)] = E4[B. = E5[B,

a-1(1), EZ(q(x(s) E db, e-!(a))]

t,PX()(x(s)Edb,

s)]

= Ea[B', P .(0(x' (s) E db)]

where P;, [x' (s) E [0, + oo)] = 0 was used in the final step; this finishes the proof. Because the

GREEN operators ma,

Ga I= E; f e'"t / (x') dt

10)

0

Ova

= E.

[In,

e-"i / (x) d tl

o

J

= E. {fe-1 d f f e'"' 1(x) d tl t21

E.

(Jdt + f edf] J

= E. f f Lo

e-"` e-t f (x)

dtl J

map C (R1) into itself (use the method of 5.2.14) ), D' is a (non-singular) coo

diffusion, and for the rest, it is enough to compute its scale s', killing

5.6. Killing times

181 tin

measure k', speed measure m', and killing rate x, using the probabilistic formulas of 4.4. Given 0 S a < b < + oo, the hitting probability

11)

A. (se) = PE (mb < ma) = PE (Mb < M. A M.)

=

a<
EE [mb < ma, e-t(me)]

is continuous, and noting that

m=maAnib, a
t(m,r1)

is invariant under time substitutions, it follows from the evaluation EE [t (m, 7)] = G

a<

i7)

,

VII

( -a)(b - +))

G

i
b -a

(see problem 5.2.1) that b

f a

.(y

12)

b (.M

= EF[f (f pba(xt) r

t(dt,71))k(dy1)]

L0m

= EE I f pba (xt) f (d t)

=Jim EE I ' t'ba(xlt-a) t[(l - 1) 2-",12-")] t2 $m

= imo lZE?[l 2-" < M. f[(1- 1) 2-", 12-"), E5(ls--) [mb < ma, e-k(1011

=" lZEt[12-" < m, t[(1 - 1) 2'", 12-"), Z

mb (wl2-a) < m a

e

`(m ( --) .

+ ="Jim lG t+co '` EE[mb < ma, a-=(m), 12-" < in, of (Is-*) t [(l - 1) 2-", 12 -n)] T

= EE [nib < ma, e-1 (m) (' et(t) f(dt)

f

o

= EE[rnb < ms, 1 - e-IM)

= -a i.e., 13a)

- pba( )

b

pna(d ) = paa( ) k(d )

a < < b.

00

5. Time changes and killing

182

pab() = PE (ma < me) satisfies

a < f
k (d f)

pab (d E) = pab

13 b)

for similar reasons; esp.,

a < f < b.

0 < pb a () pab ( ) - pab () pba ( ) is a constant

14)

As to the mean exit time,

a
eab (f) = E; (ma n in; n moo)

15)

= EE (man Mb A mOO) m=maAmb -f-

= EE [ m e-<<'') + it

e-I dtJ

0 m

r

= E, I m e-I(m) - t e-11m + f e-I dt 0

L

=

EE[ f

e-ldt

1/1

is continuous, and computing as in 12), it is found that b

16)

f G ( , n) eab(n) k (dn) a

= El

V

= EE

[ff(dt) f e t(`'10t)ds

f (di) m(1Di)

m

0

o

=EE[ f e-fds f e!dfJ

11

0

OM

= EE f (1 IO

e-t) ds

= eab(') - eab(')

i. e., 17)

-

M

- [eab (d t) - eab

tab(') = EE (ma A mb),

k (d E)] _ - eab (d') = m (d e)

a < f < b.

Because of 14), 18)

s' (d') = constant x [pea (dpab (E) - pab (d') pba (')]

= constant x s (d ') ,

5.6. Killing times

183

and putting the constant = 1, 13) implies k' = k and 17) implies m = m on (0, + oo). It remains to compute k (O) and m (O) in case 2a) and x in case 2b).

But, in case 2b), PO(mom > t) = PO(m. > 1)

19)

E0[e-x!0.0)1 = e-x1'

=

as it should be, and in case 2a), the hitting probability

p,() = P! (M; < +oo)

20 a)


0

and the mean exit time eb (

0<
) = EE (mb A m;,)

A

20b)

can be computed in terms of the GREEN function IV\

- b 0 SS j7 < b

G

G

much as in 14) and 16), with the result that f G(E,rl)pe(ii) k(d,) = 1 -pb(

21 a) 21 b)

f

(0, b)

c-)

(0. b)

G

7) ee (71) k (dn) = eb

ea (e)

.

i. e.,

22b)

m' (0)

P;+(0)

..e

0=

22 a)

pe (0)

= k (0)

[eb+ (0) - eb (0) k (0)]

_ -ee (0) = m (0),

as desired.

It is apparent that the dot motion has local times 23 a) 23 b)

t' (t, b,

measure (s : x (s) E db, s

t)

m(db)

t(tAm. ,b,w).

Problem 1. Check the formula p' (t, a, b) = at Ea [t' (t, b)]

(t, a. b) E (0, + oo) X Q X Q',

Q' = [0, +00) = (0, + oo)

in case 2 a) in case 2 b)

(see 5.4 for the conservative case and 4.11 for p (1, a, b)).

5. Time changes and killing

184

[Given a > 0 and (a, b) E Q' x Q', Ea[Te_tuit t o(dl,b)] 0 /^w

f(t) -a f tdm+k

Ea[Te_'(O t(dt, b)j 0

r+ Ea !

fe-4t(f-(dt).b) '

,

-.2

and using problem 5.3.1 and the chain rule for time substitutions of 5.5, this can be identified as the GREEN function for the dot motion: VII

(a S b)

G (a, b) = B-' g, (a) g. (b)

g =gi,ga gi (0) = gl (0) [a m (0) + k (0)]

in case 2 a)

in case 2b);

g, (0) = 0

now use +00

Go =jeatp'dt.] Problem 2. Evaluate P; [m:, E di, x' E db] on (0, + co) x Q' x x Q' as i' (t, a, b) d t k (d b) using the formula of problem 1. [Given a > 0, a E Q' , and / E C [0, + oo) vanishing at 0, +00

8+.C

400

e-°'dt f p'(t,a,b)/dk 0

+00

-.l

= f Ea f e -as t(dt, b) f dk 0

0

+00

= Ea I r e-"t e-t f (d i) l (xt), 0

= E. e-ameof(Z. (mao ))

now invert the LAPLACE transform.)

Problem 3. Give a precise meaning to the statement:

jo t-1m(db) P,(m:, S t) = k(db)

b> 0.

5.6. Killing times

185

[Given 0< a < b< c, if I = (a, b] , then lim W t-1I f m (d ) PE (m S t, x (m; -) = 0) S n t-1 Pa(mo S t) m(I) = 0

in case 2b),

(to

r-.

lim t-1 fm(d) PE (m, S t, x' (m,, -) z c) µo I

S Jim t-1 Pb (mc S t) m (I) = 0, Ito 'AI

and so t-1 f m (d ) PF (m,o S t) Jim 410

I

=hint-1 Ito C

_`

a

f-1 f m (d E) f ds f p' (s. E.') k (drl) I

r

0

o+

I

=limi-1 f dsk(drt)P; [x'(s)EI] Ito 0

= k (I),

if neither a nor b is a jump of k.] Problem 4. Consider a non-singular diffusion on RI with generator u+ (d b) - u (b) k (db) CS

m(db)

(a) = u +(a)

u (b) - u (a) bya

b-a

.ti

sample paths w', killing time m, and probabilities Pa (B') , and introduce the composite motion s(D) r

where m1 +

4(t- MA_,--- MOD + m;,-1 = O if n = O and, condi-

v+0

tional on x (t) : t < mi +

Aoo

x(1)

+ m;,, the motion

xi+1(t) : t<m;,+l is identical in law to x'(t) : t<m

starting at the place l = x;, (m,, -) (see the diagram).

The problem is to show that

and to identify the composite motion as the conservative diffusion with generator (53u = t u + (d b)/m (d b) .

Diagram ,

5. Time changes and killing

186

[Consider the conservative motion with generator 6, sample paths local times t, and probabilities Ps (B) , and kill it at

m : t --a. x (t),

time mm with conditional law P.(m..> t B) = e-i(0 (f = f tdk) to obtain a (non-standard) description of the dot motion. With this model,

it is clear that

E.

n z 1;

f(dt), x(t) E dl]]

[e-<
.'y

indeed, the formula is evident for n = 1, and assuming it for n - 1, it is found that

try

E:[m;+mi+...+nil St,I(1,j] Ayr

= E , f e-t(q (1(s).-1 W - 2)! f (ds) E=(.) f e-t(e)I (xe) f (d e) to

o

[f ,-I(-) (n1(9--t - 2)1 f(ds)f re(B-''10j )I Ayr

= E.

(xe) f(d8)

t

o

t

= E.

11

t

e

f (n-2)! 1(S).-t

0

i

f(ds)f e-I(e)I (xe) f(d0) .

t

E. f e-t(s (I(O) -I! I(xe) f(d0)

[0

.

But then

lira P:[mi+ni;+...+.m;St]

pt+oo

t

I'_1

= nfm E. f e-I (n -1)! df = 0, and for the rest, it is enough to note that the conservative diffusion starts from scratch at time m..) 5.7. Feller's Brownian motions

Here are some special cases of the results of 5.3 and 5.6. Consider a reflecting BROWNian motion on [0, + oo) with sample paths w : t - x (t) , local time t (t) = lim (2e) -1 measure (s: z (s) S e, j s S t), and probabilities Pa (B) , and let CS' be 1/2 x the second den-

5.7. FELLER's BROWNlan motions

187

vative applied to u E Cs (0, + oo), subject to the conditions:

pi u (0) - p: u+ (0) + pa (0'u) (0) = 0

1)

0 sP1,P21pa

PI + p9 + p3 = 1.

If ps = 0, then 2)

x' (t) = x (1)

t < mo = min (t : x (f) = 0)

=0

m P.

is a non-standard description of the motion associated with C3', while, if p2 > 0, then a possible (non-standard) description is x' (t) =

3)

1 < M.

x(f-'(t))

I ? M""

= 00

f=1+ ,P3t1) P.(m.>tI B)=e P. If p3 = 0 < p2, then x' is the elastic BROwxian motion as described in 2.3.

Consider the case p, = 0 < p3, and let $ _ (1: x(t) = 0). Because f (to - f (tl) > t, - ti if (ti , to fl 8 + o, the clock f -' counts actual time oll g' = f ($) but runs too slow on 3', i. e., x' = x (f -') looks like a reflecting BROwNian motion oll g' but lingers too long at the barrier as it the going were a little sticky at that point. W. FELLER [4, 9] discovered that 1) is the most general (local) boundary condition that can be imposed upon the restriction of D'/2 to [0, + oo) ; it had not been noticed before him that conditions involving ((SS' u) (0) were possible.

FELLER studied the case in which the sample path is permitted to jump from the barrier back into (0, + oo). (SS' is then D2/2 applied to u E C'[0, +oo) subject to the conditions: 4)

8`O

+00

pi UM - p2 u+ (0) + p3 (05' u) (0) = . f [u (1) o+

0

- u(0)) P(dl)

5P1 ,p:,p3,p(dl)

PI +p2+p3+ f IA1p(dl)=1, o+

5. Time changes and killing

188

and the entering sample path jumps out from 0 like the germ of the onesided differential process with generator +CO

5)

p2U+(0) + f [u(l) - u(0)]p(dl); 0+

see 7.20 for an explanation of 5), and K. ITO and H. P. McKEAN, JR. [2]

for a complete picture of the sample paths associated with W'.

S. WATANABE [1] obtained similar results for stable processes. 5.8. Ikeda's example

N. IKEDA pointed out to us an interesting example of a simple MARKovian motion with continuous sample paths which is not a diffusion (see 4.1 for the simplest such example). Consider a reflecting BROwxian motion with sample paths w : t

iW -x(t-t, - c, -tl-e . iu,

t3-t"'(c

)

nib

z(t) -x(t)

i(0) - x(0)-0 Diagram 1 a

4.1

x (t) , local time t = lim (2e)-1 measure (s: x (s) S e, s S f) and probabilities P. (B), let e (n Z 0) be independent exponential holding times with conditional law P.(e > t I B) = e-t 1) n Z 1, let t-' (t) be the inverse function t-1(t) = min (s : t (s) = t), and let k be a positive number. Define new sample paths x' as in diagram 1, and note that since 2)

P. (ti > t I B) = P. (e, > k t (t) I B) = e-k t (t).

5.8. IXEDA's example

189

the excursions alternating with the black intervals on which x' = 0 are independent copies of the elastic Brownian motion corresponding to u+(0) = ku(0). D' = [x*, P.] is simple MAxxov because P.[x (t) = 0] = 0 (t > 0) ; it cannot be strict MAxxov because Eo (e-'",+) is neither 0 nor 1 [see 3.3.3 a)].

F

I I

0)-x(0)-0 Diagram t b

+w

Given

/ E B[0, +oo),

G;/ = E.[ f e-`t /(xi) dtl o

belongs

to

J

III

D = B [0 , + oo) fl C (0, + oo) f1 (u: u (0 +) exists) *, and using the

obvious Ga - G; + (a - fl) G. G; = 0 (a, P > 0), it is clear that G; applied to D is 1 : 1 and that a generator (W : D ((Y) = G1D -+ D can be introduced as in 3.7. (i' is similar to the elastic BRowxian generator; in fact, D (05') is the class of functions u E D f l C' (0, + oo) such that 3 a)

3 b)

u+ (0+) = Cu (0+) - u (0)] k

u" (0+) exists,

and 4 a)

4b)

OAS' = 2 D2

on

(0, + oo)

(OP u) (0) = u (0+) - u (o)

Diagram 2

B(o, +oo) is the class of all bounded BoREL measurable functions defined on (o,+00).

5. Time chapges and killing

190

D. B. RAY [3] has found that the state space of a simple Markovian

motion can be ramified so as to make the sample path start afresh ,,r

at each MARKOV time; the ramification for the present case is shown in diagram 2.

5'0

'"'

Problem 1. Give the detailed evaluation of W. Problem 2. Check that the motion of diagram I is not even simple MARKOV if the initial black interval in diagram I b is suppressed. 5.9. Time substitutions must come from local time integrals

III

Given a standard BROWNian motion with local times t, it is to be proved that the time substitutions t --> f -1, f = f t d m of 5.2 could not have been otherwise; to be exact, if t - r-1(t) maps the standard BROWNian motion into the conservative diffusion with the BROWNian

scale and speed measure m on R1 and if, as in the case r = f tdm, 1)

r(t)=r(s)+r(t-s,w.)

2)

(a 5r (t) < b) E Bi

3a)

O S r(t±) = r(t) < +oo

3 b)

r (O+) = 0,

tZs t?0

then

4)

r(t)=f(t)= ftdm

t?0

up to a negligible set of BROWNian paths; a similar result holds for the time substitutions of 5.3. Consider, for the proof, an additive functional r of the standard BROWNian path as in 1), 2), 3) and introduce the sample paths 5)

x° = x(t)

t < m'O

t z m". ,

= 00

with m. distributed according to the conditional law 6a)

P. (M.0 > I B)=ir(1)

tz0.

[x°, P.] is a diffusion as the reader will prove using method of 5.6, and, as such, can be expressed as a conservative diffusion x' with local times V, killed at time m:. with conditional law. 6b)

P:(mom>tI B')=e-1'(r)

r'= ft- dk, dkz0.

5.10. Shunts

991

is identical in law to [x' (t) : t < m;, m;,] , Because [x(t) : t < M., x and x' are identical in law, i.e., x' is a standard BaowNian motion

(see problem 5.6.4).

0, A E Bj,

But now x and x' can be identified, and choosing t and using 2), it follows that 10a)

E. [A , e-i fit)] = P. (A ,

E. [A ,

which implies ...

10b)

m > 1) = P. (A , m;, > t)

P.[r(t)=t'(t)= ftdk,Qt 011,

thanks to 3 a).

Now suppose that the time substitution t -+ r-' maps the standard BxowNian motion into the conservative diffusion on R' with speed measure m. Because x (r-1) has to be continuous, O J < r (t=)

(tl < 4), and

computing the mean exit time 11)

eab (E)

= Ee[min(t : x(r-1) ( (a, b))]

=

Ed [r (mm A mb)] b

= f Ee [t (ma A mb, ,I)J k (dry) a

b

= fG

a < E < b,

rj) k (drt)

a

Srj

(b

G(.r1)

as in 5.2, it follows that k = m; in brief, r = f tdm as stated. 5.10. Shunts

Consider non-overlapping subintervals Q. = [1,,, rn) of Q = [0, 1]

with scales s = s and speed measures m = m attached, let K, Q - U (t,,, r,J, let s+ be a (shunt) scale with

_

nal

I a)

0 < s+ [a, b) = s+(b) - s+ (a)

I b)

s+ U., b) = f m [ln, t) s (d)

a
b

1 c)

s+ (0, 1) < -(- oo.

in S b < rn

5. Time changes and killing

192

let D (CS') be the class of functions u E C (Q) such that, for some u' E C (Q), 2a)

f u' (ln) m (ln) = U4 (ln)

3)

f u's+(dc)= [a,b)nx+

f ud

a
(a.b)nx+

u' (1) = 0, and let (SS' be the differential operator u --> u'. Given a conservative singular diffusion on Q with generator (SS and 4)

E0(ml) < +oo Pi (mi- A m = + oo) = 1, 5 b) if In (n ? 1) are the right isolated shunts of Q, if r is the smallest shunt > ln, and if pba and el are the hitting probabilities and mean exit time P.(mb < ma) and E.(m,), then 3(5 can be identified with the differential operator 0' on putting 5a)

6)

7 b)

8)

au,

7a)

s (d ) = constant x Pba (d ) i, < a < < b < r,

,n(d) = -et(d)

In < t < r

m (l.) _ -ei (1,.) s, (d c) = -e i (d t)

0St51

(see 4.8 and 4.9). On the other hand, each such differential operator (SS' generates a conservative singular diffusion with E0 (ml) < + oo and Pi (mi A M. = + oo) = I as will now be proved; this will show that the differential operators (SS' are in 1 : 1 correspondence with the conservative singular diffusions with E0 (m1) < + co and Pi (mi A M. = + oo) = 1 as suggested in 5.1. On Qn, (5' agrees with the generator of a conservative non-singular diffusion stopped at rn with sample paths t -,. xn(t), probabilities PQ (B) (ln S a < re) , and 9) E" (m.) (in) = f [s(rn) - s()) m(d. ) Q.

= f m [1n,)

s (d )

Q.

= S. (Q,.) < + 00

(see 5.2, 5-3); on a shunt interval [a, b) C K, @' generates the motion

of translation at speed +1 in the shunt scale: 10)

5.10. Shunts

193

and the idea is to hitch these motions end to end, arranging little hesitations as the particle crosses OK+ so as to have 11)

measure (s : x (s) E K+ fl [0, 1) , s S t) = s+ ([a, b) fl K+)

a = x(0), Consider, for this purpose, a point 0

b = x(t).

1, choose sample paths

a

n21

W. : t --s X. (t) according to the law

P = x P" (x stands for direct product), nZl

let 12) to(b) _

VII

mt.Va (weX + s+([a. b) fl K.) a < b S 1,

r. E (a. bl

and, noting 13)

E(ta(b)) S E(to(1)) *

s+(K+) nZ =s+[0,1)<+oo

a
introduce the continuous sample path In S a = xn (0) < rn

t < to (rn)

X. (t) = X. (t)

_ X. (t - to (In))

t < to (rn) , A

14)

to (In)

=b

t = to (b)

=1

t Z ta(1)

a S In

aSb<1

III

as in diagram 1 below and let us prove that the motion D' with probabilities P; (B') = P (xa E B') (0 S a S 1) is simple Markov, i.e., 4(b)/t1

4(t)=1 t-ta(1)

.r,(t-is(4)

t- t"(4)

0-t (4) t- tnli,)

`'l

a z

I,

,

ti

.

Z

b/ra

Diagram I

* E is the expectation based upon P. 10fMcKean, Diffusion processes

13

5. Time changes and killing

194

that P[x.(t + s) E A I x0(0) : 0 S s] = P[xb(t) E A]

15)

b = x. (s) .

Given a < b, x. starts afresh at the passage time m.b = min (9: x. (t) = b) , i. e., x. (I + mob) = xb (t) (t 0), and using the obvious im E

16)

am

E (e-'"*+)

it follows as in 3.6 that the GREEN operators +*o

(Gal) (a) = E [

f

e-a'

l(x.) dt]

0

map C (Q) into itself. Choosing s > 0 and B E B [xo (6) : 6 S s] - B., if 17) b = ([2" x. (s)] + 1) 2-, m =mob

then x. (s) E J+ = K+ n (0, i) - U l implies "Z1

m>s

18a)

"lim in= s,

18b)

and it follows that M00

19)

E B , x. (s) E .J+, f e-a' l (x. (t + s)) d t] ="u+moxs

*E[B,x.(s)E[(k-1)2- ",k2-")flJ+,b=k2-", +oo

S < Mob, f d-10'/(X.(9 + mob))

lim F S Q*

"t + cc

dt]

E[B,x.(s)E[(k-1)2-",k2-")f1J+,b=k2-",

/

s < m.b (Ga l) (b)] = E[B, x.(s) E J+, b = k2-", (G /) (x.(s))];

in brief, 20)

P [xo(t + s) E A I B41]

=P[xb(t)EA)

b=x.(s)EJ+=K+fl[0,1) -"Z1 Ul".

and now it remains to check 15) in case xo (s) = 1 and in case xo (s) E Q".

But, if x. (s) = 1, then 21)

P[x.(t+s)=iIB.s]=1=P[xl(t)=1],

5.10. shunts

195

1<.

as desired, while, if b = xa (s) E Q., then, conditional on Bas, xa (t + s) = xn (t + s - is (ln V a)) t < to (en) - s 22) t ? to (rn) - s = x,. (t - [to (rn) - s]) is identical in law to x6 as is evident from the diagram on noting that to (ln V a) and xa (t) : t S to (ln V a) are independent of x,,,(t) (t ? 0) and that xn (t) (t ? 0) starts afresh at times s - to (1 V a) . Ir.

Because the GREEN operators of D' map C (Q) into itself, D' is a diffusion, and for the identification of its generator with the differential operator W, it suffices to note

Ss(a)

23 a) Pt(mb< ma) =P(mEb < mEo) = s(b) _ s(a) 23b)

EE (m1)

= E(te(1))

=z EEV,,,(mr.) + s+([[,1) flK+)

.

0

= s+ [e, 1)

is

'CD

and to use the formulas 6), 7), and 8). If K. contains no interval and Z s (Q,,) < -}- oo, then a second nZ1

method is available : changing scales so as to have

OSES4,

Zs+(QnfldE)=dt

25)

f(t)=27 f t(mr.,$)m(df)+s+([a,b)f1K+)

'!1

24)

`.-

n!1 if x is a BROWNian motion with reflection at 0, a conservative trap at 1, and local times t, and if f -1 is the inverse function of +4i

a
a=x(0),

b = avx(t),

then x' = x (f -1) is a (non-standard) description of the desired motion; in the special case +4O

26a)

m (d f) = 2 d f

26b)

S. (K,) = 0

26c)

f (t) = T measure X(O)
(s:x(s)EQn,s

:_5'-

f m,.),

'C7

the effect of the time substitution is to shift the black arcs of diagram 2 down on the time scale until a continuous path is achieved (see 2.41 for (L)

a special case).

DIsgnm a

13*

S. Time changes and killing

196

Problem 1. Give a complete justification of the time substitution t -+ f -1 in the manner of 5.2 and 5.3. Problem 2. Consider x' = x (f -1) in the spe(-9L ial case 26), taking for K+ the standard CANTOR set [0,1]

- (j--)

19 9 /

etc. ; the problem is to evaluate the shunt scale and Ee [e-. [s+[0,

)

E0 [e-:min(t:e(t)-1))

bX

rnA

OS S1

102

= fl 1cosh3_n./ /-r-1 n2:1

5.11. Shunts with killing

Consider non-overlapping Q = [1n, r,a C Q = [0, 1] as in 5.10 with

scales s = s,,, killing measures k = k, and speed measures m = m attached, let s+ be a (shunt) scale with 0 < S. (a, b) = s. (b) - s+ (a)

I a)

`11

ib)

a
f

b

s+[1,,,b)= fp( )-'s(dt)f p(rl)m(dil) 4-

1.

i3.

p + (d 4) = p (t) k (d f) 1,, < t < r,,,

p (r, -) = 1

p+ (1,,) = p (Qk (ln)

s+[0, 1) < +00,

i c)

let k+ be a (shunt) killing measure with jumps o l magnitude S 1

2 a)

f

b

2b)

1
k+[1., b) = Jp(i)-'s(dt) p(rl) k(drl)

.

I.-

r

k+ [0, 1) < +000

2c)

let D be the class of functions u E B(Q) such that 3 a)

u(a±)=u(a)

aEU(is,r,)

3b)

u(a+)=u(a)

aEK+fl[0,i),

n

let D ((J') be the class of functions u E D such that, for some u' E D,

f u'm(d ) = u+ (b) - u+ (a) - f uk(dt) (a. b)

4b) 5 a)

u' (IA) m

f

[a, b) A S+

1,,
(a. b)

u+

u' s+ (d e) = f

Ea, b) fl E+

u

k

u (d i) -(a. b) ffl uX+k+ (d e)

...

4 a)

a
5.11. Shunts with killing

197

u(1-) _ [1 - k+(1)] u(1) (1) = 0,

5b) 6)

and let (Y be the differential operator u -+ W. Given a singular non-conservative diffusion on Q = [0, 1] with

d.0

7a)

Eo(m1 A maa) < +oo

7b)

P1(m1-Amaa=+oo)=9. h''

its generator 0 can be identified with (SS' on introducing the hitting probabilities and mean exit time 8a) 8b)

Pab W = PI (ma < Mb)

8c)

e, ($) = EE (m1 A ma,)

a<

Pb W = Pf ON < + oo)

and putting 9a)

s (d E) = constant x [pea (d ) pab

Pb a(d i) Pba


k(d) =

9 b)

k

P.+, (d 0)

Pe+bV.)

=

- p%(i*)

P.b(".)

9c)

m (d

1 < < rrt

p.. (d E)

p..ft.)

) _ - [el (d ) - e1(E) k (d 4)]

m (ln) _ - [ei (lw) - el (1,,) k (ln)] p1(df)

10a) 10b)

s+ (d

l<
) _ - [e1(d E) -

°.,

(see 4.8 and 4.9). As in 5.10, our purpose is to prove that each differential operator (13' generates adif fusionwithEo (m1 A mom) <+ ooandP1(m1 A ma,=+ co) =1; this will complete the program suggested in 5.1.

'r'

Consider, for this purpose, the modified differential operator 0 arising from (3' upon the suppression of both killing measures, and appealing to 5.10, introduce the sample paths w: t -. x(t), BOREL algebras B, probabilities Pa (B), and local times measure (s : x(s) E db, s S t) IAA

t(t, b) =

11)

t ? 0, 1n S b <

m(db)

it Z I

of the standard description of the corresponding (conservative) motion.

P.(B) is now extended to B x B[0, -boo] according to the rule

j 12b)

t?0

P. (m,, > t I B) = e (t)

12a)

e(t) = e

t(t.E)k(dt)

x

n

(a.biflK+

[i - k,(df)] a = x(0), b = x(t),

S. Time changes and killing

198

where, as in 4.8, fl [i - k+ (d9)] is meant as a suggestive notation

(a,bins+ for e-f+((a. bin K+) (1 - x1) (1 - xs) etc., j+ is the continuous part of k+,

and the numbers 0 < x S 1 are the jumps of k+ (dE) (a < E S b) ; introducing the new sample paths 13) z' (t) = z(t) t < ma, = oo 1Zmm, it is claimed that D' = (x*, P.] is a (non-standard) description of the desired motion.

D' is simple Markov as the reader can check using the method of 5.6 and the obvious multiplication rule

tZs0.

e(t)=e(s)e(t-s,w;)

14)

Besides, 14) combined with the method of 5.2.14) implies that the GREEN operators MOO

CdE.Ir f

15)

l1

dt1

0

=E.[f e-aeE.(mm>tI B)1(x)di] = E. [Te--s e(t) f(x) dtl

p,'

Vv-,

map B(Q) into D, showing that D' is a di//usion and for the rest, it suffices to compute its invariants s', k', m', s+, k+. But, on Q,,, s' = s, k' = k and m' = m as is clear from 5.6 and 12) ; thus, s* = s+ and k' = k+ inside Q,,, and since mi = min (t : z' (t) = i pi (a) = Pa (m; < + oo) 16) o`1

= Pa(m1 < mm) = E, (e(m1))

r

-Z

al

e

t(ms.1)k(dE)1

n [1 - k+(db)] (a.ilnx+

S)

(10

f

t(m...!) k(df)Ea

fl [1k+(db)]

l)nB+

7P 11

a
E)

< + oo]# E4Va [e NV

x

[1 - k+ (d b)] (1 (a,1)nx+

a P, [t%v ..(+ oo] = t .

A.`i

(tw

r')

J

= P, v (mr+< mm) = ((a V a)

5.11. Shunts with killing

= 17 p:. Y. V a)

fl

_a
(a, 1]f1X+

199

[1 - k+(db))

(a. 1]f1 K+

a < r.

fl

[1 -k+(db)]

=fl[1-k+(db)]. (a. 1]

one has

0< a 5 1, '1I

k+(da) _,pl*

17)

k+(da)

while since 18)

ei (a) = Ea (mi A mom)

= Ea(m1 A m.o) 8

8

= Ea [Ea (moo , m,, < m, I B)] + Ea [m1 Pa (moo > m1 I B))

=EaI- f tde+m1e(m1)J 0

(L

1

= Ea l f ledt1 r

Otr.

1

r

f - u 1.)

e (ma) d me

K+

m,.

a
Ea e(mlwva)EI.va f edi 0

+ Ea (

f f

e(me) s+(db)l**

lta.1)f1$+

...f

ZP6[mLva<+oo]El.va(mr.)+ _a
JJ

Ea[e(ma)]s+(db)

ta. 1)f1 S+

`,C

f

..E

pe (a) S+ (d b) _ 1' pi. v a (a)*** f pe (l V a) t s+ (d b) + l.va (a.1)f1%+ a
r.

_ a
f

pe (a) S+ (d b)

(a.1)f1S+

1 .^O

a

* P;(m <+oo) on each non-singular interval; see 4.8.14) and the footnote attached.

** din = s+(d6) on [a,1) f l K+ ; see 5.10.12) and 5.10.14). *** pC(a) = P.(n4' <+oo). t See 4.8.18).

200

5. Time changes and killing

one has s; (d a)

19)

0S a< 1,

[ei (d a) - k; (d a) ei (a)] = s+ (d a)

completing the proof. 5.12. Creation of mass A special model for the creation of mass will be described and used

._,

to discuss a non-linear parabolic equation (see G. HUNT [2 (3)] for a different approach to creation). I Given an integral f(t) = ft (t , ) k (d ) If of standard BRowNian local times relative to a non-negative BOREL measure k (d t) on RI (k (R') > 0), consider the shower pictured in diagram 1 , where, for each k = 1 , 3, ... ,

e..

2n - 1 and each n ? 1, WkIe is a standard BROwNian path starting from scratch at the indicated ramification point, mkle S 5 + oo is subject to the conditional law (t z0), P. (Ink/2- > t I and, conditional on the ramification point, All

WkI2% and mkl2% are independent of the path

up to the ramification and also independent of all the subshowers splitting off before the ramification time. Given en = 0 or 1 (n 1), the path )U: i _+x(t 1) with 2)

to e,...

toe, ...e.-1+2-

S t < too, ...

L m.e,...e.+2-.

(n Z 1)

M;5 ft

=0

(n=0)

is standard BROwNian and the ramification times

1

are the jump times of a POISSON process A subject to the conditional law 3)

P. [/ (t)=11ro]= e-`r'l1(

j? 0, f=f(t+, w);

the proofs are left to the reader (see problem 5.6.4). In addition, the shower is strict MARKOV in the sense that if m is a MARKOV time (i.e., if the indicator of (m < 1) is a function of the shower

for times s < t), and if, up to time M, n < + oo particles have been created, then the future shower is identical in law to the superposition of n independent showers springing from the points that the n particles occupied at time t = m. Here is a grander model combining creation and annihilation. Given

o5 kn(dl) (0 5 n + 1) such that k = X kn is finite on compacts, Osn+l

5.13. A parabolic equation

201

"C1

select a standard BxowNian path w1, a ramification time m1, and an integer n, subject to the conditional law V11

tZ0,05n+1,

'et

4)

P.[mEdt,n,=nIw]=e-t(c+)in(dt)

`s"

and, in case n, = 0, stop; otherwise, select n, independent standard Bxowxian paths w, starting at 1, = x (m, , w,), to each of these attach independent ramification times m2 and integers n2 subject to the law ^..

400

PJmEdl,n,=nIw=w,],and so on.

....

Below, simple creation refers to the first model, creation and annihilation to the second. Problem 1. Compute the generator 0 of the model of creation and annihilation, viewing it as a MAxxovian motion on the space Q of points G"+

l":1=(11,11,...,ld),11<12<...<1d,n=(n1,its .... ,nd), VII

1 S n,, n,, ..., nd in the special case k," (d) = 2 d (0 S m + 1). Here l stands for the C."

positions occupied by the particles and n for the number of particles at those points. ""T

9:a

[Given u : Q - RI such that u (l") E C2 (Rd) as a function of (11,1,, ..., ld)

for each d and n, one finds

(05 u) (l") _ d u(l") +'

ni[u(in+(m-"el)

+.o

isd

-u

Osmx$1

where A is the d-dimensional LAPLACE operator, and in the sum,

e,=(1,0,.. ,0),e,=(0,1, .., 0), etc.]

5.13. A parabolic equation

Consider the model of diagram 5.12.1 (simple creation) with k(dl) = 2 I l ly d l (y > 0). Given (1, a, b) E (0, +oo ) x R2, if # (t, db) is the number of particles E db at the instant t, then 1) Ea[#(t,db)]

=

SF

n z 0 el,e,....,ew

Pa2-.< i <

x(t -

w.e,...e.+2

_" A.d 2"Eale-tn ,x(t)Edb = Ea [e1, x (i) E d b)

- e(t, a, b) db.

e (t , a, b) is the fundamental solution of au = yu, ,=D2/2+I

at

lIV

-s) E db]

S. Time changes and killing

202

in the sense that, if 0:5,' / E C (RI) and it the problem 2a)

Vo u

at

u(0+, .) _ / has a non-negative solution u for t < il, then u - f e/ (t < t,); the 2b)

proof occupies the rest of this section. Given n ? 1, the eigendifferential expansion of 4.11 can be applied +-b

to the kernel e-s*y x e" (t, a, b) : 3)

IIx(ppd.

e"(t, a, b) db = Ea[e

, m-" A m+. > t, x(t) E db]

t>0,

this justifies 4a)

at = (goe"

4b)

lim a" (t, a, b) = lim e"(1, a, b) = 0 6j-" at+" t+o

l

4d)

1> 0, bI < n

6>0

f e"(t,a,b)db=1

lim

4c)

fal,JbI
Ib-a I <e

f e"(t,a,b)db=0

6> 0;

Ib-al>o

in brie/, e" is the fundamental solution of the problem U

5a)

a = (Bog

IbI < n

5b)

u(t, fn) = 0

t> 0.

Consider now a solution v < + oo of 2) for t < tl ; 6)

if Ia < n , then

r e"(t-s,a,b)v(s,b)db= - i e'v as J

"

0,

ibl
and since e" t e (n t + co), it follows that 7)

v a f e"/db

f e/db

to

IbI<"

nt +oo, t< t,.

Because u can be expressed as

rlx(a 'd u (t, b) = Ea e d

l (xt)

+00

f-1

8)

t

jlx(e)I''de

= f g(t,a,b)/db+Ea fei-. -00

Jx(t-s)1rds/(xt)

o

+0

+00

= f g(t,a,b)/(b)db+ f ds f g(t - s, a, b) IblYu(b)db, -CO

* W is applied to a or to b.

o

-00

5.13. A parabolic equation

203

...

it is a solution of 2) as long as it is < + oo, as straight-forward differentiation shows, and combining this with 7b), u is found to be the smallest solution of 2).

Given y > 2, then e (t, a, b)

+ oo (t > 0)

because the tied

Bxowxian motions [Z (S) : s S t, Pab(B) = Pa(B I x(t) = b)]

9a)

[a+x(s) + (b - a)

9b) t3.

are identical in law, and so 10)

j

: s S t, Pool

e(2t, a, b) +00

= f e(t, a, ) e(t, , b) d + 00

= f E. -00

exp f I a 1

(

+ x (s) + (t -a) i I v d s I .(t) = 0 X X g(t, a, )e(t, E, b)d

(ii a + x (s) + ( - a) = s - constant x

? constant x E0 [TexP

IYd

,n,,

X Vde x (t) = o III

+ 00

because the inner integral explodes; esp., t1= 0 unless / = u - 0. + 00

On the other hand, i/ y < 2, then f e (t, a, b) db < + oo (t > 0) -00

because

f e(t, a, b) db 'et

11)

f I z(i)I3'da

= E. leo'

/14 +x(.)IYd. t(Ia+maxz(s))7 egg

S 2E0 [e +00

=4.f

e-+'

0

< +00,

2xt

e(lal+b)''db

5.' Time changes and lulling

204

while, if y = 2, then

a (1, a, b) < + 00 or = + 00 according as t < n/ y2 or not

12a) +00

12b) f e (t, a, b) d b < ± 00 or = + oo according as t < n/2 1 or not (see problem I below), so it remains to prove that if y S 2 and if v solves 2)

for t < tl , then v = u (t < t,) ; the method used below goes back to A. TIHON ov [1].

Given such a solution v, w = v - a is a non-negative solution of 13 a)

w(O+, ) = 0,

13 b)

and 14)

z(t, a) = f [w(s, a) + 2 f db f IcIYw(s, c) dcJds -a

t

b

+ fIw(s,-a)+2f dbflclYw(s,c)dc Jds is an even function of a and has the properties 15 a)

a z w (t , -a) + w (t, a)

= f [0-w (t, -a) + OYw(t, a)] ds 0

=,z$tZ0

1
.l]

zZ0 z(0+, .) 0. According to 6) applied to w, 15b) 15c)

t

w(t,a)= f -jesw10,ds

16)

jai
t
0

and using the bound 17)

Feat, a, ±n) S e,"Y

4e

g(t, a, ±n)

a i < n,

which the reader can easily check from 3), it follows that i

18)

w (t, a) < constant x e*"Y e-'138 f [w (s, -n) + w (s, +n)] ds 0

< constant x es"Y e- 2/38 z (t, n) .

i.'

< constant x egnV

5.13. A parabolic equation

205

Also, n

as fg(t-s,a,b)z(s,b)db

19)

+n

.a-

=- gsz+igz2l;n+ f (-Iz22+z,)gdb 0,

z(t,a)Z f g(t - s, a, b) z(s, b) db,

20)

-00 VII

and since 0 S zs (see 15 a)] 4n

z(121 ,0)? f g(6,O,b)z(3,b)db

21)

n

> constant x e-14n'It, z (4, n)

.

art

Returning to 18) with this bound in hand, it develops that for I a I < n and t < t1/3 , w (1, a) < constant x ei"Y e-'13' e121 % ,

22)

'ti

111

which tends to 0 as n t + oo provided t + 12/t1 < 1 /3 t; in brief, w - 0 for small t, and this can be propagated, proving that w 0 (t < t1) . yam.,

Problem 1 after H. TROTTER (see also R. C. CAMERON and W. T. MAR-

TIN [1]). Prove that, if y = 2 and 0 =)I t, then om"

e(t,a,b)=

In j sin 0

1<}/2

tZ n

- f2

and

f e (1, a, b) d b = }1see 0 e-al tane/V$ -00

al?

+00

=+o0 III

[e(t, a, b) = e-oo°B etc. satisfies 2), so it must be the fundamental solution ; another method is to express the fundamental solution

ea(t,a,b)db=Ea[e

,x(t)Edb

5. Time changes and killing

206

in terms of eigenvalues V. = -2-a (n + f) and eigenfunctions

of (=2D= - a b2 :

fx = e-1 - b'I2 H (} 2a b) s

ea(l , a, b)

(x+})Ie-Y8`(a sue )

H»( 2 a a) H. (V2 a b)

»ZU

fx- 2"n1IY2

- /s° e

2nsinh 2a1/y2 (see A. ERDELYI [1 (2): 194 (22))), and then to continue both sides

from a > 0 to a = -1 in the complex plane.] 5.14. Explosions

Consider the model of diagram 5.12.1 (simple creation), denote by

e the explosion time sup (t: # (t, R') < + oo), and let us check that P. (e = + oo) is constant: P. (e = + oo) - 0 or - I. Given a < b, if w is a (standard BROwxian) branch of the full shower starting at a and if m is its passage time to b, then the subshower beginning with wm is a replica (in law) of the full shower starting at b; this implies Pa (e = + oo) S Pb (e = + oo), and now the 01 law

P. (e = + oo) - 0 or I is immediate from 4 = x (mi, w}). P. (e = -+- oo) = E.[PI, (e = + oo)1 1) P. (e = + oo) is to be computed in some special cases. Given n 1, if S >t + for each (e1, es, . . ., ex_ J, then # (t, R'):5. 2)

P. [# (t, RI) > 2x"1]

2x-1 so that

P. [tielex...@*-I+$-" S t]

2x-1 P. [t2-w S t] = 2x-1 E. f e-1 l

M;-."

1

= 2x-1 E. (n

I )!

fsn_1e_srds 0

I

= 2 E [e!

(n(2V)

f sx-1 a-(& +1)Ids !

0

2

(2nfe) E E. en) ( (n-1)!

Sw-1

f (s+ I). 0

Use

Y`e-a+.)I

(n/e(1 + s))".

ds

m !!

,

f = 1(1+)

5.14. Explosions

-1E. 2

207

1/2

(ej

(2n/e)" fdt (n-1)! 1-t

5 E. (et) (!/e)" n!

-

0

E.(et)

V2nn '-.

and now the Maxxovian nature of the shower shows that if E. [et(t+)] < + oo for some t, then

1 = P. [# (t, R1) < + oo] = P. [# (21, R1) < + 00]

=P.[#(3t,R1)<+oo]=etc., or, what is the same, P. (e = + oo) - 1. Suppose now that e

t(t) = 2 f t(t, l) Ill Ydl = f 1x(s)lYds 0

R1

and let us prove that 3)

P. (e = + oo) = I or 0 according as y;5 2 or y > 2.

If y S 2, then 4)

E.[et(t)] < E.le

t (1 + msx z (s))s

.S8

J

+w

etll+(1al"M

S 2 V-2

nt db

- b-121

0

<1(y2,

<+00 which implies that P. (e = + oo) = 1. Consider for the proof that P. (e < + oo)

i for y > 2, the en-

velope e+ (t) = the smallest l such that the shower lies to the left of l up to

time t, take j < fl S 1 such that (I - j9) y > 2fl > I and then take I < a < 2#, define

l*=0 (n=0)

_l$m$nm-e

(nZ1),

and note that the event

B+ :

e+ (2 X m-4) ? l.

(n Z 1)

2$m
and the resulting e+(2 £ n-°) = + oo cannot occur unless #( I, n-`, nZ2

_ + oo, i.e., unless e S 2

/

"-a< + 00.

nZ2

R1)

nZ2

* e < +oo if and only if the shower is unbounded at t = e; the picture is that of a line storm moving out toward ±- and raining down harder and harder as it travels.

5. Time changes and killing

208

01

Our task is now to find a manageable event B_ C B. and using the law for P. (e < + oo), to conclude from P, (B-) > 0 that

P.(e<+00)=1.

Consider for this purpose a single BRowNian path w starting at In-1 = Z m-0 (n Z 2), let new BxowNian paths split off it at the

15m
[p (t) = j I B] =

e-cu)

f (t)1H I

1 ? 0,

let jnbe the number of such paths splitting off before timen-'A in

notice that the chance that none of these hit 1 before time 2n-1' is

and let B_ be the event

< E 1[Po[m((n - 1)-a + n-4) >

'.7

that some of the is paths splitting off w3 before time 2-'n in (lo. w,) hit Is before time 2 - 2-4; that, if wQ is the winner in point of time of this race to l,, then some of the j9 paths splitting off w3 = (w2*)m(,,s) before time 3 A M (11, we) hit Is before time 2 .3 -'; that if ws is the winner in point of time of this race to la, then some of the j. paths tr

13

t Diagram I

Ir.

0

splitting off w, _ (w8 )m (ti,.?) before time 4-'A m (ls, w1) hit 11 before time 2 - 4-"; etc. as in diagram 1, depicting part of the original shower starting at 11 = 1.

Bt ) B-, and P1(B4) Z P1(B-) Z (1 - E,1 [P0 (m(1 -a + 2-0) > 2-" x (1 - El, [P0 (m (2-P + 3-0) > 31t2])

5)

x etc.

is positive if 6)

E,,[P0(m(1-fi+ 2-Q) > 2-°`)]

+ E4 [P0(m (2-"+ 3-a) > 3-")''] + etc.

< + co. * in (1) = nt (1, w) is the passage time min (1: x(1) = 1) .

209

5.15. A non-linear parabolic equation

But since a < 2/3, 7) PO (m ((n - 1)-P + n1 > n a) +00

1'

[(n-1)

n

a

e-,rsr Y2nrsdt

+x-]-: a

< i

for n f + oo, and since (1 - /3) y > 2/3 > a,

1 E1

8)

_,

[2-is]

X2

,II ,II

nZ3

nys Erg

na

mZo

[t(n-°Am(l,-s))]"El.,

e

ml

nZ3


w1+


-I-F nZ3

< Z e-1 stn + Z etR 9(n-1) nZ8 nZ3

_

e

r

-}n-a

nZ3 c, e-,]nu-1)y

i-1'y

I+

e

-(n-lj ( E t a)yi2

nZ3

+ Z e_nJ(1-d)y-P

na1

nal

<'rt- oo , which, in conjunction with 6), implies P, (B_) > 0 and completes the proof.

5.15. A non-linear parabolic equation

Given a non-negative number y, let u = u(t,1) be the solution of 1a)

OU

ar

_

1b)

t>o,IERI

+-u(1 -u)Illy

1

2

(9t=

0
1 c)

it - 0 it y S 2, while it y > 2 an additional solution is u = P. (e < t), e being the explosion time of the model of 5.14 with i = I I x,y ds; jor this second solution,

0

0
2a)

2b) ItO/McKean, Diffusion processes

(1L> 0)

ut1

(It + oo) ; 14

5. Time changes and killing

210

the proof is indicated below. A. KOLMOGOROV, I. PETROVSKn, and N. PISCOUNOV [1] discussed a similar problem.

N. LEvINsoN made us the following simple proof that u m 0 if then y 2. Define v = e - 2 3)

ar

=2

MV

+ al [411 y]

+

+ [-2t + 80 12 + (1 - u) Ill" - 2(1 + P)],

and since the last bracket is SO for t S 1/3,

+f

D

da)

fvdl

-A-1

A

Sfds(j+j) [ vs(s,l)+4slv(s,l)]dl -AAal

-A

t

e}'

A

S f ds 2e-2A, 0

< n-' 10 (n t + 00).

But then u in 0(t:9 1/3) and this propagates to all later times. Suppose now that y > 2, put u = P. (e < t) and v = (1 - u)' 11 IT, and use S)

1-9

=P.(e?t) 1

= P. (mlr: z + E.

yP.

(minE dsIw1H)P5(,)(e Z 1- s)=1

+ E. [f e-t(,) [1 - u (t - s, x (s))]= d tl

= E.

0

t = E. [e-'('*] + f ds E.[e-1(s) v (1 - s, x (s))] 0 c

=et 1+ 0

to conclude that u solves i a) :

- et8u= W eia' i + f ds

at v(t - s, .)

ii.

i

0 s

= Wetv 1 -

v(t-s,.)

ids

0

u) - ids as [e14*' v(t - s, )] 2'I

6)

0

=(SS'(1-u)+v.

5.15. A non-linear parabolic equation

211

Because u = P. (e < t) < P. (m112 < t), u(0+, ) = 0, and the same bound proves that u < 1(t L>- 0) ; also, u > 0 (1 > 0) because if u (1, l) = 0 at some point (t, 1) E (0, + oo) x R1, then 6) would make

u - 0 up to time t and u m 0 would follow, contradicting lim u = tt+W P. (e < + oo) - 1 as proved in 5.14; this completes the proof of 1) and 2).

Problem 1. Consider the explosion time e for the second model of 5.12 (creation and annihilation) with kn (d t) = 2 cfl d 1, 0S c.(0;5 n + 1),

and 7 cn = 1, and let c (8) = V c 0". Check that u = P. (e < t) is n$1

fl$1

a solution of 8u

I

8=u

+[1-u-c(i-u)]

at = 2 8A

esu<1.

Problem 2. Give also a proof that if f is the time of extinction, then

u = P. (f < t) is a solution of

at =

j-p + [c (u) - ul u(0+,.)=0, 0..i;u<1. Problem S. Deduce or give a direct proof that the chance 2

u = P. (f < + oo) of ultimate extinction is a solution of [c (u) - ul = 0. Problem 4. Prove that if 0 < co X c,, then the solution of problem 4 Aa2

.+.

III

is u m 0 where 0 < 0 S 1 is a root of c (0) = 0. (1 is a root ; a second root occurs if Z n cn > 1.) na2 n c, S 1 implies c (0) = co +n I c fl 0" < I and na2 na2 C, (0) n C. 0"-1 < 1(0 < 1), flag

so 0 = 1 is the sole root of c (0) = 0 and (1 /2) u" = u --- c (u) S 0, proving that u (< 1) is constant (= 1). Contrariwise, if n c, > 1,

J2

n

then c (1) = 1 and c' (0) > 1 near 0 = I so that 0 = c (0) has a second root 0 < I. But u is still constant (= I or 0), for if 1 > u (1) > 0 and u' (1) Z 0 at some point 1 E R1, then u" (l) > 0, so

(E>1) r¢¢11+

contradicting u < 1; a similar contradiction is obtained if u' (1) S 0. is < 0 is likewise impossible.] 14'

6. Local and inverse local times

212

6. Local and inverse local times 6.1. Local and inverse local times

We now take up the fine structure of the local time t(O = t (t, 0) and its inverse function t-1 (t) for a persistent non-singular diffusion D* on an interval Q containing 0 as an inside point or as a left end point, with -u+ (0) + m (0) ((li u) (0) = 0 in the second case. A number

of the statements made below hold for transient diffusions also (see esp. 6.3, 6.5, 6.6); the necessary modifications of t-yt the proofs are left to the reader. t-' turns out to be differential and homogeneous in time just as in the standard BROwNian case (see 6.2), and the connection between it and the intervals of the complement of .3 = (t: x (t) = 0)

comes through in part (see 6.3). t can also be interpreted as a HAUSDORFF measure (see 6.5b), and the down-crossing estimate of 2.4 is unaltered t/t (see 6.5 a). t-1 is useful for computing the HAUS-

t=ay

'C7

'"'

DORFF-BES,covrTC} dimension of 3 (see 6.6 and 6.7).

Remove from the path space a negligible set of sample paths so that the following hold with no exceptions: 1)

t E C [0 , + oo)

2)

mo = min (t : x (t) = 0) < + 00

3)

t(+oo) = iimot(t) = +00

4)

t (ti) > t (t1) t (t$) = t (t,) + t (1s - t, , wig)

5)

(ti , t£) ^ S + 0

t2 > t, ? 0,

define the inverse local time 6) t-' (t) = max (s : t (s) = t) and note for future use the following simple facts: 7a)

t-1 E t,

7b)

t-1(t +) = t-' (t),

7c)

t-1(0) = MO

7d)

+1+00 t-' (t) = + 00 t-1(+ oo) = lim 13

7e)

the range of t-1 is the set of points t E 8 not isolated above.

t-' is to be split into 4 parts : 8)

t-1(t) = mo + I m (0) + t_1 (t) + t+' (t) , * See problem 4.6.6 for a discussion of persistent diffusion.

IZ0,

6.1. Local and inverse local times

213

where

t=1(t) = measure (s x (s) < 0, mo S s S t-' (t))

9a) and

t hl (t) = measure (s x (s) > 0, ma S s S t-11 (1)) 9b) are differential, independent, and independent of mo.

Given t > 0, it is clear from 5) that t-1(t) = max (s, : t (S.) = t)

10)

I = max (s, s, = MO + ss, t (mo) + t (Ss' wnl,) = t) = mo + max (ss : t (ss, wme) = t) = mp + t-1(t , win)mo

is therefore independent of t41 (t) = measure (s : x(s, ww,,)

11)

0, s S t-1 (t, wm ))

and t-1 (t) - MO = measure (s : x (s,

12)

t+1(t)

0, s S t-1 (t, w*ie))

= t (t-1 (t, wme)) m (0)

= t m (0), All

'w^

establishing the decomposition 8). Consider for the proof that t- is differential, two times is ? 11 > 0, let m = t-' (tl) , and use 5) and 13) t (M) = tl to check a'+

t-1(ts) = max (s, : t (sl) = ts)

14)

I'1

i"+

= max (s1 : Sl = m + ss, t (M) + t (S2, wm) = ts) = m + max (s9 : t (SS, w+) = t2 - t,)

=m-}-t-1 ts - tl.wns Using 14) and wo(w;,) = 0, it results that 15)

t*1

(t8)

= measure [s : x (s, w+) < 0, mo S s S m]

+measure [s:x(s,w+)<0, mS sS m+t-1(12wm)] = t;1(tl) + measure [s : x (s, wm) > 0, s < t-1 (is - tl , wm)] = t*1(t,) + t-1102 - tl, wm)

But 16)

(w:m<s)=(w:1,
sZ0,

6. Local and inverse local times

214

so that m is MARKOV. Because of 17)

f= t-1, t+1, t+1 - 0,

C Bm+

B (f (t) : i S

000

it follows that 18)

12Z t1

P.(f (ts) - f (tl) S S I Bm+)

t=t2-tl

=P.(f(t,wm)SsIBm+) = P=(m)(f (t) S S) = P0(f(1) S S),

proving that t_1, t+1, and t?1 - t -I are differential, and, to complete the proof, it is enough to note from 9) that t_1 and t+1 are sums of jumps

and to conclude from the differential character of t+l - t_1 that the sum (t_1) of its negative jumps is independent of the sum (t+1) of its positive jumps. Problem 1. Give new proofs of the results of the present section using time changes. [Given a standard BROwxian motion in the natural scale with local times t, if t-1 is its inverse local time at 0, then o-

f t (t-1(t)°I , E) in (d

f t (t-1 (t), E) m (d ),

),

and f t (t-1(t) , t) in (d E) o+

are identical in law to the inverse local times v -1, t' *1 for the diffusion D'

with speed measure m; now check the independence and differential property using problem 2.8.4.] 6.2. Levy meseares We will now check the LEvy formulas: 1 a)

E0 (e--t_1(t)) = 6-411-MINI(O) +00

-t f (1-s `1)n_(dt) 0+

n-(dl) = li m s(-e, 0] -1 P_1(mo E dl), I b)

Eo(i .t+l(()) = e+tp; (o)1V.(O) +m

If

= e 0+ n+ (dl)

(1-e °7)n+(dt)

l o s[0, e)-1P.(moE dl),

and 2)

Eo(e-at 1(1))

= e-1l00, 0) 11

=e

-t ,(0).+ f (1-s °')n(dl)1 0+

n(dl) = n- (dl) + n. (dl).

6.2. Lfrvv measures

215

where G = g, g, is the GREEN function for D and m (0) is the jump of the speed measure at 0. A part of the above can be got from 4.10, but we prefer to derive it all anew. Because of

-

G(0, 0) -1 = g; (0)

3)

=

g9 (o)

g.(o)

g, (0)

g (o) + m(0) a - gi (o) go(o)

gs(o)

and the fact that t_1 and t+1 are independent, it is enough to prove I b).

Given I > 0, P. (mo S l) is convex in the scale s (e) fore > 0 (see problem 4.4.2), so s[0, E)-1 [P.(m0 S i) - 1] _ -s [0, e)-1 P. (mo > 1)

4)

n+(1)<0

e+0,

and it results from 4) that 5)

gi (0) /g: (0) =1i s [0, e) -1 [g2 (e) - g: (0)] gt (0)

=lims[0, e)

1 f e-01 P. (mo E dl) - 1J 0+00

+00

=lims[0[`of P.(mo>1)d(e'ar-1)J +00

f n+(1) d(e-°' -

1)

0

+00

f (1

-

e-at) n+(d1),

0

where, for the justification of the final step, we use +00

6)

0 -- lim it +00 n+ (l) ? f n+ (1) d (1 - e-°) 0

ga (0) /92 (0)

=

g, (f)

g2(o)

_ a i g.() m (d ) J

(0..1

g.(o)

> s[0, e)-1 g2(e) -g1(o) - a m(0, e) g,(o)

= s (0, e) -1 [E. (e-a m.) - I] - a m (0, e] -+0

a4.0

and

. 7)

0 ? e n+ (e)

e f n. (1) e-' dl > - oo. 0

6. Local and inverse local times

216

Coming to t+1, the time change t -> J-1(f) based on the local time integral f (t) = f t (t, e) m (d ) takes D into the diffusion D+ on [0, + oo) o+

with invariants

m+ =M

s+ = s,

8)

on

(0, +oo),

m+ (0) = 0

and since, for t> 0, f grows just to the left of 0 < s = t-1(t), one finds

i.e., f (0) < j (s) for

III

t+1(t) = W-110))

9)

= min (s : s

(t-1y))) A11

= min (s : t-1 (S)

t-11 W)

= min (s t (f -1(s)) ? t), .+a ova

proving that t+' (t) is identical (in law) to the inverse of the local time t+ (t) = t+ (l, 0) for D+ .

But now, taking y > 0 and using the GREEN function expansion of problem 5.6.1, one finds 10)

f

r +cc e-'3'8 di

r e-Y, ea'+ () dt

Eo(e--a t+'(t)) = Eo

[

+00

= Eo If

e-a' e-yt+(t) t+ (dl)]

0

= G_(0, 0),

where G_ is the GREEN function for the diffusion D_ on (0, +oo) with invariants i1)

s_ = s+,

k_ (0) = y, and since G_ = gs ga with 12)

m_ = m+, k_ (0, + oo) = 0,

g(d) = m(dj)

( > 0)

a ga W

gi (0) = y ga (0) g3+1 (0) ga (o) - ga (0) gs+ (0) = 1

I-'

it follows that 13)

gt(o)

G- (0, 0) = ga (0) ga (0) =

A(o) go(o)

yg:( )

g# (o)

1

Y

completing the proof of I b) apart from undoing a LAPLACE transform.

6.2. Lavv measures

217

Not all non-negative mass distributions n+ (dl) with f l A 1 d n+ < + 00

can arise as LEVY measures of inverse local times; in fact, it is neces-

sary that

0+ n+ddi)

14)

= fe'/z2(d,)

0 s fYY (dy) ,

-00 as will now be proved.

Consider the generator @' for the original diffusion starting from [0, +oo) and stopped at 0, write G = gl g! for its GREEN function, and 15)

y50, a,bz0,

e (y, a) f (dy) e (y, b)

for the eigen-differentials of (' where e = (e1, e2) is the solution of 16)

(;e=ye e(0)=(1,0),

a+(0)=(0,1)

and f is a BOREL measure on (- oo, 0) to 2 x 2 matrices with entries f11, /i t, /21, f!, such that 17)

/12 =

0 S /111 /!Y,

/2 1 .

fl l

/1Y = 1221

f!!.

as in 4.11. Owing to g, (0) = 0,

18)

gl (0) g! (0) = gi (0) g! (0) - gl (0) gi (0) = 1

cad

and so 19)

G. (o t, 0, e) = gi (0) g! (e) = g! (e) /g1(0) +00

= f e-"IP,(m0E dl)

> 0.

0

Inverting the transform, 19) goes over into 20)

P,(m0 E dl) = J e1Y a+(y, 0) f (dy) e(y, e) dl

e> 0,

-00

.+.

and, computing the gradient ate = 0, it follows from the definition of n+ [see I b)] that 0+

0+

21)

f e'Y a (y, 0) f(dy) e+(y. 0) = felV/11(dv)

1> 0.

-00

-co

It is an open problem to describe the class of non-negative mass distributions / (dy) on (- oo, 01 such that n(dl) = dl x f e"Y f (dy) -00

is the LEvy measure of the inverse local time of a diffusion. Problem 1. Check that

n+[l,+-0)<

-g;(;1,0)

g!(I- , o)

6. Local and inverse local times

218 and that

n [l, + oo) < G (1-', 0, O)-'. 0

[n+ [1, + oo) = f 1Y I-' e1 f (d7)

with 0S f (d7)

i.e., n+ V. + °D)

-00

is convex, so +00

- 92(a, 0) =a r le-"cn+[l, +oo)dl g:(a, OF 0 +00

> n+ [a f 1 e ' dl, + oo) = n+ [a-', + oo) 0

the proof of the second part is similar.] Problem 2. Check that n+ (0, + co) = n (0, + oo) = + oo +00

[n (0, + oo) = Jim

f [1 - e-" ] n (dl) = "line I

+00 f+GO n

f e-": p (1. 0, 0) d t

\n

t

= +00.] 6.3. t and the intervals of [0, + 30) -8 We saw in 2.2.4) that in the case of the standard Bxowxian motion,

if 8 = (t : x(t) = 0), then ...

i) l o V

2 x the number of intervals o t [0, t] - 8 o t length ? e= t(t) tL> 0, +00

in which t is Bxowxian local time at 0 and 2/x e =

r ,1

dl y2n 1'

is eke

,.r

^,w

the expected number of jumps of t-' of length Z e per unit time. n(dl) = (2n13)-112dl is the LEvy measure for t-'. With the general n [e, + oo) in place of V 21x c, 1) is still true : in fact, -0-

letting .3 - designate an (open) interval of (t : x (t) < 0), 8 + an interval of (t: x (t) > 0), and n-, n+ , n the LEvy measures for t_', t', and t-', #(8-: 8- C [0, t), I8-1 La E)

2 a)

P. [lim

= t(t), t z

0J

=1

2 b)

P. [lim # (8+: 8'n+C [o, +0, 18+1 L> e) = t (t) , t L>-

OJ

=1

8j0

n_[E,+00)

!o

and

po[lio #(8* :8*C[o,t) 18*1?O =t(t),tz 01 = 1. ...

3)

Consider, for the proof of 2b), the counting function 4) # (t, e) = the number of jumps o/ magnitude ? e of tom' up to time t,

6.3. t and the intervals of [0, + oo) - 8

219

and keeping in mind that n+ [e, + oo) is continuous (e > 0) and that n+ (0, + oo) = + oo, let n+1 (s) = min (t : n+ [t, + oo) S s).

5)

[# (t, n+1 (s)) : s Z 0, P0] is differential and POISSON distributed. Because

E0[# (t, n+1(s))] = t n+ [n+1 (s), + oo) = t s, it is homogeneous in its temporal parameters, and thanks to the strong law of large numbers, 6)

P.(li1840 m

7)

# (t , E)

n+[E, + oo)

-

= Jim # (t, SKIM) - t, t:>- 0 = ! . J

000

But # (t (t), c) = #(8+: 8+C [0, t], 18+1 ? C) if t-1 (t (t), 0) = t, i. e., if t E is not isolated from above (in 8), and since # ($+ : $+ C [0, t), 18+ 1 z e) and t(t) are increasing in t, since .8 ,,a

;:n

8)

is closed and perfect, and since t is continuous and flat outside 8, the strong law PO [Jim # (8+ : 8 C [o, t) I 8+I ? E) .-.

9)

= t (t), t-1(t) =I, t Z 0J = 1,

acs

which is clear from 7), implies the desired 2b). The proofs of 2a) and 3) are similar. We also saw in 2.2.6) that for the standard BROwxian motion, 10)

1`m

iE x the total length of the intervals of [0, t] - 8 t> 0, o t length < e= t (t)

in which V2e/n is the expected sum of the jumps of t-1 of magnitude <e per unit time. A natural conjecture is 1i)

Ii m

(f l n+ (dl)1 -1 x the total length of the inkrvals .+ C [0, t] ti,

0

o l length < e= t (t) i.e., substituting t-1 (1) in place of t and n+1 (s) in place of e, '1 -1 (t) ri+l (1) 12)

Jim

+t+=

f l n+ (dl)

t Z 0,

f 1# (t, dl)

E Jim

ftiao +OD f n-+1(O)dO

=t

t?0,

6. Local and inverse local times

220

in which the i) are the jump times of the (homogeneous) Poisson process # (1, n-.1 (s)) : s ? 0.

Because of strong law of large numbers, ....

lim n-1 S. = t-1,

13)

at OD

and it is a simple matter to check 12) from 13) if n+1 (s) is a power of s

times a function f (s) such that f (t s) - f (s) as s f + oo for each t > 0 (see 6.7.4) for the special case: ds = d E, dm = 2 JE 10 d t (fi > -1), do = constant x l-0+°> dl, a = (fi + 2)-'). But 11) is false in general as the counter example of the next section proves. 6.4. A counter example: t and the intervals of [0, +oo) - 8. 6.3.11) is false in general: for example, it is possible to have 0 < cl < e n+' (s) < c= < -I- oo as s t + oo and then 6.3.11) would force lim sup e'* X e-" S c,/c, < + oo, violating the fact that the law of at+oo kZn e'" X e-'+ is independent of n and has as its support the whole of [1, + oo) .

kan Consider for the construction of such n+ , the function y, (t) _

(1/2) f OF (s + 1)-1 ds and its inverse function 0 and let CS be the 0

generator with the natural scale and the speed measure dm = (1/4) (01'dE on (0, + oo).

Consider the increasing and decreasing solutions g, and g= of (si g = a g with g , g, - g1 g+, = 1, gl (0) = 1, and gi (0) = 0. +m

gs = g, () f g1E drl satisfies g$ < 0, 05 g3 = a g,, and g8 (0) =

= gE (0) _ -1, so that g= = g=, or what interests us, +00

..r

i)

g:(0) = f gi-2dE. 0

Given t > 0, 0 < ip'(t) E l so that 0 < 0'(1) E t and 0"(t) > 0. Now is convex so that Ig 0 (t) is concave and 0" (0') -2 < 0-1. Besides y,' (0+) = + oo so that 0'(0+) = 0, and putting g, (0) = g, (E), it follows from 0 g, = a g, and the facts just cited that y, (et)

+.+

2)

g4" (0) + 0" (0') -' g4, (0) = 4 g. (0)

g, (0+) = I g (E) i g(0+) =1im 40 W(e)

lim4a g, (e) O' (r) = 0 .{o

6.4. A counter example: t and the intervals of (0, +0o) - 8

221

and

0> o.

g4< 91
3)

But then 4)

10 (ir 0/2) < g4(0) < e}Vae

and i) implies 5)

+00

f

+00

0

0

I0(2 )ao)-2d +00

<

-} f I0(2 )a 0)-$dE

1

ig /ya

<

+00

lga y._

+J eVya°d

a

,

0

where y t I as a t + oo and is independent of eE 10 y2n as t + oo). Working out the estimates with the aid of +00

+00

1

fe-Y0d = f e-Va8yp'(t) dt =

6)

(use the fact that

0

0

2

f acI2 ds 0

1 - a-'/2

lg a

one finds 1 - a-1j2 < lga

7)

g2 (0)

< 1 - (y

a)-112

a

1gy a

t + 00 ,

and thanks to +W

8)

g2 (0) -' _ [g1(0) gs (0)]

' = f (1 -eat) n+

(dl)

0 +CO

= a f e-al n+ [1, + oo) dl, 0

it follows that

+00

9)

1g y a

9 -
1 - (y 00-11 11

19 °`

a t +00.

1 - a'1M

0

But n+ (dl)/dl is the LAPLACE transform of a (positive) spectral 0+

measure / (d y). n+ [1, + oo) = f ly 1-1 ed" / (d y) is therefore convex, and -00

6. Local and inverse local times

222

the conclusion is that +0D

lg a

10)

a-ils

> a f e-at n+ [1, + oo) dl > n+ [a f e at l dl, + oo) 0

0

1

On the other hand, since now

g

n+ [1, + oo) is positive and

1

1-

1 ma-.

since n+ [1, + co) and 11g 1 l are both decreasing for small 1, 11

2e'1 [

i

1a a -1/2

- n.

1

f

L 2a '

+00/

i/a

< e Jf A

l_'gI

,

1/2a

l - n+

[1.

+00)1 dl

+00

A

< e 13 -af e-al [(gl) (1 + 2 ni +n+[l, +oo)] dl 0 +OD

A

+

< e-mIS + lga - f e-i lgl dl +00

+

0

r e-1 ill dl

0;

+00

-

r e4 Viigl dl vat

0

1

lgy x

0

+OD

A

< constant X V


re

J

a

lgl dl

0

at+0o,

12)

O_'

and therefore

1lgl1 -4<

lg21 -v1T2

-3e
proving the desired estimate 13) c, a-' < n+1 (s) < cs e-' with c, = e-4 and c2 = e

140,

st+00

6.6a. t and downerossings Because the statement 1) lim s [0, e) x the number of times x (s) : s < t crosses down from e to 0 840 = t y)

is correct for the standard Bxowxian motion (see 2.4) and invariant under the time substitution t f -1 [see 5.4.1 b)], it applies to the present case also.

6.5d. Excursions

223

6.6b. t as Hausdorif measure

Given t=> t1, if 6- and

x(s) = 0: tl S s S is

are the first and last roots of (a- = A+ = 0 if no such roots occur),

and if t(t) is local time at 0, then much as in 2.5, x (s) : s ? t,] E0 [t (t=) - t (tl) x (s) : s S tI, 1) = Eo [t (t) I x (t) = 0]

Jp(o. 0, 0)p(t - 0,0, o)dO

p(t,0.0) 6 h (t) e'+

where P (t, E, rj) is the transition kernel of 4.11, and this implies (compare 2.5) that

urn V h (diam $ n [(i -1) 2-" t, i 2-" t]) = t (t)

2)

tssw

t L> 0,

where = (I: z(t) = 0); the proof is left to the reader as a problem. t is therefore something like HAUSDORFF h-measure cut down to 8* ;

it is plausible that h (0+) = 0 and that h (t) E f for small t but the proof eludes us.

f3.

.t7

.ate

ago

6.6c. t as diffusion D. B. RAY [4] proved that for a class of MARKOV times e, including independent exponential holding times and passage times, the conditional local times [t (e, ) : E Q, Pap (B) = P. (B I x (e) = b)] can be described in terms of the 2 and 4-dimensional BESSEL processes as in 2.8. t (m, , t) is invariant under time substitutions, so the passage time case is as before, and the description of t (e, t) for an exponential hold-

ing time e with conditional law P. (e > t I B) = e_c is not changed except that the end points of the state interval come in, and for the scaling of the BESSEL processes, the solutions of 05 g = g take the place

of the ef1 used in 2.8. 6.6 d. Excursions

O(3.

C3.

H. P. MCKEAN [5] described the excursions [x (t) : t E ,S", Po] (n ? 1) corresponding to the open intervals of the complement of 3 = (t : x = 0) in case Q = [0, + oo) and u' (0) = 0, u E D ((Si) ; the standard BROwNian scaling is not available, but otherwise the description of 2.9 and 2.10 is unchanged, except that the 3-dimensional BESSEL process is replaced by the diffusion associated with OY = h-1 (3S h, h = h (y, ) being the See note 2.5.2 for the definition of HAUSDORFF measure.

6. Local and inverse local times

224

solution of

Qh=yh, h(0)=0, h'(0)=1

1)

with

y = yi = the greatest negative root o/ Jr (y, + oo) = 0

2a) or

y=0

2b)

according as + oo is entrance or not. (M' can also be described as the generator of the original motion conditioned so as not to land at 0 at a positive time. 6.6. Dimension numbers 1 We use the inverse local time t-I to prove that the dimension number

y = dim8 of 8 = (t: x = 0) is constant; for example, dim,$ = 1/2 for the standard BROwxian motion as we saw in 2.5.

w.:

Rya

Consider, for the proof, a partition £1 of [0, 1] into a finite number

of non-overlapping intervals Qj : i S n with rational endpoints, let IZI = ax jQ; I, and let e { = 0 or I according as 3 fl Q{ = 0 or not (i:9 ft). Given 0S a:5.- 1, it is clear that Z e;jQ{ j` is a BOREL function ism of the sample path x (t) : t S I. Because the class of possible partitions £1 is countable,

A ($ fl [0, 1]) =

1)

840

ee I WO

0.0

014

is also BOREL, and recalling the definition of the dimension number, it 6'1

results that dim (5 fl [0, 1]), and with it

y=dim(8) = sup dim($f1[n-1,x)), Mki

is a BOREL function of the path. Given m (O) > 0, measure [8 fl [0, t-I ft))] = measure (s: x (s) = 0, s S t-I (t)) ,c°,

2)

= t (t-I (t)) M (O) =I M (O) > 0

dip

for each 1 > 0, so that y = 1. Consider now the case m (O) = 0 and let 8 stand for the range (t-I (t) : t > 0). 3) %9 and 8 - 9B is countable, so that dim (8) = y I See note 2.5.2 for the definition of numbers.

dimension

6.7. Comparison tests

225

is BoREL, and it follows from the properties of the POISSON integral +Cc

f l p ([0, t] x dl) for t-1 that y = dim (9) is measurable over 0

fl B[p (dt x dl) : t Z 0, l S n-lj and so is constant, thanks to the

n21 differential character of p and the KOLMOGOROV 01 law.

Problem 1. y = dim 8 = dim 8 n [0, e) for each e > 0 for paths starting at 0. 6.7. Comparison tests C7'

Consider y = dim8 as a function of the speed measure m. Because the time substitution t -. e I maps t-1 (t) -. E-1 t-1 (e t) and 3 -+ e-1 $,

it is clear that dim ,3 = dim (t-1(1) : i S 1) = dime-' = dim (t-1(e i) :1 VII

1)

1)

= li m dim 3 fl [0, t-1e)] ,

and since t-1(e)

'ti

mo as e 4 0, y is a local /unction of m, i.e., it depends only on the behavior of m in the immediate neighborhood of 0. Given a pair of diffusions D and D' with speed measures m and

m' g, m, it is to be proved that y'

The proof is not hard: in

y.

fact, using the time substitution

t - f -1(t) (f = f t d m') mapping x into x' = x (J-'), it is clear that J-1 maps 8' = (t: x' = 0) into $ _ (1: x = 0), and since f (f-1) = t, it follows that 8' C f (,3). Using 2)

S f t(dt, )m(d ) =dt

f (dl) = f t(dt, )m'

and the result of 0. FROSTMAN cited in the note placed at the end of this section, it follows that 3)

dim,3' S dim f (,3) d

sup (a

elfin o

,l e(f(8))-1 1(8)xf(8)

S sup ra : inf e

O

r

J

c(8)-1 8x8

S sup (a: eino

a (l aa, eb b) < + oo)

e(da)e(db) 11(a) - f (b) la

1

< +00 l

f e'daa)e(db) < +oo)

e(8)-1 8x8 = dim (8). completing the proof. 1tA/McKean, Diffusion processes

15

6. Local and inverse local times

226

CA9

VII

At no extra cost, it is found that y' S y it m' - m near 0. V1 and y will be computed for the special case:

P> -i:

Q=R1,

.;.

4)

''^

the result is that t-1 is stable with characteristic exponent y = dim 8 _ = (f + 2)-1, and this stock of dimension numbers will lead to a comparison test, permitting us to read off the dimension number of the zeros of most diffusions encountered in practice. 'C"`

Given 9 > -1 and a > 0, the solution ga = ga(a, ) of 5)

a g1 (() = 4 1 t I

-1

ga (t)

ga E 1, ga (+ oo) = O, gs (0) = 1,

is invariant under the substitution

- /cy

a - C X,

6)

Y = ( + 2)-1, c > 0;

i. e., 7)

ga (a,

) = g2(1, a''

),

and therefore

(dl) = 2 f (1 - e'°`) n+ (dl) `e7

.'!

f (1

8)

0

0

-2 g2+ (a,0)

2a yga(1,0)

g1(a, 0)

a

ga(1, 0)

- 0,

`art

all

ago

E0(

is - ti-°°t(ds) t(di)

!

/1

= Eai f

t-1(1)) 1

f

11

.°i

8n10,

+3g

(Y'

VII

9)

op,

proving that t-1 is stable with characteristic exponent y = ( + 2)-1. Computing p (t) = p (t, 0, 0) = constant x ty-1 from 8), we find that the HAUSDORFF function h = p-1 p ®p of 6.5 b is a constant multiple of P. dim 8 S y is now clear from 6.5 b.2), and since

It-'(s) -t-1(1)1-°dsdtI 1

+00

1

= constant f ds f di f o

Eo(e-0t-1(It-aI))

0

o 1

0-1 d9

1

+oo

= constant f d s f d t f O-1 d9 e-CO-t--t 1 t -, I ey o

0 1

0

1

= constant f f Is - t1a"y ds dt

<+00

00

a
FROSTMAN'S lemma implies

if

dim 8 = sup ra : in tt

Co.

10)

(

e(8))-15xJ8

and the proof is complete.

ei aa) 6(d b) < }-

? y,

6.7. Comparison tests

227

aid

Our comparison test now reads as follows : if s (d

) = d , it fl > -1,

C1.

VIII

',e

and if m (d .) Z>= (;5) constant x 1 I" d E near = 0, then y = dim 8 z (S) (fi + 2)-1; the proof is clear. Additional information on the computation of dimension numbers can be found in BLUMENTHAL and GETOOR [1].

Problem 1. Can y = 0 serve as the dimension number of the visiting set 3 of a diffusion? [Given s(d) = d. and m (d E) = e-IEI-' d , m (d E) S i d near = 0 for each fi > -1, proving y inf (fl + 2) -1 = 0.] v>-1

Note 1. Dimension numbers and fractional dimensional capacities. We need the result of O. FROSTMAN [1], that for compact BC [0, + oo),

l-

dim (B) = sup (fi: inf f e aa) eb db) < + co) , 0

1)

where e runs over the class of non-negative Borel measures of total mass

+1 loaded up on B. Here is the proof.

Given ft > 0 such that + oo > f la - b I -a compact Q C B such that e (Q) > 0 and

e (d a) a (d b) , select

l l a- b -O e (d b) 5 x<+ oo

(a E Q), and cover Q with closed intervals Q. (n Z 1). Because

.-.

0<e(Q)S Ee(QfQ.)

2)

nZ1

V IQ"Ilinf OEQnQ

nZ1

f

l a- bl-ae(db)

QnQ.

S I IQnlax it is immediate that

AP(B) ? A (Q) ? e(Q)Ix> 0,

3)

proving

dim (B) Z fJ,

4)

and to complete the proof, it is enough to check that "..I

+oo>inf f Ia-bl-Pe(da)e(db)

for

fl
Given fi < a < dim (B) so that A ° (B) = + oo, A. S. BESICOVITCH [2]

showed that it is possible to select compact Q1 C B with 5)

0< Aa(Ql)<+oo .

Next, according to BESICOVITCH [1], it is possible to select compact Q2 C Q1 with 6)

0 < A (Q,) < + oo 15

6. Local and inverse local times

228

and

li r (2e) -a A* (Q2 fl (a - e, a + e)) S I

7)

a E Q,.

Now select compact Q3 C Qs such that

0 < A (Q3) < + oo

8)

and

(2e)-aA°`(Qsfl(a-e,a+e))<2

9)

a<6, aEQs

for suitable 6 > 0 and define e(db) = ,1s(Qs) -1 A°(Qs fl db);

10)

then 11)

f e(da) e(db)

J

Ia - bjP

BxB

f e(d4)

f

e(dl)

e(db)

+

L-bI6 s

l

2-'ASIs-bI<2-.+Id

VII

fe(da) [6-B +xZ (2'" a)-P 2(2.2-a+18)'1 A &(Q3)-1 l

Q,

= 6-P+ 22+6&-P X 2-n(a-d) A

a(Qa)-1

n21

< +00, completing the proof.

6.8. An individual ergodie theorem 1

Consider a persistent diffusion D with state interval Q, scale s, speed measure m, and local times t, and define the local time integral (additive functional) e(t) = f t(t, e) e(d¢)

1)

for any non-negative measure e, and let us prove that 2)

P. [Jim e'(t)

e.(Q)1 =

1

in case

0 < e:(Q) < + oo.

C. DERMAN [1] proved a special case of 2) :

f /dnl

f /[x(s)]ds

P. lim

a.'

3)

G

t+ f g[x(s)] ds

=

Q

f dm

=1

0

1 7.17 contains similar results for the 2-dimensional BROWNian motion.

6.8. An individual ergodic theorem

229

for the standard Bxowxian motion by a method of W. DOBLIN [1) used below; see also M. MoToo and H. WATANABE [1]. f<9

Given a < b, let ml < m2 < etc. be the successive passage times to a via b, i. e., m" = m" -' + M (W", -1) m° = ma, m = mb+

4)

Because the excursions x (t) : m"-1 S I:< m" (n Z 1) are independent

and identical in law, so are the increments of e:

(n z 1),

e" = e(m") - e(m"-')

5)

Using the strong law of large numbers and the evaluation of problem 1:

Ea[t(m, )] = s(b) - s(a),

6)

one obtains

P. I[ lim

7)

nf+oo

a (m")

n

_ [s (b) - s (a)] a (Q), = 1,

e., 8)

P. [,lim

[s (b) - s (a)] a (Q)

t+oo min(n : m" ? I)

and 2) follows. Besides 3), there are several interesting special cases of 2) : 9)

measure (s:x(s)EA,sSt)

lim t t+oo measure (s : x(s) E B, s S t)

0 < m(B) < +oo,

A, BE B(Q).

lim e(t) tt+ao t(t, a) = e(Q)

11)

t(t, a) = 1 it+oo t(t, b) I-0

10)

1im

e (t)

12)

tt+.

- m(A) m(B)

_

I

e(Q) m(Q)

aEQ t(tta) = m(Q) it (in case m (Q) = + oo, 12) and 13) are to be understood with the convention I/+ co = 0). If D is a standard Brownian motion, then 13)

b

'[

5b

re-0,2 do

V n2 0

b? 0,0<e(R')<+oo; N..

lim P

tt+oo

VII

14)

2e(t) e(RI)Yt

6. Local and inverse local times

230

in the special case e (d law (1]:

d , this becomes the KALLIANPUR-ROBBINS

_

t

lime

15)

It+00

f /[x(s)] ds °

yt f /d

bz0,

b

2 f e-,'12 do

Sb

Yn.j0

0< f/dZ<+oo.

14) follows from 9) and the properties of the BROwNian local times as follows: 2e(t) b Pa

It

e (R') V

= It+00 Pa

-

e(t)

2t(t, a)

t(t, a) e(R')

Yt-

= mPa(2t( a) S

S b]

b]

I, Sb]=P0[2t(1,0)Sb]

=tlimPO ll

b

1

Vii

=P0[t+(1)Sb]=P0[t-(1)Sb]=Vn f e-0I2

dc

0

-}-

(see 2.1 for t' and t-). Given a (Q) < + oo, lim t-' e(t) = e (Q)/r t (Q) by 12), and it is an gt+C* interesting problem to describe the behavior of e' = e - i e (Q)/m (Q) as t t + oo. H. TANAKA [1] solved this problem in the special case Q = R', s (b) - s (a) = b - a, m (Q) < + oo by showing that if 02 m (Q) m(0, b)sdb< +co, then

00.0

0

b

im P. (a S e' (1)/aV2-1 < b) = 2n f e-0/2 do

ft

On

and

P. (fin ,t +00e' (t)/a y2t lgst = 1) = 1. Problem 1. Derive 6) from the result of problem 2.8.3. [Because t (nt, ) is invariant under the time change, it is enough to prove 17)

E6[t(m, )]=b-a,

m=mb+m5(w ),

a
for the standard BROwNian motion. Using problem 2.8.3, one obtains

E5[t(m, f)] = Ee[t(mb, )] + Ee[t(ma, f)] = (b - se) + ( - a) = b - a

in case a < < b, and the other cases can be treated similarly.]

6.8. An individual ergodic theorem

231

Problem 2. Give a new proof of 15) using the scaling properties of the standard Bxowxian motion.

[Given a standard Bxowxian motion starting at 0, the scaling --,. I x (s/t') shows that t (t, $) is identical in law to VF t (1, i#); thus 2 f t d e/e (R') Ii is identical in law to

x (s)

e (RI)

ft(i

,

$/t) a (d e) .# 2 t (1, 0) . ]

Problem 3. Show how to compute the invariants of a persistent diffusion from the sample path. [9) shows how to compute the speed measure from the path up to a constant factor, and fixing it, one can obtain the scale from the special case of 7a) : t(mN. F) Blimp = s(b) - s(a).]

`d + + dr)

Problem 4. Consider the BESSEL process with CAS =

Use the law 18)

nf +oo

2

P.< 11 = 4-11I' (t> 1),

= 0(i

1)

.-.

for the successive passage times m" to 1 via 2 to check that 19)

lim P. [2e(1) < u e(0, +00)] = 1 -

this formula is an analogue of the KALLIANPUR-ROBBINS law 14) and

will be used in 7.17 (see problem 4.6.4 for a related result). (Use the passage time formulas E l (e-an't) =

10(1'2aa)

,

10(21'2 a)

Es

(e-an`')

=

Ko(2V2a)

Ko(V2a)

and the estimates K.(x) - -lgx and la(x) -1 ti x2/4 for x 10 to check E. (e-at "I")

1

1g4

nlgt

4-1/19',

which implies 18). To prove 19), it is enough to check lim P. [2e(i)/1g t < u] tt+M

= lim P. [2 a (m")/lg m" < u] nt+oo

= nt+OD lim P. [2e(mfnlg
"t+00

< u] ,

noting that s (r) = igr; now replace u with u e (0, + oo) and use 18) to complete the proof of 19).]

7. Baowxian motion in several dimensions

232

7. Brownian motion in several dimensions 7.1. Diffusion in several dimensions Given a nice topological space Q (for example, a several-dimensional differentiable manifold), let oo be an extra point which is either isolated or the one-point compacti f ication of Q according as Q is compact or not, let C. (Q) be the space of bounded continuous functions / : Q U oo -,. RI

with f (oo) = 0, introduce the (continuous) sample paths w : t -. x (t) E Q U oo with x (t) E Q (t < + oo) and x (+ oo) m co, define Markov times m, shifted paths w" and Borel algebras B and Bm. as usual, take probabilities P. (B) (a E Q U co, B E B) with the usual properties in-

cluding P. [x(t) - oo (t ? 0)] = 1, and call the associated motion a diffusion it it starts afresh at each Markov time, i.e., if 1)

P.(w+EBIBm+)=Pb(B) aEQ,BEB,b=x(m),m<+oo

for each Markov time m and it, in addition, it is smooth, i. e., it the Green operators G

-+ E

+

e-

dt

map C

into itsel :

0

a> 0.

2)

1) and 2) are not unrelated. R. BLUMENTHAL [1] proved that if the motion is simple Markov, i.e., if it starts afresh at each constant time t Z 0, and if it is also smooth as in 2), then 1) is automatic (the proof can be modelled on the standard BROwxian case 1.6). Bearing in mind the result of 3.6, it is natural to conjecture that the GREEN operators map into itself a space similar to C (Q) under the sole condition 1) (see E. B. DYNKIN [7], H. P. McKEAN, JR. and H. TANAKA [1],

and especially D. B. RAY [3] for information on this point). Given a diffusion as described above,

Ga-Gp+(a-fl)GaGp=O

3)

as in 3.7.1), permitting the usual definition of the generator:

0 = i - G-I

4a) 4b)

D ((5) = G,

and the derivation of Dynkin's formula : m

E.[u(xm)]-u=E, f (@u)(x!)dt

5)

uED((3S),

0 III

where m = min (t: x (t) & B) is the (MARKovian) exit time from a suffi-

ciently small open region B c Q, and it follows that if B is a neigh-

7.2. The standard BROWNian motion in several dimensions

233

borhood of a E Q, then either P. [m = + oo] =1 for all B or E. (m) < + oo for all small B ; in both cases (ti u) (a)

6)

E D((3),

=B

just as in 3.7. (i is a local operator, and ((l u) (a) z 0 if u E D ((s3) has a local minimum at a E Q. Q$ shares both these features with the classical elliptic differential operators (see chapter 8 for additional information about elliptic operators). BLUMENTHAL'S useful 01 law

P. (B) = 0 or 1

7)

B E BO. m n B, 6>0

follows from 1).

Problem I. Give a complete proof that if B C Q is open and A C Q is closed, then MB = inf(t : x(t) E B) and MA = inf(t : t > 0, x(t) E A) are both MARKOV times.

MB is MARxov, and, choosing open Bl j B. etc. I A, the MARKovian nature of MA is clear [(w: MB < t) = U (t v: x (in-') E B), so jn-1
from

MA = inf sup (j -1 + mB. (w 1)]

jZ1nZ1

and

(W: mA < t)

=jZ1iZ1nZ1 U U n (t v: mB,.(wl-1) < (w: x (kl-1, wj-1) E B.).) =y Un U jZ1 ial n21 kt-1
7.2. The standard Brownian motion in several dimensions

Consider the standard d(z2) dimensional BROwNian motion of section 2.10: it has continuous sample paths, its GREEN operators 1)

(G« t) (a) 2

rr l Q-1Kd-1( 2aIb-aJ)l(b)db* J llb? all Bt 2

map C (Rd) into itself, and so it is a diffusion as defined in 7.1. DYNKIN'S formula will now be used to compute its generator 0.

_

t 2 (2nc)

a I Ib - aI I s

Kt

(}/2a I b

- a I) is the LApLAca transform of the

jam

111

2

see A. ERD$LYI (I (1): 146 (26)). Kt the usual modified BESSEL function. GAUSS kernel (27Lt)

is

7. BaowNian motion in several dimensions

234

Because the GAUSS kernel (2r t) -d12 e- I b - a l'!s a is invariant under the °°,

1"r

group of euclidean motions, it is clear that, for a E Rd, the distribution P. [x (maB) E d b]* of BROWNian hits on the spherical surfacecIB : I b-a I=E

is invariant under the group of rotations about its center; as such it has to be the uniform distribution do on 8B, an observation that was already used in 2.7. Ea (ma B) can also be computed with ease : the radial motion r(t) tin

a I (t ? 0) is the BESSEL process with scale (d ? 3) lgr (d = 2) -rs-d/(d - 2) and speed measure 2rd-1 dr. maB is the BESSEL passage time to r = E, and so s

e

Ea(maB) =f m[0, r] s(dr)

2)

=.f

d

0

0

dr = a

(see 2.7).

DYNKIN'S formula now reads f u(b) do - u(a) 03 uE D(i); (e u) (a) = lim 3) Old eto esp., if u E C2(Rd), then 5 u = (1/2) d u, A being LAPLACE'S operator,

as the reader can easily check using the MncLAURIN expansion of is about a. 0 can be described as the operator

(Y is = liftn0 de -I (of is do - u)

4)

B

applied to the class of functions is E C. (Rd) such that (S S' is E C00 (Rd).

(33* ) 0, as has just been proved, and now to obtain (Y = (SS it is enough to check that D((33*) C D((33). Given u1 E D((33*), let f = (1 - tb') ul ; then f E COQ (Rd), u2 = G1 f E D (G), and us = u2 - u1 E D ((33) is a solution of (J' u3 = us. But then us E C. (Rd) attains its greatest value at some point a E Rd, and it follows from the definition VII

of 0' that us S u3(a) = (W us) (a) S 0. us ? 0 for a similar reason 11!

and so us = 0, i. e., ul = u3 E D ((33) as desired.

Problem 1. 0 is the uniform closure of d/2 acting on the class CO

..a

C. (d) of functions is E C (Rd) with d u E CO, (Rd), i. e., if is E D ((3) , then it is possible to select un E CO, (d) (n ? 1) such that I I un - u 1100 and I Id u I I. tend to 0 as x t + oo. Problem 2. 03 is a proper extension of d/2 acting on C. (d). [Consider the class C. (Rd) of continuous functions f with I l f I I. = I I (1 + r) /1100 < +oo and let G0, f= c (d) f la b I2-d f (b) d b x'

-

* aB means the boundary of B.

7.2. The standard BROwxian motion in several dimensions

with 4c (d) = 7r 12 r(d/2

235

- 1). 0 is closed and (d/2) Go, = -1 on

c. (Rd) n Cl (Rd), so 0 Go, = -1 on C. (Rd) , and D (GJ) C C.. (A) would mean that 02G0, could be viewed as a (closed) map from C. (Rd) to C (I a I< 1) for each second partial 02 = 82/8x18x1 (i, j S d). As

such 82Go, would be bounded, and it would follow that (02 u) (0) = = f (d u) a (d ) for u E C2 (Rd) vanishing far out with a signed Rd

measure e such that f d I e I < + oo. But then, changing u into b-s u (b

)

R`

and letting b l 0, it would follow that (82u) (0) = e (0) (d u) (0) which is ridiculous. The above proof is adapted from K. DE LEEuw and H. MIRKIL [1].]

y'0

'l6

Problem 3 after P. Li:vY [3]. Use the scaling and differential properties of the 2-dimensional BRowxian motion to show that its sample path has 2-dimensional LEBESGUE measure 0. [Consider the arcs

B1=(x(t):0
BQ=(x(t):0
B3 =(x(1 -t)-x(1):051<1) B, = (x(1 +t) -x(1):0
1)

and their indicators e1, es, e3, and e, for 2-dimensional BROWNian paths starting at 0. B3 and B. are independent, and B1, Be, and B, are identical

in law, as is clear by the differential property and the translation and reflection invariance of the motion. Bs and Y2- B1 are identical in law, as

is also clear on using the measure-preserving scaling x(t) - x(2t)/j/2. Using B to denote the area of B C R2, (B3 U B4 I = (B2 1 by the translation invariance of the LEBESGUE measure, and so 2Eo(IB1 I) = Eo(IB21)

=EO(IB3UB4I) = EO(IB31 +I B41 (-rIB3 nB4!)

= 2E0(IB11) - Eo1J e3 e' db) = 2Eo(IBl1) - f Eo(es e,) db

= 2Eo(IB11) - f Eo(ei)3db which implies

EO(IB11) = f E.(el)db=0 as desired. Another solution due to H. SATO is as follows. 2.7.9) states that the 2-dimensional BRowNian path never hits a point singled out

236

7. BRowKian motion in several dimensions Cs.

in advance at a positive time. Because of this PQ[ sup eb(x(t)) = 0] _ 1>o

= i (a E R2, eb = the indicator function of b E R2), and using FUBINI'S theorem, it follows that E0 [measure (x (t) : t ? 0)) = Ea [ f sup eb (x (t)) d b 1 = 0.] jo 9>0

7.3. Wandering out to co

One feature of the BROWNian motion that will be used below is

that in d

3 dimensions P. [ lim I x (t) ct+00

while in the

+ oo]

2-dimensional case, the BROWNian sample path x(t) meets each little circle at an infinite series of times increasing to + oo. Here is the proof (see problem 4.6.3 for a different method). Given a standard BROwxian path starting at 0 E Rd, let A C B be C11

two spheres with centers at 0, let m, < m2 < etc. be the successive passage times to 8A via 8B, i.e., define ml =mad, mn = mn 1 + m(wmn-1) (n z 2), m = MOB + mad (Wb1aB).

S t < mn (n ? 1,

let en be the time the path spends in A for

ma = 0), and using the isotropic nature of the BROWNian motion, let y be the (constant) value of P. (mad < + 00) on ,3B and e2 the (constant) value of E. (e2) on 8A A.

Because the m, < m2 < etc. are MARKOV times and the motion 1 < + oo (n Z 2) so

is isotropic, en is identical in law to e2 if +CO

1)

.~i

+00 5

f dt r ,1

o

d 52

(2x t)" db

A

= E0[measure (t: x (t) E A)]

= EO(maA) + E EO[en, mn-, < +oo] nZ2

=EO(moA)+ Ze,yn-2, nZ2

and using

E0(maA) + e2 S Eo(m0B) < +co

2)

[see 7.2.2)], it develops that 3)

X yn z + oo (i. e., y z 1) according as d 5 2.

nZo

cod

But then, if d L> 3, the path cannot meet A for large times and + ool = I as is clear on letting A f Rd, while if d = 2, then mn < + oo for each n 1, lim M. = + oo, and selecting Pa [gf m I x (t)

"f+00

-I'

?

7.4. GREHNian domains and GRaaN functions

237

a disc D C R2, the obvious estimate sup P. (mD = + Co) = y < 1 coupled with 4)

OA

y = sup lim E. [mD >

Pz(m.)(mD = + oo)]

OA 0+0o

Y sup"t"o lim P. (MD > m*) 04

= Y2

justifies

Po(mD = +oo) = Eo[Pzlmedl (mD = +0 0 )0,

5)

proving that the path meets each dksc an infinite number of times as desired.

7.4. Greenian domains and Green functions

D C R4 (d Z 2) is said to be a domain if it is open and connected. D is said to be a GREExian domain if, in addition, there is a function G = G (a, b) = K + H defined on D x D and < + oo off the diagonal such that 1) G ? 0, G(a, b) = G(b, a),

2)

3)

K=- (23r) -l lg I b- al

- r(2 -1) lb - al=-d

d= 2

d

4n4M

d> 2 ,

d6H =dbH = 0.

4)

Given a GREENian domain D, there is a smallest such function G = G'. G' is the so-called GREEN function of D. G' (a, b) should be considered as the potential at the point a due to a unit positive electrical charge placed at b inside the grounded shell 8D. G' has a probabilistic as well as an electrical significance : 2G' (a, b) d b +CO

is

the expected time f d t Pa [maD > t, x (t) E d b] that the standard 0

BROwNian path starting at a spends in db before hitting 8D, and this aspect of G' is used below for the proof of the basic formula +Cc

5)

f g'(t,a,b)dt=G'(a,b)

if D is Greenian

0

M +oo if D is not Greenian where (a, b) E D x D, and g' is the absorbing BROwNian kernel: 6)

2g' (t, a, b)db=P.[moD>t,x(t)Edb]

t > 0, (a, b) E D x D

7. BxowNian motion in several dimensions

238

(that g' exists as a BOREL function

is

clear from the estimate

P.[MOD >t, x (1) E db] S 2g d b*; it will be found below that g' can be modified so as to be smooth and unambiguous). Because D = Rd (d > 2) is GREENian (its GREEN function is 4 X-dJ2 r (Z t) b - a 12-d) , each d > 2 dimensional domain is

-

GREENian.

D = R2 is not GREENian; in fact, it will appear that the 2-dimensional

GREENian domains are those for which P. [mOD< + oo] = 1 (see problem 7.8.3 where this is seen to be the same as R2 - D having positive logarithmic capacity). Here is the proof for d = 2 (see G. HUNT for a similar proof); the d > 2 dimensional case is much easier and can be left to the reader. _ g' is the first item of business. Given a subdomain B of D such that B C D, if (t, a, b) E (0, + oo) X B x B, then

Pa[x(i2-11 t)EB,i<2",x(t)Edb] = Pa[x(t) E db] - T Pa[x(j2-" t) E < i, x(i 2-* t) f B, x(t) E db]

7)

i<2"

"...

.,.

0

0 off the diagonal of R2 x R2, it is found that the corresponding x

t

4«+

decreases as n t + oo to P. [x (s) E B, s S t, x(t) E d b] , which, in turn, increases as B t D to Pa [meD > t, x (t) E d b] . and using the fact that the GAUSS kernel g is continuous on (0, + eo) x R2 x R2 and . 0 as ,.-,

(unambiguous) densities

8)

f g(2-" t, a, b,) 2db1 g(2-" t, bl, b2) 2db2 ... g(2-t,

b)

he" -,

=g(t,a,b)-

r--,

i<2"

g((1 - i2-*) t, x(i2-* t), b)] ..r

decrease as n f + oo to some limiting function, which, in turn, increases

as BfDto 9)

g(t, a, b) - Ea[t > rid D, g(m@D, x(moD), b)]

9 >0,(a,b)EDxD.

sa g'(t,a,b) VII

Because f g (2-* t, a, b1) etc. S g is bounded for each t > 0, 6) is im-

a glance at 8) shows that end

mediate. Because g (t, a. b) = g (t, b, a),

* g is the GAUSS kernel J (2nt)-1112 e-16-10121 for the standard Bntowxian motion

(note the nuisance factors 1/2 in g and 2 in 2db).

7.4. GREENian domains and GREEN functions

239

g' (1, a, b) = g (1, b, a), and using this and the smoothness of g described above, it follows from 9) that g' (1, a, b) is continuous on (0, + co) x D x D. g' (t, a, b) also L 0 as i 10 off the diagonal of D x D as is clear from g' S g. +00

Consider G' = f g' dt. 1) and 2) are automatic for G', and the next 0

item is the proof that G' = + oo in case P. [maD< + oo] < 1 at a single point of D.

a E D and a closed disc B:

Given

b- a j S e (BCD),

P. [maD > maB] = 1, and using the notation m = man,

P,[MOD = +ooJ = P,[mOD> m, maD(Wi) = +00]

10)

= f P.(MOD =+oo]do* 0B

r-.

implies that u = P. [maD = + oo] is continuous on D . P. [MOB < + oo] =1 (see 7.3) now implies

u(b) = impb[maD> m, maD(TOM) _ +00]

11)

_ limo Eb[maD > m, u(x(m))]

u(a), showing that u is constant on D, and using

u = P. [maD > n, maD (w;) = + oo] = E. [maD > n, u(xw)]

12)

n t +oo,

U P.[maD > n] I u2 f/.

it is seen that u = 1 in case P , [maD < + oo] = 1 - u < 1

at a

single point of D. But then g' (t, a, b) = g(t, a, b) at each (l, a, b)

E (0,+oo)xDxD, and +0o

G' =f

13)

4n t

dt rm + oo

0

as desired. 'L3

6.v

Coming to thecase P. [m,ID< +oo] = 1, the next step is the proof that f G' (a, b) d b < + oo for each a E D and each closed disc B C D. B

Given concentric closed discs B1 C B! C D, if m0 = 0 < m1 < etc.

'>d

are the successive passage times to 8B1 via 8B, for the standard BxowNian path starting on 8B1 and if e. = measure (1: x(t) E BI, * do is the uniform distribution on 8B.

240

7. Bxowxian motion in several dimensions

m,-15 t <

(n Z 1), then

2f G' (a, b) db

14)

B,

= E5[measure (1: x(t) E B1, i < me D)] taro

supE. (ma B) + supE. [measure (t : x (t) E B1, t < MOD)] IVY

B,

O B,

1=

d + sup

ZE.

(el,

ml-1

l1 = diamB1

< maD)

5 d + upE.(e1)iz )tOB,

y=

BPP.[moB, < MOD]

alb

I+ upE. (mOB) (1 - )-1 OB,

11- -{-

a d

12 = diam B2,

(1 - y) -1

much as in 7.3, and to complete the proof, it is enough to check that y < 1. But, if u = P. [mOB,< MOD], then for each a E D - B1 and

each closed disc B: I b-a15e(BCD-B,), u(a) = P5[m0B,(tm) < m0D(wm)] = f udo

15)

m = MOD,

OR

and either y < I or u = I at some point of 0B2, in which case u = 1 on D - B1, contradicting P. [MOD < + oo] =- 1. Continuing with the case P. [mOD< + co] = 1, it is now possible to show that G + (2ar)-11g I b - a I is harmonic on D x D as stated in 4).

Given a E D, a closed disc B: b - 41 5 s (B C D), and a point b E D - B and using the smoothness of g' as described below 9), it is not difficult to see that g' (t, a, b) = EQ[t > maB, g' (t - ma B, x (MOB), b)], and integrating this over i to get 16)

17)

G' (a, b) = E.[G' (x (MOB), b)] = f G' ( , b) do, OR

it is found, using f G' db < +oo and 2), that B

18)

and i9)

G'<+oo AaG' =dbG' =0 NCO

off the diagonal of D x D. G' + (27t)-11g b - a has now to be discussed on a neighborhood of the diagonal.

7.4. GREExian domains and GREEN functions

241

Given an open disc D, C D with closure D, c D, the corresponding absorbing BROwxian kernel gi satisfies 20) gi (t, a, b) = g' It, a, b) - Ea[t > maDt, g' (t -- ma D,, x(m0D,), b)]

t>0,(a,b)ED,xD, r-,

[see 9)], and integrating over t, it is found that +00

Gi (a, b) = f gi dt = G' (a, b) - E. (G* (x(mOD), b)]

21)

o

(a,b)ED1xD,.

But if a is the center of D, : jb - a j < e, then gi (t, a, b) depends upon i and jb - aj = r < e alone; in fact, gi (t, a, b) = (2 ar) X the transition kernel p(t, 0, r) for the BESSEL motion jx(t) - a (t ? 0) with scale Ig r and speed measure 2 r d r starting at r = 0 and killed at the BESSEL passage time m1 = MOD,, and using the BESSEL local time t to compute GI* (a, b) = (2a)

22)

f t(t, 0, r) di = (2 r)-' F.0 [t(m1, r)J* 0

= -(2n)-' Igr/e +O+

[see problem 5.2.1], 21) goes over into

G' (a, b) + (27t)-' lgjb - al = E0[G'(x(m,D,), b)]

23)

'''

+(2,z)-'Ige lb - al < e. - a j is now seen to be a solution of 4) near as well

p'-

G' + (2 a) -' lg j b

i'°

as of/ the diagonal, as desired. G' has now to be identified as the smallest solution of 1, 2, 3, 4). Given a bounded domain D such that OD E C2, P. [m8D< + oo]

1,

G' (a, b) < +oo off the diagonal, and for each b E D, G' (a, b) tends

to 0 as a E D tends to a point c of OD as is clear from the estimate

24) G (a, b)

88

V.1

E,[MOB < man, G' (x(maB), b)] sup G' (., b) P.[nioB < ma D] OR

a.°

and the diagram ; this depicts the positions of the points a, b, c, and the circle aB enclosing b and shows, in addition, a closed disc B, touch-

JD Diagram I

ing 8D at the single point c and a concentric closed disc B. excluding B; indeed, Pa [m0B

MOD] is smaller than P. [mOB, < maB,] on the annulus

* E. is the BESSEL expectation. 10/McKean, Diffusion procesaea

16

7. BROwxian motion in several dimensions

242

..0

between 8B, and 8B2, and thanks to the isotropic nature of the BROWNian motion, 25) PPLmeB, < me B,] = Pd[md, < ma,]*

d 4 d,

WAN

as noted in 2.7 already. Consider now the general domain D and a subdomain D, such that D, C D is bounded and 8D, E C2. Given a solution G of 1, 2, 3, 4), if gi is the absorbing BROwxian kernel for D, and if Gi = f g1' d t on D, x D,

26)

0

=0

'''

on DI X BD,, then G - Gi E C (D, x D,), A [G - G1] = 0 on D, x D1, and G - GI z 0 on D, x0DI. G --> G'1 on D, x D, follows, and using the fact that g 1 t g' as D, t D [see 9)], it is found that G Z G' on D x D; in brief, D is Greenian or not according as P . [meD < + oo] -1 or not, and in the Greenian case, G' = J g' d t is the Green function of D. Problem 1 (CHAPMAN-KOLMOGOROV identity).

g' (t,+is,a,b) =2,'g'(t,,a.c)g'(ts,c,b)dc b

t,,ts>0,(a,b)EDxD.

Problem 2. g' (t, a, b) is of class C°° [(0, + oo) x D X D] and a solution

of 8u/8t = d u/2 (t > 0). [g' E C°' (D x D) as is clear from 9) and g' (t, a, b) = g' (t, b, a); now compute 8g'/8t from 9).] Problem S. Compute P. [x (meD) E d b] , where a is a point of the open

unit disc D : I a I < I and d b is a little arc on 8D. [Pa [arg x (meD) E d V] = d+ i x the classical

POISSON kernel

(1 - rt)/2x(1 - 2r cos (0 - V) + r2), where r = I a I and 0 = arga.] 1a-b - ab = (2x)-1lgGoth[a, b]*** is the GREEN function of the disc a I < 1 and is invariant under the Problem 4. G' (a, b) = (2x) -1 lg

group r of non-euclidean (PoINcARi) motionst of the disc onto itself. COURANT and HILBERT [1: 326-337] compute a number of other 0.i

GREEN functions; see also problem 7.6.2 and MAGNUS and OBERHETTINGER [2].

Pi is the law of the BzssEL motion starting at d, < d < ds. ** b is the conjugate of b. [a, b] is the non-euclidean distance from a to b (see C. CARATRHODORY

[1:69]) t r is the group of all f : 1 conformal maps a-+ e

-b T-

ab (0 S +e < 2m,

0o-

ibl<1) of the disc onto itself and their conjugates (see C.CARATRSoDoRY [1: 81]).

7.5. Excessive functions

243

7.6. Excessive functions

Consider a non-negative function u S + oo defined on a GREExian domain D and excessive in the sense of G. HUNT [2 (1)], i.e.,

u < + oo at some point of D E.Cu (xt),t<MOD) tii

i a)

1b)

tt0.

Because

v - 2 f g' (t, a, b) (u An) (b) d b is excessive for each n ? 1 and t > 0, u is a limit of bounded excessive functions v E C°° (D) ; for such v, 2) 05 v = limt-1(E.[v(xe), t < maD] - v) S 0 and

f vdo - v(a) = Ea[v(x(moB))] - v(a)

3)

8B m8 B

Ea

D)B, B:lb-al;5e,

0,

(05 V) (XJd1 0

and it follows that u is superharmonic in the sense of F. RiEsz [1(1)], i. e.,

u < + oo at some point o/ D lim u b) = u(a) b-'a

4a) 4b)

aED

f udo Su(a)

4c)

aB

On the other hand, if u is superharmonic on D, then so is v - u An

for each n Z 1 and likewise v (a) = f u (b) db on the smaller Ib-aI 5

domain D' of points of D at distance >e from OD, and using this, u is seen to be an increasing limit of bounded superharmonic functions v E C2 (D) on each domain D' with closure C D. But for such v,

f vdo-v 5)

and 6)

Q$v - lymaBE,ld So

S0 -

E. [v (xs), t < maD-] - v

S

v LAmoD-

=E.

f G3vds SO 0

and it follows that u is also excessive; in brief, excessive is just the same as superharmonic. 16*

7. Bxowxian motion in several dimensions

244

Given a non-negative mass distribution a (d b) on D, its potential u = f G' de is excessive because D

E. [u (xe), t < m0 D]

7)

av.

= 2 f g' (t, , a) da f G' (a, b) e(db) D

D +00

'!t

= f e(db) f ds 2 f go (t, , a) g' (s, a, b) da D

D

0

= f e(db) f+00ds g' ft + s, , b) D

0

+00

=.f f g' (s,

, b) ds e(db)

D9

t f G'de=u

t40.

D

As to the general excessive function u, the mass distribution 8)

-

e> 0,

e1(db) = e-1 [u (b) - Eb(u(x1), e < MOD)] X 2db

is non-negative, its potential 9)

f G' de, D

MOD

r

= e-' E.

N

[u(x,) - E:ul(u(x.), e < me D)] dt)

0 +OD

v(t) =E.[u(xg),t<meDI III

..a

=e-1 f [v(t)-v(t+e)]dt 0

"+I

e

f v dt -- lim e-1 r v dt 0

nt+co

n

=e-1 f vdt - h

h=v(-3-oo)

0

t/1

increases to u -- h as e 0, h is harmonic (see problem 9 below), and, as is not difficult to prove, e, converges (in the sense of mass distributions)

as e l 0 to e = e" > 0* such that 10)

u =.f G' de" + the greatest harmonic minorant of u; D

see 7.7 for the proof in a special case and 7.18 for an application. e(B) = lim e,(B) .+0

for compact B C D with e(OB) = 0.

7.6. Application to the spectrum of d/2

245

e" is the Riesz measure of u, so named for F. RIEsz [1 (2)] who r.,

discovered 10); it leads to a simple expression for the standard BROwxian generator

0u =li 0

de-2

I Judo-ul ...

lOB

`1'

of 7.2-4): it 0 z W u E C (D), these u is excessive on D and -e" (d b)/2d b

= di u, while i f u is excessive on D and it u' = -e" (d b)/2d b E C (D), then 0 Z (3S u = u'. Problem 1. Given an excessive function u on a GREENian domain D, h - lim E. [u (xe) , t < maD] is harmonic. it+00 [h is superharmonic, the isotropic nature of the BRowNian motion

implies that, if a E D and if DDB: lb - a 5 e, then E. 1k (XI), t < IndB] S h (a) P. [t < MO DI,

and using h = F.. [h (xx) , t < mo D) , it follows that maD

h (a) = lim E. a J e-al h (xj d t + e-a meB h [x (m# B)] 0

= f h do.] aB

Problem 2. Prove that e = e" if e Z 0 and if it = f G' de up to a harmonic function.

+'1

7.6. Application to the spectrum of d/2 Given a compact surface 8D C Rd of class C$ bounding a domain D with volume D I, the operator d /2, applied to the class D (A) of functions it E Ct (D) (l c (D) such that A u E C (D) and it = 0 on 8 D, has a countable spectrum

0>,'

VsL> etc..-oo* and as H. WEYL [2: 41-45] discovered, C+1

1)

-y,,

2)

n

constant X (-ffj-)

2 /d

'n t +00'

the constant being independent of D [its actual value is 2 irr(d/2 + 1)d12]

M. KAC [1: 205 -209] derived 2) using the BROwNian motion, and it is his proof that will be explained below (see COURANT and HILBERT .-r

[1: 373-3871 for the classical proof). * y is listed the same number of times as d1/2 = y/ has independent solutions / E D(A).

7. BROwxian motion in several dimensions

246

Consider the operator H, / = 2 f g (t, a, b) / db acting on the D

(real) HILBERT space L2(D, 2db) with inner product (/,, /_) = 2J / /2db and norm 11/11, = y(/, /) and use the CHAPMAN-KOLMOGOROV identity

(problem 7.4.1) and the fact that g (t, a, b) = g (1, b, a) to construct, as in 4.11, a BOREL measure p (dy) from (-oo, 0] to projections such o+

that H, = e'D = f e'Y p (dy) (t > 0, p(-oo, 01 = 1), m being the opeo+

-00

o+

rator f y p (d y) applied to D (ti) _ (u: f y$ I I p (d y) u 11121 < + oo). -00

1

Because g S g S (2n t)-d12/2 and I D I < + oo, H, is of finite trace 21 g' (t, b, b) d b S (27c t) -d/2 I D I and so compact, and this means that D

has a countable spectrum

l.-oo.*

3)

4) e.,

Iu

12 f g uu d b 1;5 constant x t-d/2

0

e^.

Given an eigenfunction u corresponding to y, = 0,

(It + 00),

y, < 0

5)

as in 1). Consider the eigenfunctions / = /1, /g, etc. corresponding to

Y= Y1? Ys? etc.

(f/=y/,Il/11:=

=0,n<m).

Because eY' / = 2 f g / d b, problem 7.4.2 plus the bound g' S (2= t) -412/2 ensures that /E COO (D), and using the smoothness of OD, it is seen that 6)

eV` I/(a) 15 2 f g (1, a, b) 1/1 db

2fg'(t,a,b)/$db

s

(2n 1)-414

Y2

2

II/II:YPa(moD> t)

tends to 0 as a tends to 8 D, i.e., / E D (A). But m = d /2 on D (d) C D (a) (use problem 7.4.2) so d l/2 = y as desired, and using MERCER'S theorem', one finds 7)

g' (1, a, b) = ' e'-1 &(a) / (b) nZ1

t > 0, (a, b) E D x D.

* y is listed the same number of times as El/ - y/ has independent solutions E D(D). ' COURANT and HILBERT [l: 117-118].

7.7. Potentials and hitting probabilities

247

which implies

t>0.

F

2f

8)

nZ1

D

Because

9) g' (t, a, a) _ J(2= t)-d/2 - Ea[t > meD, g(t - mOD, x(m8D). a)] _ j (2x t)-d12 + an error of magnitude < e-''fit 14 0, a E D, where e is 1 /2 the distance from a to t 9D, 8) implies

t j 0,

Z eYat , (2x t)-d12 I D I

10)

Rai

and a standard TAUBERian theorem' converts this into

x 1 ^'

11)

Y.>Y

IDId

lyld1l

y 1 -oo,

(2n>=j9r2 +1

from which 2) is clear on putting y

yn and letting is 1 + oo. ".T

Problem 1. Compute the eigenvalues of d/2 for the 3-dimensional domain D : 0 < tt < a, 0 < t2 < b, 0 < to < c and check WEYL'S law 2). s : Pnl 12 (l , m, n z 7) are the eigenvalues; the number (--= + M9 MIN

+ ,I

2

of these that lie above y (l - oo) is (about) the volume } x -j x abc (2 Iy I/x2)312 of the ellipsoidal octant

+ b + E'S2iyI/xs

&S f2, t,>0,

as it should be.) v.%

Problem 2. Compute the GREEN function for the domain D of problem I (see COURANT and HILBERT [1: 328-333). 7.7. Potentials and hitting probabilities

Given a GREENian domain D with GREEN function G, if B is a compact subset of D or an open subset with compact closure B C D, then the hitting probability pB = P. [MB < mOD] is the potential f G d eB .,r

of a Borel measure eB ? 0 concentrated on 8B; indeed, pB can be identified as the Newtonian electrostatic potential of B, meaning for compact B, that it is the greatest of the potentials f G do < 1 (e 0) , 'VA

B

1 G. DoETSCR [I: 203-216).

7. Bxowxian motion in several dimensions

248

new

and for open B, that it is the smallest majorant of the potentials

fGde S1 (ez0,ACB) A

We give the proof in the case d = 3, D = R3, G = (4x) -' Ib - a I-1. Given bounded B C R°, P. [mB (wf) < + oo] 4 0 as t t + oo because 111im Ix(t)I

= +00 =1

(see 7.3), and introducing the measures 1)

a>0,

e,(db)=e-'2dbxPb[ntB<+00,MB (W )_+00]

it is evident that 2)

p, (a) = , f G (a, b) e, (d b) Ra +fW

= E.

\0

e-' P:(O[mB < +oo, mB(w:) _ +oo] dt) 1

+00

+ oo] dt

= e -' f Pa [mB (u'i) < + oo , mB (wt 0 +CO

= e-' f [P.(mB(w+) < +00) - P,(mB(wi+.) < +co)]di 0 a

= e-' f Pa(mB (w) < + oo) di 0 (MB

t Pa

e10.

< + oo) = PB (a)

Consider a bounded open neighborhood A of 8B: 3)

e. (R11 - A)

=e-'2 f Pb[mB<+00,mB(w.+oo]db R'-A

e-' 2

f db

f R'-A

f

e-0129 dt*

V2nt'

S constant x e-1 f dc f ,J } mini 0

e3

e-a'12e dt

Yin $3

i0

e +0, coo

where l is 1/3 of the distance from b E R3 -A to 8B; in addition, 4)

` p e, (A) 5 I / ff G < + oo, (w:m, < +00. ma(w,) - +oo) C (w: mom S s) is used at this point.

7.7. Potentials and hitting probabilities

249

and it follows that limo e, = eB exists, is concentrated on 8B, and has pB

as its potential; indeed, selecting e = el > e! etc. + 0 such that lime, = e a0

exists (in the sense of mass distributions) and using the fact that f G de is an excessive function, 5)

PS[mB(w) < +00] = 2 f g(t. a, b) pB db

=1im2 f gp,db

=lim2 fgdb fGde, its

= limf de,2 f gGdb 1140 As

= f de 2 gGdb M

=2

gdb f Gde

implies PB = f G de, and the stated facts follow at once. The potential PB has now to be identified as the NEwroNian electro-

static potential of B. Given compact B, a potential p = f G de !!!g 1, and a point a J B, D

if A is the open 1/n neighborhood of B and if the distance from a to 8B

is >I /n, then P (a) = f G (a, b) a (d b)

6)

B

=,f Ea[m8A < +oo, G(x(m8A), b)] e(db)* B

= f P.[maA < +oo, x(maA) E db] P 0A

SPatm0A<-1-ao] = PA (a) 4 PB (a)

n t +00'

and since PB=1?ponB-8B, P:5pe on R3-OB. On 8B itself, either PB(a) = 1 ? P (a) or pB (a) < 1. But in the second case, Pa [MB = 0] < I, which combined with BLUMENTHAL'S * See 7.4.17).

250

7. BRowNian motion in several dimensions

7)

rat

.CS

01 law shows that P. [mB > 0] = I. and using this, it is found that AB (a) =s{0 lim P. [m8 (w:) < + ool

= liO Ea [pB (x.)]

=J'

Ea[e < ms, pB (x.JJ

lijn Ea [e < mB, p (xj] = li+m Ea[p (xJl

= p(a), completing the proof. o00

Given open B, it is clear from the probabilistic picture that pB is the smallest majorant of the potentials PA as A runs through the compact subsets of B, so pB is the NEwTONian electrostatic potential in this case also. Problem 1 after F. SPITZER [3]. e, (Rd) = C (B) for each e > 0, where C (B) is the NEwTONian capacity of B defined in the next section. 7.8. Newtonian capacities

Given a GREENian domain D and B C D as in 7.7, the NEwTONian

electrostatic capacity of B (relative to D) is defined to be the total mass C(B) of the NEwToNian electrostatic distribution eB; if B is compact and if 8D is considered to be grounded, then C (B) is the greatest

positive electrical charge that can be placed on B and still have the potential of the corresponding electrostatic field S 1. C (B) has the following properties: 1)

2) 3)

4)

C(A) S C(B) C (A U B) + C (A fl B) S C (A) + C (B),

A C B,

C(B) S C(OB) = C(B), B compact,

C (B) = dinnf C (A) OVOR

d)B 5)

C(B) = sup C (A)

B open,

A aompad

ACB

6)

C(mB) = Imld-'C(B)

if D = Rd (d > 2) and m is a rigid motion of Rd (I m 1) or a magnification by the factor Im I > 0. GAUSS [1] observed that if the potential f G do of a signed meaB

sure e is non-negative at each point of D, then the total (algebraic)

7.8. N$wToxian capacities

251

charge e (B) is likewise non-negative, as is clear on selecting some open

A B and noting that e (B) = f PA de = f p d ed ? 0 (use 7.4.2)]. B

8A

GAUSS's observation, combined with PA S PB (A C B), proves 1). 2) is likewise clear from

tea;

7) PAUB - PB = P. [mA U B < m8 D, mB Z m] S P. [mA < ma D, 11A n B ? MOD] gyp.

=PA-PAnB, ii.

'L7

and using PB S POB = pa, another application of GAUSS's trick proves 3). Given compact B and open At B such .that limed = e exists, AtB

p= f G d e S 1

, e (D

-0B) = 0, and so C(A) te(8B) S C (B) S m C (A),

_

GAS

proving 4).

Given open B, compact A t B, and open D ) B, PA t pB, and so

C (A) = f PD deA = f PA deD t f PB deD = f PDdeB = C (B), proving 5). 6) is immediate from the fact that G is a constant multiple of b -a 1II_d

if D=Rd (d> 2). ...

G. CHOQUET [1: 147-1531 proved an infinite series of inequalities similar to 2). Here is G. HUNT'S neat derivation [2 (1): 53]. Given A, B1, B2, . . ., B. C D and using I for subsets of 1 , 2 , ... , n,

I I for the number of integers in I, and Bl for U B1, it is found that lei

OSP.[mAUB.<MOD, lSn,MA ZnI D]

8)

= P. [MA U B. < MOD, 1 S n] - P. [mA < MOD]

_ -msn (-)n` 11XP .[mAUB1< MOD]* - P.[nA< MOD] mX (-) `I PA U Bl - PA,

I

and an application of GAUSS' trick establishes CHOQUET's result:

0 S - S' (-)m I C(A U BI) - C(A);

9)

mgn

X11-m

this reduces to 2) for n = 2 on replacing A, B1, B2 by A fl B, A - B,

B-A.

Problem 1. Let G = GD (a, b) denote the GREEN function of a bounded domain D C Rd (G m 0 outside D x D). S. BERGMAN and M. SCHIFFER [1: 368] showed that GD,UD1 + GD1nD, a GD1 + GD,.

woo

P.(f E,,,) = - E (-)" E P.(E1) is used at this point; this is the dual

"$w "S, Ill-" of the classical inclusion-exclusion formula of Po1ncARB (see W. Fnu R[3:61-62]).

7. BROWNian motion in several dimensions

252

Use the method of the proof of 2) to prove ...nDt, ? 0. GDiuD,...uDR + X (-)k E GDSt fl D, a 1Sk$n 11
UD.,sS t))

Pa(A n(x(s)ED1

z P.

A n (x (s) E Dk, s s- 1)) n

1

Z Pa(A n (x(s) E D1,, s S t) n .. . Z (-)k11<1i<...
1$k$n

= -15

n(x(s)EDI.,sSt))

Sn(-)

<4Pa(A

tf «

n (x(s) E

D1,

n ... n D1t, s S t));

now let A be the event (x (t) E d b) and integrate over 1.1 Problem 2 after G. CHOQUET [1]. Check that if B1 is open, if Al is compact, and if B1 ) Al (1 S n), then C(1U B1)

- C(1Un A 1)S15 [C(B1) - C(A1)]

ISO

muA,m8D] ISO

8:50

Sts P.(mBB

< MOD, MA,? MOD] =1I [PB, - PAJ; now use GAUSS'S trick.)

Problem 3 (see the discussion of 2-dimensional GREENian domains in 7.4). Use the connection between NEwroNian capacities and hitting probabilities in 2 dimensions to prove KAKUTANI'S tests: that for com-

pact B C R2, P. [mB < + oo) = 0 or I according as the logarithmic capacity

f lglb - aldede '!!

l(B)=exp( sup

BxB

a'0

is 0 or positive (use problem 7.9.1). [l (B) 5 0 according as sup f lg b -a 15 - oo, and in this eZo BxB e(B)-1 alternative, the negative of the GREEN function G of some disc D B can be used in place of Ig Ib - a'. KELVIN'S principle (see problem 7.9.1)

now shows that I (B) 5 0 according as the NEwTONian capacity C (B) 5 0, i. e., according as P. [mB < mmD] 5 0. KAKUTANI's alternative follows from the fact that P. [mB < + oo] is constant (- 0 or 1) on R2 - B (see the discussion of GREENian domains in 7.4).] Problem 4 after F. SPITZER in collaboration with H. KESTEN and W. WHITMAN; see F. SPITZER [3] and W. WHITMAN [1]. Given a compact d ? 3 dimensional figure B, let c (t) denote the volume swept 1 S. KAIUTANI (E).

7.9. GAUSS'S quadratic form

out by B + x (s) up to time

t;

253

the problem is to prove that

c(t) = C(B)/2, = t .

lim t P010+00

[Consider / (£) = volume (a : a E B + x (s) at some time s < e but at no time later than e) and use problem 7.7.1 to check Eo[£-' 1(e)] = £-' f Pa(mB < £, mB(we)

+ oo) da

= e. (Rd)/2 = C (B)12.

Using w; to denote the Po-preserving shift x (t, w;) = x (t + s) --- x (s),

one finds the lower bound:

w(k_1)') S c(ne), and since the

r5*

flow w --> w; is ergodic (use KOLMOGOROV'S 01 law), G. D. BIRKHOFF'S

ergodic theorem implies lim t-1 c (l) ? C (B)12. But also £fsoo

C(n E) S 'Y C(£, w4 -1)e),

k$n

so that

lim

tt+00

1-1 c (t)

S £-1 E [c (£)] _ £-1 f Pa (MB < £) da A

= C(B)/2+£-1 f P.(mB<e, C(B)/2 + constant x £-1 f E'((1 + I x(e)1)2-d, mg < £) da = C (B)12 + constant x £-i I P'(m° C E) da

0+

aU`-s

problem 7.7.1 and the obvious Pa (mB < £ j x (e) = b) = = Pb (MB < £ x (£) = a), and this last integral tends to 0 as e t + oo.] lo,

using

7.9. Gauss's quadratic form GAUSS [1: 41-431 tried to base the definition of the NEwTONian electrostatic potential of a compact subset B of a GREENian domain D

on the fact that the quadratic form

G(e)=J f Gdede - e(B)

1)

e Z0*

BXB

has a strict minimum at the NEwTONian electrostatic distribution

e=eB. `.1

Here is a modern proof. Given en Z 0 such that G (en) t inf G (e) as n t + oo, it is clear from 0Z0

+ oo > limo G (en)

2)

nj *o

n tni

that 3)

sup [ (inf C) e,, (B)2 - en (B)J

+00

t

BxB

sup en (B) < + 00, "Z:1

* G is the GREEN function of D.

7. Bxowxian motion in several dimensions

254

permitting us to suppose that lim en = e,,, exists. nt+oo G (e) is then least for e = e,,. as is clear from 'CS

G(e) Z lim G (en) ?nt+oo lim j f G A m de. den - en(B)

4)

nt+ao

BxB

_ J f GA mdeodeoo-eo(B)tG(eo)

m t+oo,

BxB

and it follows that for e ? 0,

0lime-1[G(e.+ee)-G(eo)]

5)

f G de de.,, - e (B) BxB

=

f

p = f G de,o,

-1) de

B

B

provided all the integrals converge, with the = sign in case e5 e.O. Given B such that 8B is of class C2, pB = 1 on B [see 7.4], and using 5), it is seen that 6)

f Gd (eB - e,,.) d (eB BxB

= .f pB deB - f p. deB -f PB deco + f pao de

_ - f (poo - 1) deB + f (p - 1) deco 0,

which is impossible unless e,,,, = eB, as is clear from 7)

e=eB - eao

f G de de BxB +oo

=f

dt f

e(dit) go (t, ,71)*

BxB

0 +oo

= f dt f e(dE) e(dit) 2.f g' (t/2, a, ) g' (t/2, a, rt) da BxB

0

D

+oo

= 2.f dt f da ( f g' (t/2, a, b) e(db))!. Compute y (B) = inf G (e) for this special case : the result is - j C (B), o and coming back to the original case, it is clear from K'+

9)

A D B,

y(A) s y(B)

8)

y (B) = x f poo d e

- e o (B) _ - j e, (B),

g' is the absorbing BRowNian kernel for D.

See problem 7.4.1.

7.10. WIENER'S test

255

and 7.8.4) that

y(B) z -jC(B),

10)

which justifies 11)

f Gd (eB BxB

-

=f

d (eB - e,o)

f

,5BdeB-2 f

S C(B) - 2f (P - 1) d eB - 2C(B)

VII

1) de. + e (B)

VII

S e (B) - C (B)

VII

S 0.

e = eB is now established as in 7), and the proof of GAUSS'S statement is complete. Problem 1. Give a proof of KELVIN'S principle: that if B is compact, then f Gdede = C(B)-1# inf e(B)-1 BxB

and, if C(B) > 0, then

e0,e(B)=1

f Gdede>C(B)-I BxB

unless e = C (B) -1 X eB .

[Given e !!0 with e (B) =1 and t> 0,

2

f G de de - t e (B) BxB

- C(B) with the > sign unless to = eB; putting .7m

f G deB deB BxB

t = C(B) completes the proof if C(B) > 0, and if C(B) = 0, f G de de BxB + oo is immediate from 12

2BxRfGdedeie(B)=t.] 7.10. Wiener's test Given compact B C R4 (d 2), the event mB = 0 is measurable Ba, , and so (use BLUMENTHAL'S 01 law) P. [mB = 0] = 0 or 1. WIENER'S test states that 1)

PQ[mB=O]=0or1 * C(B)-1 = +00 if C(B) - 0.

7. BROWNian motion in several dimensions

256 according as

In C (Bn)

(d = 2)

n,> 1

converges or diverges

n21

2" J-2, C (B.)

(d

3)

where C is NEwroxian capacity relative to a spherical surface I b - a I = y Z 1 (y < + co in case d = 2) and B. is the intersection of B with the spherical shell 2-A-1 < I b - a I < 2-n (n Z 1) .*

Consider for the proof the case d = 3, a = 0, y = + oo. 2_,,n Given n ' 2"C(B") <+or it is clear from 2

B 2)

2 -4n-2

2

4n-4

2-4n-

e`n-

P0 [MB. < + 00] = YB.(0)

=

i

0

(42r)-1

('

OB.

S

I

In.1

Ib1-1 eB. (db)

(4=)-12A+1C

(Bn)

and the first BOREL-CANTELLI

Diagram I

lemma that, for all but a negligiable class of BROWNian paths, m. = + oo from some n = n1 < -}-

< + oo on, so that P0 [MB = 0] = 0 as desired. Contrariwise, if

,

nal

2" C (B") _ + oo, then T 24n+1 C (B4. +j) _ + 00

nal

.ti

for some 1 = 0, 1, 2, 3, and confining ourselves to the case y = 2, it is clear from the diagram that for mA = min (t: jx(t) I = 2-4") and all

.ti

I..

In=x(m,,) (nZ1),

P0[x(t) E B4n.s for some t E [mn+I m") I Bm.+,] [x (t) E B4 n+. for some t < m"] = Pi. + I [x (t) E Bon+$ for some t > 0] - P1. + I [x(i) E Bon+s /or some t L> m.] +.a

3)

1

= Pn4.+1 (1n+1) [lb - 1n+11-1 - E1.+1(Ib - 1Al-1)] eB4.+s (db) (42c)-1

f

0 B4.+2

[(2-4n-2 + 2-4n-4)-1 - (2-4" - 2-4A-')-1]

(42r) 1 C (B4n+2)

_ (42r)-1C(B4n+2) 15

24"

r

> 24"-3C(B4n+2) * See COURANT and HILBERT [2: 284-286], 0. D. KELLOGG [1: 330-334], and N. WIENER [2], for the classical proofs of WIENER'S test as applied to poten7`0

tials and the DIRICHLET problem. WIENER himself was interested in the DIRICHLET

problem and did not give a probabilistic interpretation to his test. S. KAKUTANI [2]

was the first to discover the connection with the BROWNian motion; see 7.12, where the DIRICHLET problem and BROWNian motion aspects Of WIENER'S test

are brought together.

7.11. Applications of WIENER'S test

257

But then, using 3), 2 t E [mn t1, m")

= E,[P,[x(t)

x(t)

I

+41

do - Po[x(t) f B41. 2, t E [m1+1, m1) 9 ? n]

4)

B4t+,,

IE[mf,1,m1),j>n]

"-s

S [1 - 24"-a C(B4n+2)] (1 -- 2t("+1)-a C(B,(w+l)+,)] d"+2 S etc.

S fl [1

-

24i-3

C(Bg1+2)1,

jan and since X 24n+2 C (Ban+,) was infinite, this product is 0, and nZ1

Po[mB = 0] = nJimo Po[mB < m"] = 1

5)

Cs.

completing the proof of WIENER'S test. J. LAMPERTI [1] found another proof Of WIENER'S test using a nice
BOREL-CANTELLI lemma.

POINCARE'S test' states that Po[m8 = 0] = 1 if B C R3 contains a `+°

little cone A with summit at 0. WIENER'S test can be used for the proof; 'C3

indeed, C (A,) is proportional to 2-" (n ? 1) (see problem I below), causing WIENER'S Sum X 2" C (Be) to diverge. nal

VII

Problem 1 after G. HUNT [1]. Use the isotropic nature of the BROwNian motion to give a direct proof of POINCARE'S test. [A C B is the cone with summit at 0. Po [m8 = 0] = lim Pu [m8 S "t+00

lim sup Po [l" E A] = a > 0, where a = (4x) -' x the solid angle "j+00

(1,

of A, and an application of BLUMENTHAL'S 01 law completes the proof.]

Problem 2. Prove that for the BROwxian motion in d Z 3 dimensions,

P. (Z) = 0 or 1 according as

may

Y 2-"(a-2) C (B.) converges or diverges, "Z1 where B is a closed set clustering to oo, B" is the intersection of B with the spherical shell 2"-' S I x I < 2", C is the d-dimensional NEWTONian

capacity, and Z is the event that (t: x (t) E B) clusters to + oo (see K. TTo and H. P. McKEAN, JR. [1] for a similar fact for the random .SC

walk).

7.11. Applications of Wiener's test Consider a JORDAN arc B C R2, i.e., the image of the closed unit interval [0, 1] under a topological map a : t --),. R2, and let us use WIENER'S 1 0. D. KELLOGG [1]. Itd/HcKean, DWu oD prooes es

17

7. BROwNian motion in several dimensions

258

test and also a direct probabilistic method to prove P.[m'U = 0] = 1

1)

at each point of B. Consider the case b (0) = 0. Given n so large that 2-" < 16 (1) J, if tl = max (1: 16 (t) = 2_" -') and if t$ = min (1: t > it , 18 (t) = 2-"), then A,=(b(t):t,StSts)CB.=Bn(2-"-'
2)

in addition if Al is the reflection of Al in the straight line A connecting b (t1) and S (Q, then A

lies in the (closed) complement A1, of the unC3+

bounded part of R2-Al U At and 8A11 C Al U As (see diagram 1). KELVIN'S principle: Diagram I

C(B)-' = inf 6>0

3)

f Gdede*

e(B)-1 BXB

.y1

(see problem 7.9.1), coupled with the fact that G for small b - a 1, now justifies the estimates C (A1)

4)

-(2x)'' lgI b - a

C (A=)

and

C(A)-1< f -IgIb-aldede

5)

AxA

< -2lg[length of A] < 2lg(2"+') for nt + oo, where d e is the element of arc length divided by the total length of A, and using diagram 1, it is found that 1/2n

6)

< C(A) S C(A11) = C(8A1q) S C(A1 U A9) S C(A1) + C(A,)

< 3C(A,) 5 3C(Bn) n t +00. causing WIENER'S sum Z n C (Be) to diverge. 1) is immediate from this.

na1 VII

mar.

Coming to the probabilistic proof, let Z. be the event that, for the BROwNian motion starting at 0, x(t) : 0 < t S n-' winds about 0 and cuts itself in the sense that the (closed) complement of the unbounded part of R2 - (x(t) : 0 S t < n-1) contains a disc IS I < e. ins

* G is the GREEN function of 151 < 1.

7.11. Applications of WIENER'S test

259

PO (ZI) > 0 (see diagram 2), and using the measure-preserving scaling x (t) -- Yn x (1/n), it is found that 7)

0 < Po (ZI)

=

P0[ yn x (t/n) : 0 < t < 1 = Po [x (t) : 0 < t < n-'

winds about 0 and cuts itsel/] winds about 0 and cuts itself] i.'

nZI n t +00.

= PO (Za) I PO (+aZ fl

z'a)

But fl Z,a E Bo., and using BLUMENTHAL'S 01 law, 7) implies M ,Z 1

P0(aZ1Za) = 1 = P0[mB = 0],

8)

thanks to the topological fact that the BROwNian path cannot wind about 0 and cut itself for small times without meeting B.

Coming to d ? 3 dimensions, take a positive function h E C [0, + oo) such that b-' h (b) 10 as b 4 0, let B be the thorn

Diagram 2

B:(b2,bt,...,bd)ERd, b;+..-+ba_1Sh(bd),bd?0, o0,

and let us use WIENER'S test to establish the alternative P0[mB = 0] = 0 or I according as 9)

Jig (h (b)Ib) I-1 db

(d=3)

0 ago

converges or diverges;* I

I (h(b)/b)4_3

db

(d> 3)

0

for example, in the 3-dimensional case, if h(b) = e_1Igbl', then

1 ore < 1, III

10) f I lg (h (b)/b) I -1 db converges or diverges according as e 0

""'

and, in the d > 3 dimensional case, if h = b I lg b I', then 1

11) f(h(b)/b)h1_3_. bconverges ordiverges according as e < 0 or e z 0.** 0

pox

* See K. ITO and H. P. McKEAN [1] for a similar alternative for the d-dimensional random walk. ** 11) is to be compared to POINCARIi'S test (see problem 7.10.1). 17*

7. Bitowrtian motion in several dimensions

260

Because b-1 h (b) 10 as b 10, B. = B fl (2-'" -1 S Jb I S 2-"') is long and thin for large m, and this makes it possible to obtain favorable

estimates of the capacity of B. in terms of the capacities of the ellipsoids: bi h(2-`-1)i

+

+

bid-,

h(2-0-1)'

- 3.2---')' S 1 + (ba 2-20-4

and

+... +

E+' 2nh(2 .)s

+ (ba - 3.2---2)' 2nh(21-)'

2- 2W-2

S

1

Iv'

E_:

E+ ) B,,, ) E_ . G. CHRYSTAL [1: 30] presents a neat proof that the NEWTONian capacity (relative to Rd) of the ellipsoid

W

e?

is a constant multiple of the reciprocal of the elliptic integral dt

e= 0

'

y(el + 1) (ell + t) ... (ea + t)

and using .v..

e ,.+ 2 (ell)s-d

12)

(d = 3)

2

.-.

N17

d-3

(d> 3) e'/e..j0

el=es= Vat

tad

to estimate the bounds C (E_) S C (B,,,) S C (E+) as m t + ao, it appears that WIENER'S sum ' 2-'"(d-2) C(B.) diverges or converges according as '"a1 0.0

`-'

can

13)

Ig [2m h (2-`)] -1

(d = 3) diverges or converges.

Z [2`

MZ1

h (2-+*)]d-3

(d > 3)

9) is immediate from this. Incidentally, the proof beginning with 7) above shows that the 2dimensional BROwxian path has an infinite number of double points; the same holds in 3 dimensions but not in 4 or more, as A. DVORETSKY,

a'0

P. Eiu) s, and S. KAKUTANI [1] proved, using the fact that a small segment [x(s) : f :!g s < t + h] of the BROwNian path has a positive capacity in 2 and 3 dimensions but not in 4. DVORETSKY, ERDOs, and KAKUTANI [2] also proved that the 2-dimensional BRoWNian path has

an infinite number of n-fold multiple points for each n = 2, 3, 4, etc.

7.12. DIRICHLET problem

261

and, in collaboration with S. J. TAYLOR [1], that the 3-dimensional BROwxian motion has no triple point; some open problems are to prove a) that the n-fold multiple points of the 2-dimensional BROwxian path have dimension 2 for each n z 2, and b) that the double points of the 3-dimensional BROwxian path have dimension 1. Problem 1. Check that for the 3-dimensional thorn llg.cls

VII

0 S C< eni's Po [m.B = 0] = 0 or i according as e < I or e a 1, and use this to B: Va' _+029 e-l1eelg2c...III._icl

.mob

prove that for the 2-dimensional BROwxian motion, e-llgtlg,t...lew-oil 1'9*t1`, t 0] = 0 or 1 Po[I x(o I > according as e > 1 or not. This answers a question of A. DVORETSKY and P. ERDOS [1: 367]; see also 4.12 for the more precise alternative of F. SPITZER [1: 188]. t

(f (ig.-... 1

-'do

+00 f c1

converges or diverges according as e > 1 or not,

proving the first statement. Consider, for the second part, the projec-

0.I

0..N

tions x$ (x') of the 3-dimensional BROwxian motion onto c = 0 (a = b = 0)

and let &(t) = e- j

19"

(t < e;'). x2(xl) is a 2 (1) dimensional BROwxian motion, and using the law of the iterated logarithm for x', I

o-1

I (t) it is clear that, for e = 1, I xs (1) S f [lx' (1) F S / [1/3 t lgs 11 ] for an infinite number of times t l 0. On the other hand, e > I implies Po [+ x2 (t) I > / [ x' (t) 1), 1 .0] = 1, and using the independence of x' and xs, it follows that, if 0 < Po [ x2 (t) < &(t) an infinite number o l times t t 0], then it will be possible to select tl > 1= > etc. 4 0 such that P. [I x1 (Q I < t., i.o.] = 1. But limP0[1x'(t) 1 < 1] = 0, so this is absurd, and the proof is complete.] t1o

7.12. Dlrichlet problem Consider the Classical DIRICHLET problem for a bounded domain D C RI (d ? 2) : given f E C (8 D), to find a function u E C (D) such that

d u = 0 inside D and lim u(a) = f (b) at each Point b E 8D. a-+b aED

H. LEBESGUE [2: 350-352] discovered that the DIRICHLET problem

is not well-posed in d > 2 dimensions; his example, called Lebesgue's thorn, follows. * Ig, - Ig, Igs = lg(Ig). etc. ; el = e. es = e, etc.

7. BRowNian motion in several dimensions

262

Consider a d L> 3 dimensional spherical surface, push a sharp thorn into its side, deforming it into the surface OD of diagram 1, and think of the domain D enclosed as a little chamber and the tip of the thorn

as a heater, the walls (8D) being held at temperature / E C (8D) (0!5.- tS 1). Consider an extreme case: / m 0 save near the tip of the thorn, at which point f = 1. As t t +oo, the temperature u inside D should converge to the solution of the DIRICRLET problem

with u =./ on D. But the heat radiated from the thorn is proportional to its surface area, and if its tip is sharp enough, a person sitting in the chamber will be cold no Diagram t

matter how close he huddles to the heater; in other words, lim u (a) < 1 = /(0) 1) a-+O

aED

(see COURANT and HILBERT [2: 272-274] for a simple proof of this). Define b E 8 D to be singular if Pb [mRd _ D > 0] = 1, and imitating

J. DooB [2], let us prove that u (a) = Ea (/[x (moD)])

2)

aED

is the solution of the modified Diriehlet problem : to find a bounded /unction u such that d u = 0 inside D and lim u (a) = f (b) at each non-singular a--+b

aED

point b E BD.0

Given a BROWNian path starting at

n

a E D, if m is the passage time to the surface of a little closed sphere B C D centered at a then m8D= in .-f- m8D(m+) m 0

and u (a) = Es(u [x (m)]) is the arithmetic average of u over 8B, i. e., u is harmonic Diagram 2 inside D. Consider next a non-singular point b E 8D(Pb[mRd _D = 0] = 1), let BI be the sphere I b - a I < el, let G be the GREEN function of B,, and select B' : b - a I S es (es < eI) as in diagram 2. Because P. [mB,-D < MOB] is the potential f G de of a non-negative BOREL measure on 0 (B, --- D), it satisfies r < MOB,] = 1 . lim PS[mB,-D < m,3 ,) Z Pb[mB,-D 3) aED

and making es and eI j 0 in that order, it is seen that 4)

s m u(a)

(b)

aED N. WIENER [2] discovered that the modified DIRICHLET problem (with singular point defined in terms Of WIENER'S test) is well-posed. WIENER'S Solution

is the same as the older solution of 0. PERRON [1].

263

7.12. DIRICHLET problem

completing the proof that u is a solution of the modified DIRICHLET problem.

(if possible) a second solution v, select domains D. t D with -D. CGiven D and ODn of class C= and use POINCARE'S test (see 7.10) to check

that a ED,

un(a) = Ea(v[x(maD,)])

5)

converges to v (b) as a E D. approaches b E BD,,. Because the DIRICHLET problem in its classical form is well-posed for such smooth domains,

un = v inside Dn, and granting 6)

P. [x (MOD) singular] = 0,

the fact that v is bounded and approaches i at the non-singular points of OD implies 7)

v = nt+oo lim u = E. ( lim v[x(maDJ]) et+ao

r0.

= E.(/[x(maD)]) = u. Coming to the proof of 6), select open spheres D1 C D C D2 C D3 as in diagram 3 (DI C D C D C D2 C Ds C D3) and, for e> 0,

Diagram 3

VII

let B, be the points of OD at which the electrostatic potential p = P, [mD,-D < MOD, A maD.] is S I - e. B. is compact, U B, is a>o

the set of singular points of aD, p, = P. [MB. < maD, A maD,] S P S I - E on B,, and using GAUSS' principle (see 7.9), it is found that 8)

C(B,) = f Gde,de,= f p, de,< (1 -e)C(B,), B. x B,

B.

where G is the GREEN function of D. - Dl and e, is the electrostatic distribution for B,. But then C (B,) = 0 (e > 0), .K.

9)

Pa[x(maD) singular, MOD < MOD,]

=limP4[x(man)E B., maD= mB.< maD,] .4o S lim .to P4[mB < maD, A maD,]

=0

aED - DI,

and 6) follows on letting D1 shrink to a point. Coming back to LEBESGUE'S thorn, its complete explanation is now before us; it is enough to prove that if b E 8D is singular and if

7. BROwxian motion in several dimensions

264 ..n

f E C (8D) is 1 at b and <1 elsewhere, then lim u (a) < I = f (b) ,

10)

aED

where u is the solution 2) of the modified DIRICHLET problem'. But e with center at this is clear, for if B is the closed sphere I b - a the singular point b, then I > Eb(f[x(m8D)])

11)

Eb(meB < MOD, f [x(maD)])

= Eb(meB < MOD, 9 [x(m8,01) Pb(maB < MOD)

Ib

a'f

s

u(a).

aE

and 10) follows on letting a 10. Given a E D, the distribution h (a, d b) = P. [x (moD) E d b] used in the solution u = h(-, d b) / of the modified DIRICHLET problem is the 8 of

so-called harmonic measure of 8D as viewed from a; see 0. D. KELLOGG [1]

for its electrical interpretation. 7.13. Neumann problem

Consider the open (d z 2) -dimensional ball D : bi +

+ ba < I.

Given f E C (8D) such that f f do= 0*, the classical NEUMANN 8D

problems is to find a function u harmonic on D such that Bu/&n = f on 8D, where 8/On denotes differentiation along the outward pointing normal. N. IKEDA [1: 416-426] discovered a probabilistic method for solving NEUMANN problems as indicated below in the case d = 2. Consider the reflecting BROwNian motion D on the closed unit disc

as the skew product x = [r, 00)] of a reflecting BESSEL motion and an independent standard circular BROwNian motion run with the clock

I(t) _

r (s) -s ds (see 7.15), define a new diffusion D : x' = x

where f = t + t and t is the BESSEL local time at 1, and introduce the space C' (D) of functions u defined and bounded on the closed disc, conti' 7.11.10) and 7.11.11) provide examples of singular thorns. " do is the uniform distribution on 8D. 2 1. G. PETROVSKY l2: 192-196].

7.13. NsuMANN problem

265

nuous on the open disc, and such that u(1, .) is continuous on the perimeter.

rGiven / E C' (D1) and a > 0, it is not difficult to see that Ga f =

E' I f e-°t f(x') dt] is continuous on the closed disc; in addition, u

;-'

up,

G; -- G; + (a - fl) GO, G; = 0 (a, fl> 0) as usual, so that G; C' (D) D((a3')

is independent of a, and if Ga f = 0 for a single a > 0,

then 0 = lim a Ga f = f on the open disc, with the result that at+OO

0=a(G; f)(1,0)

1)

f

= E(1. e) [a

e'1 f[1. 0 (1)] f (dt) 1

(1:r(t)-1)

= E(1. e)

Te-- f f [1, 9 (1)] t (dt)

la 0

+00

Nf(1,0)E(1e)fare-°(t(dt)

at +00

u

-f(1,9), proving that Ga : C' (D) -. D ((1) is 1 : I. A generator 05 can now be defined as usual. D(05*) turns out to be the class of functions u E C (D) such that

= -8ul8n

r
..a

Yu = 03 u

2)

r=1

.^A

is of class C' (D), (53 being the (local) standard BxowNian generator as described in 7.2.3). Given / E C (8D) such that f f do = 0, define f to be f on 8D and 0 OD

inside and introduce u = E. [f' (x')]. IKEDA'S solution of the NEUMANN

problem for f is 3)

v=Go+f'=f*00udt. 0

Consider for the proof the BESSEL expectation E* and integrate out the circular Bxowxian motion from the definition of u: the result is 4)

u = Er [r(f-1) = 1 ,

Ini>o

e-1' 10-1)12 c. else],

7. BROwNian motion in several dimensions

266

c being the n-th FovRIER coefficient of /, and using the estimates +00

VII

try

f Jul dl S +00 f dt E,

5)

0

c-n't(t-x)12 cwI

Inl>o

0

f

=E E+

f

z e-*'"(t)12 l t;n i f (do J

'de/2

+oo

= Er

o

Icn t(dt)

e

In,'>o +00

InI>0

f

e-nst/2

l0

.°3

E;

t(dt)]

_ Z IcnI G(ns/2, r, !)* InI>O

I IcnI G(n'I2, 1 j)

InI>0

Y

T

G(U212, 1, 1)2

and

G (n=/2, 1, 1) r.. 2/n

6)

n t + oo,

+00

r e-at u dl is seen to be bounded as a + 0 and to converge

Ga o

to

+00

v = Ga+ /' = f u dt. But then v = Gs (/' + a v) as is seen on 0

0 in [G;, - Gp + (a - ) Ga G;] / = 0, and it is immediate that v E D ((33') C C' (D) and 4 v = -/', i.e.,

letting

au/

7)

W

as desired.

inside D

on aD

7.14. Space-time Brownian motion Consider the space-time BROwNian motion : 6 (s) = It - s, x (s)]

1)

S ? 0,

or.

x(s) : s Z 0 being the standard d-dimensional BROwNian motion and let P(t, a) be the measure for space-time paths starting at (t, a) E RI x Rd. DOOB [4] noticed that the generator of the space-time BROwNian motion is 01 = - a + z d and used this fact to treat heat flow 0

* G is the reflecting BESSEL GREEN function.

7.14. Space-time BROwNian motion

267

problems

- a + 2du=0

2)

(t,a)EDCRd41

u=IEC(8D) as DIRICHLET problems. Confining ourselves to

(t,a)E8D

the case d = 1,

D=(0,1)x(0,1)=((t,a):0
< a < 1) and imitating the probabilistic solu-

tion of the DIRICHLET problem for. J, it is plau-

sible that the solution u of 2) is the integral of f over 8D relative to the law of the hitting place 3)

6(m)

m=min(s:s>0,i(s)EOD)

as a function of the starting point 6 (0) = (t, a) Diagram I E D of the space-time path (see diagram 1) ; in fact, if mo and m1 are the 4-dimensional BROwxian passage times and if Ea : 0 < a < 1 is the 4-dimensional BROwxian expectation, then 4)

u (t, a) = Eca. a) U (6 (m))]

= E.[mo < m1A1, f(t - mo, 0) ]

+ Es[m1 < moAt, f (t - m1, 4)] pal

+ Ea(t < moA mi, t(0, x(t))]

=

f f(t-s,0)n-1'

nads

o

1

+ f f(t-s,1) o 1

-

' nxe-"'n'8I2sinn2r(1-a)ds

n-1

00

+ f 2 T' e-n'n'1I2 sinnn a sinnn b f (0, b) db, p

n-1

and as the reader will check, this is just the classical (DUHAMEL) solution 'LS

of 2), as presented for example in G. DOETSCH [1: 346-66]. Of course, to be candid, the problem 2) is not well-posed since u = f

cannot be prescribed on I x (0, 1) C 8D. But the integral of 4) is exL10

tended just over D - I x (0, 1) and this good fortune is due to the fact that 1 x (0, 1) as viewed from D is of Parabolic measure 0: 5)

P0, a) (a (m) E I x (0, 1)) = 0

(t, a) E D.

DIRICHLET problems where the knowledge of f on part of 8D speci-

fies the solution look a little strange, but as the reader will recall, the

7. BRowxian motion in several dimensions

268

same thing occurs in the classical DIRICHLET problem for plane regions

possessing a prime end: for example, 0 x [0, 1] is a prime end for the interior C of the mouth of LITTLEWOOD'S crocodile shown in diagram 2, the BROwNian motion stopped on 8C cannot reach it, and the values of / on this end have no influence on the VIRICHLET solution.

Given open D C R2, a point of D is singular for the space-time BRoWNian motion

the hitting time m = inf (s: s > 0, i(s) E 8D) is positive except for a negligible set of paths. BLUMENTHAL'S 01 law shows t-1

if

Ia

c°3

an,

that the alternative is m = 0 except on a negligible set : for example, all the points

Diagram 2

of I x (0, 1) C 0(0, 1)' are singular for the crocodile. Consider, now, the region D of diagram 3 to the left of the graph

of /(-t):t5 0.

KoLMOGOROV'S test (4.12) states that if / E C (0, 1] , if / E f, and if t-i / E 1, then 0 is regular or singular for D according as 1

/e-1'128 di

Z + oo: thus, KOLMOGOROV'S test t°7

f t-' o+

is the WIENER test for the space-time BROWNian motion (see diagram 3). (t)

ERDOS's and FELLER'S proofs of KOLMOGOROV's

test are long and complicated and MoToo's is not

INS

Qty

yes

VET ate:;

elementary either, so it is natural to ask if we cannot give a new proof similar to the probabilistic proof Diagram 3 Of WIENER's test, estimating the chances of meeting (parabolic) slices of the complement of D in terms of (heat) capacities, but this is an open problem. Problem 1. Consider a simple MARxovian motion with probabilities Pa(wj E B) (t ? 0, a E Q, B E B) depending upon time (violating, for a moment, our established usage), i.e., suppose Pig [x(t) E d b] is the unit

mass at b = a and

Pig (wy E B I x (s) : t, S s S t2) = Ple (wi, E B)

is> t,, aEQ, BE B. b E x(ts); the problem is to prove that the space-time motion with state space Q' = [0, + oo) x Q and probabilities Pa. (B') = PQ(w;+ E B') W: t -. (t, x(t)), a' = (s, a) E Q' is simple MARKOV with probabilities no longer depending upon time, i.e.,

that at the expense of adding time as a new coordinate, the temporal dependence of the probabilities can be ignored.

7.15. Spherical Bitowxian motion and skew products

269

Problem 2. Given a smooth positive function u' = v' (t, a) of the pair (t, a) E [0, + oo) x R', if t - x(t) is the sample path of a standard 1-dimensional BROWN.ian motion with probabilities Pa(B) and if, for t a 0, f -1 is the inverse function of the unique solution f = f (s) of

s Z 0,

f(s) = f v-2 [t + f (r), x(r)] dr 0

then the time-dependent motion with probabilities Pa (w+ E B) = P. (x (f -1) E B)

(t. a) E [0, + oo) x R1

au

a

is simple MARKOV and u (t, a) a Pta(x (t1) < b) solves at = I2 us as (t < t1) , while the associated space-time motion is a diffusion with generator

u)(a')=at + 2°= "u

a' =(t,a)E[0,+oo)xR1

(see VOLKONSKII [2] for an ingeneous alternative approach to the above time change).

7.15. Spherical Brownian motion and skew products Define the spherical Brownian motion BM (Sd) to be the diffusion

on the spherical surface S':lxI = I C Rd+1 with generator 1/2 the spherical LAPLACE operator d = dd, closed up as in problem 7.2.1: Ad = (sinp)1-d

1)

a aOF

(sin9)d-1

97 = colatitude,

a a4p

+ (sing')-2dd-1

d1 = 8'/08'.

z.?

PM(

BM (S1) is the projection modulo 2n of the standard 1-dimensional BROWNIan motion BM (R1) onto the unit circle S1. BM (S') is constructed as follows. Given a circular BROWNian motion BM (S1) with sample paths t ---- 0(t) and an independent LEGENDRE process LEG (2) on [0, n] with generator (sin

2)

sin q' a (0 < P < 2r)

2

and sample paths t --> 4p (t), the additive functional (clock) t 3)

T W = f [singq(s)]-'ds 0

converges (t Z 0) if 0 < 97 (0) < rc, and the skew product 4)

x = [q', 0 (1)]

is a diffusion since 0 begins afresh at time 1.

7. Biowxian motion in several dimensions

270

00v

Computing its generator 0, if / is the product of a smooth function of colatitude e- = e_ (qq) vanishing at 0 and n and a smooth function of longitude e+ = e+ (0), then as t + 0, 5)

E. X Ee[/(x)] = E,[e- (9' (t)) Es [e+ (e (I))]]

= E,

[e- (9' (t)) [e+ (6) + 2 e+ (e) I] ] + o (t) (sin 97) -1

= e- (97) e+ (8) +

a

sin p

- e- (4) e+ (9) t

z

+ e- (9') 4e+ (0) (sin 97)-'t + o (1). tea

2 functions, and the desired identification of =A //2 for such

a^v

the skew product and BM (S') follows at once. BM (S4) can be factored in a similar manner as the skew product of the LEGENDRE process LEG (d) with generator 6)

(0 < 9' < n)

(sing7)1-d as (sing7)d-1 a

a and an independent spherical BROWNian motion BM (Sd-1) run with t

the clock I = f (sin q )-2 ds. 0

BM (Rd) can also be factored as the skew product of its radial (BESSEL) part BES(d) with generator (a=

d-i

2ar1 + r

7)

a

(r > o)

ar)

and an independent spherical BROWNian motion BM (Sd -1) run with e

the clock I = f r (s)-2 d s ; the proof is the same. 0

The skew product formula BES (2) x BM (S') = BM (R') can be used to check the result of F. SPITZER [1: 194] that for 2-dimensional BROwNian paths not starting at 0, the total algebraic angle q7 (t) swept out up to time t satisfies a

8)

ufm P. (29p(t)/lgt S a) = (()-1

f

-00

I

+

b

Because BM (S') = BM (R') modulo 2n, q7 is identical in law to a 1-dimensional BROwNian motion BM (R') run with an independent BESSEL clock f r,2 ds; this justifies [_.j_Jr;2ds 9)

E.[eiaP(t)] = E.

[e.--

0

7.1 S. Spherical BROWNian motion and skew products

271

and taking a LAPLACE transform, it is seen that 10)

- a$2 f r(e)-1d

+00

+00

/ = f e-Bt E. [ei0 0)] di = E. [ J

e-Ot e

o

d t]

0

0

P>0

is a radial function and satisfies

[ - 2 ars8 + r a - rs 8

1

1

11)

as

r> 0.

l=1

Computing the GREEN function for 11) and solving, it develops that 12)

/(a) =Klal(t!IPa) f IIaI( 2fl b)2bdb + 0

+oo

f

+ I,

and consulting A. ERDELYI [1 (1): 284 (56)] for the inverse LAPLACE transform, +00

13)

E.[e"c(l)] _ (2t)-1 fe

all + b, Yt

Ilal(ab/t) 2bdb

a = r(0)> 0

0

into which we substitute 2a/lg t in place of a, make t t oo, and conclude as follows :

+00 (2t)-Ir

14)

E.[eix2q(I)I 1 =

e

ai+b' 21

I2Iail1at(a b/t) 2bdb

0

f

+00 e-b"12

121.1/191 -j ) b d b

0 e- IaI

_

f etb(1 + b2)-1 db.

S. BOCHNER [2] gives an instructive discussion of spherical differential processes; the spherical BROWNian motion is a special case. A. R. GALMARINO [1] has proved that the most general d ? 2 dimensional diffusion which is isotropic in the sense that its law is unchanged

by the action of the (full) rotation group 0(d) can be expressed as a skew product of its radial motion and the spherical BROwxian motion BM (Sd-1) run with a stochastic clock f which is an additive functional of the radial motion. Problem 1. Given a 2-dimensional BROWNian motion BM (R2), use time changes and skew products to check that if we take the excursions

7. BROWNIan motion in several dimensions

272

leading into the disc Es: r < 1 and rotate them clockwise as in the picture, closing up the gaps, then the resulting motion is identical in law to the reflecting BROwNian motion BM + (E2) with generator (53+ = J A,

u+(1,0) =0, 0S 0<2r (u+ = outward normal derivative).

[Consider the radial (BESSEL)

part BES (2) of BM (R=) and let f (t) be the time it spends in [0, 1]

up to time t. BES (2) run with the clock f -' is the reflecting process and BM + (E2) is identical in law to a skew product of this sampled BESSEL process and an independent circular BESSEL

BROwNian motion BM (S') .

But the angular part of this skew

e

product is 0 f r (f -1)_2 d sl with 0 = BM (S'), r = BES (2) , and t = measure l0 VII

(s : r(s) S 1, s S f -' (t)), and substituting f (s) for s in the inner intef-' (t)

gral, the resulting 0 f r-2 d f ] is identical in law to the standard io J BROwxian angle 0 f r (s) -2 d sl sampled on the excursions leading J

Lo

.N+

into E8, i.e., sampled on the support of f up to time Problem 2. Consider the plane motion with generator 05 = d/2 (r < 1) = (0/2) d (r Z 1),i use the circular inversion r -+ 1 /r to map the excursions leading out of the unit disc back into it, check that this folded motion is still a diffusion, and find its generator (SS+. [Gi is invariant under circular inversion; this makes the derived motion MARKOV. @+ =A12 subject to u+(1, ) = 0, u E D(CS+), i.e., the derived motion is reflecting BRowNian.] Problem 3. The GREEN function of the d z 2 dimensional BROWNian motion is +CO G(a, a, b) = } f e-at (2xt)-d/2e-Ib-°I`l2tdt 15) ld/2-1

0

r =(27c)-dl2llb?a)/

KdI2-1( 2aJb-aJ)

(see A. ERDELYI [I (1): 146 (29)]). Use the skew product to expand G

in the form 16)

(22r)-110( 2a Ia1)Ko(1 a Ib1)

+x-1ZI.( 2aIal)K+.(f2aIb1)cosn(02-01)

,Zl

a = flal, 01), b = (Ibl, 0,), lalS JbI

7.15. Spherical BRowxian motion and skew products

273

in case d = 2 and, in cased Z 3, in the form `A.

r(d/2)

17)

a

2(d - 2)ndf2

nz

CI(d-2)[(0I (2n + d - 2) , 00]

IaI`-I'll I#(n)( Za IaI) IbII-d/2K#(n)( 2a IbI) (IaI,01),b= (Ib1,01), IaI< IbI, #(n) n (n + d - 2) + (d/2 - 1) 2.

A. ERDELYI [2 (2) : 44 (4)] states 16), but not 17). Cj(d-2) is the GEGENBAUER polynomial figuring in the addition rule for spherical harmonics

of weight n:

r(d/2) 2n + d - 2 C}(d-f) [(01, 02)] 2at'12 d-2 SN("I) Su t V,

_I

I;s m(n)

where m (n) is the number of spherical harmonics of weight n ; see A. ERDELYI [2 (2) : 243 (2)]. [(G. f) (r, 0) I +00

Er If e-°' dt Lo

nZO

Sn)Sn(0) f S,(1V) dip f(r(1),

gyp)] .

r..

where E. is the BESSEL expectation, and so G =nZG"(a, IaI, IbI) r(d12) Znd+d2 2 Cn(d 21 +00

Cdr

2n'!2

[(eI, ds)]

where G" is the GREEN function for BES(d) killed with j.n (n+d - 2) x xrd-3 dr, i.e., G" is the GREEN function that goes with

n=

I

2

d' \dr2 +

d-1 r

d

dr

n(n+d-2)

(r Z 0) :

0

G"(a.,t,) _ I-dJ'-I#(")( 2a

2a r1)

#(n)= n(n+d-2)+(d/2-1)2,Sr).] Problem 4. Construct a diffusion on the closed unit disc E2 identical in law to BM' (E2) up to time mop and such that, for smooth u E D (03),

p2uI(1-, ) + p3(0$u) (1-, .) = p$ 8u'/80+ U*

u(1, 0),

It0/McXean, Diffusion processes

p2, pa, pi, p3 z 0,

p$

pa = 1 18

7. Baowrrian motion in several dimensions

274

'-'

(use skew products and local time). A. VENTCEL' [1] has studied this motion from a different standpoint; see also N. IKEDA [I], T. UENO [2], and 8.5.

[Given a BESSEL motion on [0, 1] with generator W' subject to ps u - (1) + p, (@ 4 u) (1) = 0 for u E D ((53 *), and an independent standard circular BROwxian motion 0, if t is the BESSEL local time at r =1 'ti

ca'

(t = (t - m,) V 0 if p= = 0), then the desired motion is the skew pro[r(t).

°1..

duct [Jr(s)_2 0ds+p;t(t)j+p t(t)1 0

J

indeed, t his motion is MARKOV, it is standard BROWNian up to time nter, GE'

and for smooth /, a brief computation shows that u = G. / satisfies

ps u1(1-, ) + P2 (@ u) (1-, ) = p:' 8u'/89 + ip; 7.16. spinning

Consider the d-dimensional skew product

t ? 0,

[r(t), 0([(t))]

.Ow

1)

of which the radial part is a non-singular conservative diffusion on (0, +oo) with scale s, speed measure m, an entrance non-exit. barrier at 0 (s (0) = - oo), and a non-exit barrier at + oo, and the angular part is an independent (d - 1)-dimensional spherical BRowNian motion

run with the clock I (t) = f t (t, r)1(d r) , where t is radial local time 0

and I is a non-negative BOREL measure finite on subcompacts of (01+00).

A brief computation verifies that the skew product is a diffusion with (local) generator 2)

((33 u) (r, 8) =

u*(dr, 8) + l(dr) }(du) (r, 8)

md rI ) 11

(r + 0)

%

and GREEN operators 3)

Ga e* e-

= E. °e1te+[r(i)] dt fPOt, , 0) e- (0) dOI

J.

where A is the (d - 1)-dimensional spherical LAPLACE operator. e* = e* (r) is a radial function, e_ = e_ (0) is a spherical function, E. is the radial expectation operator, and the kernel under the integral sign

7. 1 6. Spinning

275

is the transition function p (t 01, 02) =

4)

,

n-0

eY"` X S;, (01) S (02) !$m(n)

for the spherical BRowuian motion, y being the common eigenvalue - j n (n + d - 2) of the spherical harmonics S;.: I S m (n) of weight

n (4dS*=ynS .

Question : can the skew product be completed so as to be a di/f usion on the whole of Rd?

The answer is no if

i

f s(t)1(dE)> -oo,

5)

°».a

0

for the completed GREEN operators have to map C (Rd) into itself, and C-1

if e, E C [0, + oo), if e, (0) = 0, and if e_ is a spherical harmonic of weight n > 1, then the product e+ e- E C (RI) and 6)

(G. e. a-) (0) = li m (G,, e, a-) (e, 0) +00

= lim E, f e-"t e, eY* I dl e_ (0) ,to

L 0

+00

r = E0 I J, a-at e+ eY.. I d t e- (0)

cannot be independent of 0 unless +00

E0

f f e-at er.1 dt] = 0, 0

i.e., unless

Po[t(0+) = +00] = 1,

7)

and, as we know from 4.6, 1

8)

PO [t (0 +) = + oo] = 0 or I according as f s (E) I (d i) 0

converges or diverges (to - oo). .s4

7) means that the sample path entering from 0 comes in spinning like a spherical BRowxian motion defined for - oo < t:5: 0 as t comes

in from - oo. Consider for the proof of this statement, the angular part 8+ of the entering sample path and note that PO [0 + (t) E d 9 1 r (s) : s L> 0] must be the uniform spherical distribution. 18*

276

7. BROwNian motion in several dimensions

Given 1>11=1 -s1>ts= '- s,> ... >t,=l -s,>O and letting 1-(1) = 1(t, w'-g) (0;5 t S 1), it is not difficult to see that

Polo+(1 -sl)Ed01,0+(1 -s,)EdO2,..., 0+(1 -s,JEdO,Jr(1):OS1S 1,0+(1)]

9)

tn), wij, 0n, 0n-1) do.-,

= don

102 -t3),wy),03,02d0, P(1(tl - t,), wj), 0,, 01) d01 P(1(1 - t1), wt,) 01, 0+(1)) = P(1-(SJ. 0+(1), 01) dO1 p (I- (S2) - I- (s1) ,

01 ,

.. P(I-(se) - I-(Sn-1),

00 d8,

0n-1, on) don.

But this is just the statement that, conditional on r(t) : 0 S t' 1 A

and 0+(i) , 0+ (1 - s) : 0 S s S 1 is identical in law to 0(1-(s)):0!5'-s<1

with 0(0) = 0+(1), and thanks to um f t(t, r, wi_i)1(dr) = 1(1) _ +00,

10)

0

the proof of the proposed interpretation of 7) is complete. 1

At no extra cost, it is seen that f s d1 = - oo is also sufficient for 0

the skew product to admit a completion : in fact, if r (0) = 0, if 0 is

a'°

an independent spherical BROwxian motion defined for - oo < t < + oo,

and if the law of 0(0) is the uniform spherical distribution, then it is not difficult to see that the entering sample path

1=0 0<1S 1 1> I

0

[r(t),0(-1-(1 -t))]

11)

[r (t) , 0 (1(1 - 1, w+))]

is MARKovaan and identical in law to the skew product 1) on Rd - 0 with GREEN operators mapping C (R4) into itself. Problem 1. Give complete proof of all the statements about 11). Problem 2. Compute the rate of growth of I-(t) (it 1) for the generators + d- 1 2

aY

ar +

d)

s > 0, d Z 2,

using the ERDOS-DVORETSKY and SPITZER tests of 4.12. Problem 3. Give all the isotropic MARxovian completions of the skew I

product 1) in case f s 0

l (d.) > - oo.

7.17. An individual ergodic theorem

277

7.17. An individual ergodic theorem for the standard 2-dimensional Brownian motion

Consider a standard 2-dimensional BROwNian motion. Given an additive functional e = e (l) of the BROwxian path, we can determine a non-negative measure a (d b) such that Ea[e(,n D)] = f GD (a, b) e (d b)

1)

a E D,

D

for every GREENian domain D, where GD is the GREEN function of D. ./V

To prove this fact, it is enough to note that the left side of 1) is an ex-

cessive function of a E D and to use the results of section 7.5; see H. P. McKEAN, JR. and H. TANAKA [1] and also 7.19 for further information on such BROwxian additive functionals. As in the 1-dimensional case (6.8),

P.

2a)

im e,(t) [10 T,_(W)

=

e,(R=)

e.(RI)

I

in case 0 < e=(R2) < -I- 00,

and 1

P- [!gt e (t)
2b)

rlim

=I-e' uZ0,0<e(R2)<+oo

will be proved below. An elementary example of an additive

as

functional is e(t) = J /[x(s)] ds

3)

U

with e(db) = 2/(b) db; for this special case, 2a) was proved by [private communication] and by G. MARUYAMA and H. TANAKA [2], and 2b) was proved by G. KALLIANPUR and H. ROBBINS [1].

Coming to the proof of 2), suppose x(0) = 0 (this is harmless) and putting m1= min (1: x I =1) , define in' (n 0) to be the successive passage times to I a = I via I a I = 2, i. e., let

m° = in,, in" = m"-'+ m(wm.-,) (n z 1), in = ma + m,(wm,)

4)

0.y

Because of the isotropic property of x, the chain e = e(mn) +"O+

e (in" -') : n Z I is shift invariant and BIRKHOFF'S individual ergodic theorem implies that y = lim a (m")/n exists. y is measurable on the nt+00 tail algebra fl B [x (t) : t > n], and this algebra is the same as C's

"al

CND

fl B [I x (1 /t) : t < 1 In]. Because t x (1 /t) : t > 0 is a version of x (t) with

nal

respect to P°, y is constant by BLUMENTHAL'S 01 law, and BIRKHOFF'S

ergodic theorem implies that y = E° (e,) i.e., 5)

P°

Ie(n")

..r

,

Eo(ei)]

= 1.

7. BRowNian motion in several dimensions

278

try

Using the additive property of e and the isotropic property of x, one can easily see that 2n

6) E0(el) =E0[e(ml) - e(m°)] = 2n

0

2n

nE(2,e)(e(m, A

zn f d6 EE(i. o)(e(m$)) 0

2n

f d6 [fR G.,((i, 0), b)e(db)+lim fG1((2. 0), b) e(db) a

`O.

2n

fe(db) 2n f d@ [Go1((1, 0), b) +, to G,n((2, 0), b)] de,

R

...

where G0, is the GREEN function of the domain Ia I < 2 and G1s that

of I < Iai
.~+

err

+:.

The integrand G (b) in the last integral is clearly a radial function G (fib i) because of the isotropic property of x and is lower semi-continuous as an increasing limit of continuous functions. Applying 6) to 5) with a non-negative radial function f, it develops C1.

that nil

+oo .al

'!!

[ f f [x (s) ] d s] = 2R f G (I b I) 1(b) d b = 4X f G (i) f (r) r d r . 7)

E0

nto

-I.

On the other hand we can compute this expectation in terms of the r (t) = I x (t) I as

BESSEL process D m

8

E°[

f me

- E[T/[r(s)] ds

fx

I)ds ]=

0

( m' = the BESSEL passage time to I via 2)

= E; If (m', r) f (r) 2rdr]

(t' = the BESSEL local time)

+o0

= 2 f E; [t' (m', r)] f (r) rdr 0

,0

=2 f lg2 f(r)rdr 0

[see 6.8.6)]. .per

Comparing 7) with 8) we obtain G (r) = lg 2/2x for almost all r; since G (r) is lower semi-continuous, this is true for all r, i.e., e(mN) = lg2e(R2) = 1 9)

P lim P. [nj+oo

n

2n

and 2a) follows from 9) as in section 6.8.

]

-

7.18. Covering BROwxian motions

279 I

Because of 2a), it is enough to prove 2b) for e(t) =f f [I x(s) f] ds 0 $4.

with / a 0 and 0 < f f (r) r d r < + oo; this can be done in terms of the BESSEL process D' : r (t) = x (t) I as follows :

P. [ g t e (t) < u e (R=)

+

e

=1'0gt f f[r(s)]ds<4uf f(r)rdr u

o

=Po [.1.a1j/[r(s )]ds
0

by problem 6.8.4, this tends to t - e-" as t t + 00. For additional information and the ergodic theorem for general MARKOV processes, see T. HARRIS and H. ROBB1NS [1], G. MARUYAMA and H. TANAKA [2] and T. UENO [1].

7.18. Covering Brownian motions Given an unramified covering surface K of an open connected region a0'

D C R2, let j denote the natural projection of K onto D, and selecting a point o of K, consider the covering path' starting at o that lies over the standard BROwNian path x(t) : t < man K ()_ starting at j (o) as in diagram 9 . Given a MARKOV time in for the covering motion and projecting down via j, it is clear that m is also MARKOV for the ground motion, and

4

tad .-.

since the latter starts afresh at time m, it follows

from the definition of covering path that the covering motion begins afresh at time in also.

.w,

'ti

We see at once that the covering motion is a diffusion and that, in the small, its generator is 0 u = J [d u (j-1)] (1), where i-1 is the local inverse of the local homeomorphism j. With the help of coverings we can give a simple proof of P. LivY's result [3: 270] that if j is a non-constant regular function on the open ja;) - j(g) -j,) connected region DC R2 and if j' is never 0 on D,

then the composition of j and the standard

Diagram I

BROwxian motion on D is the standard BROwNian motion on 1'(D) with

a change of clock: in detail, D is an unramified covering of j (D) with ' SEIFERT and THRELFAU. (1: 181-1851.

7. BROwxian motion in several dimensions

280

natural projection j, the covering motion of the standard Bxowxian motion on j (D) is standard BRowNian on D with a change of clock, and inverting the clock in ground and covering paths, it is seen that the projection via j of the standard BRowNian motion on D is the standard BROWNian motion on j (D) run with the inverse clock.

We learn from P. LEvy's result that, up to a change of clock, the 2-dimensional BROWNian path is a conformal invariant.

Consider, as the simplest example, the stereographic projection j of the 2-sphere S2 onto R2:

j:(x1,x2,x3)Es

xlxs

``

(SS = 2 (1 - cosV)s IsinV

)ER2.

x=x a

v smy' app + sinsip e,

J ,

:J'

...

where 0 S V S rr is colatitude and 0 S 0 < 2xr is longitude; thus, the covering (stereographic) BROwxian motion is the standard spherical BROwNian motion BM (S2), run with the clock e

`>O

where

f -1(t),

f (t) = f [1 - cosip(s)]-2 ds and ye(t) : t ? 0 0

is the colatitude of BM (S2).

MI.

Consider, for the next example, the projection j : w - e'0 of the RIEMANN surface RE for w = lgz onto R2 - 0

q

as in diagram 2.

4 ci

2xi

(SS = 2 flw) l-2 A=

(

as

+ 89/1'

2

w = w, + i w1;

where

thus, the covering motion is the standard 2-dimensional BRowNian motion BM (R2) run e

x(t): I z 0

with the clock f-1(t), where f (t) = f e2--(I)ds 0

and x, (t) : t ? 0 is the horizontal component of BM (R2)

.

BM (R2) hits each disk 1.0.; thus, the covering motion does also, and since the covering path leads from strip i to strip j when the ground path winds (counterclockDiagram 2

wise) j - i times about 0, it is seen that

the ground path winds counterclockwise about 0 an infinite number of times and also unwinds itself an infinite number of times (i.e., the covering motion returns to the strip 0 S w1 < 2x).

7.18. Covering BROWNiaU motions

281

...

Consider, for the third example, the projection j of the elliptic modular figure' of diagram 3 onto the punctured plane R2 - 0 U 1.

w+1

j is the composition of the fractional linear map i(w) = -2 i W - i of the open unit disc 1w I < I onto w$ > 0 and the modulus j$ = k$=O Baa of the ,JACOBI elliptic functions mapping w2 > 0 onto the punctured a= + aw) for I w < 1, so now the covering plane. 0 1 = 2 I 1' (w) 1 -2 l owl, motion is BM (R2) run with a 00 clock f-1(0 that blows up as the covering path runs out to 1w I =1, and since the sheets of the covering surface correspond to the elements of the fundamental group of the punctured plane (free group of 2 generators), it is seen that 1

loo

IUD

the ground BROWNian motion

loops about 0 and 1 worse and worse as t f + oo and never gets unlooped.

00

Go

in'

S. KAKUTANI [3] is a source

of additional information on

.14 I

I

I

ii "

covering BROwNian motions. Problem 1. What is the reason that the ground BROwNian

motion cannot unloop itself in the twice-punctured plane?

[By the ergodic theorem for BM (R2) (7.17), the fraction lab

of time that the BROwxian path spends in I w I < 1 up to time t is something like (Igt)-'. Winding about a single

Diagram 3

apps' O.0

point depends on angle not radius; with 2 points, the path can wind about one and not the other ...

and (lg t) -' t is not enough time to unwind the loops thus produced.] Problem 2. Prove that for the spherical BROWNian motion BM (SI),

~i.

t.,

P.it-' f/(xs)ds= f/do]=1 [urn et+oo

/ELI(S

using the ergodic theorem for BM (R2) and the stereographic projection. 1 COURANT and HURWITZ (1: 432].

7. BRowNian motion in several dimensions

282

Problem 3. Give a new proof of the result of problem 2 for BM (Se) i

(d ? 2) assuming that e(/) = lim t-I f /(x,) ds exists. 8t+00

...

[Granting that e(6 exists, consider the tail algebra A= fl B[x (s) : :>o s Z fl. Given A E A, p = P. (A) satisfies A p = 0 so that p is constant, and we infer that P. (A) - 0 or I. e (/) is measurable A and therefore constant as regards w, and its invariance under rotations forces e w = f f do, as claimed; see 8.7 for an amplification of this method.]

s'

A

Problem 4. Consider the group 1' of non-euclidean (PoINCAxi?) motions of the open unit disc E2. Give a proof that l leaves unchanged the law of the geometrical curve x (0, ma$.) that the standard BROwNian traveller sweeps out for t < ma8e and find the most general topological map of the open disc onto itself that does this. [Given a non-euclidean motion g and a standard BRowNian path, g OA A map) is a standard BROwxian motion stopped on 8E2, up to a change of time scale, and so the law of x [0, max,) is invariant under g. Given a topological mapping g1: E2 - E2 leaving the law of x[0, mar) unchanged, select a non-euclidean motion g2 : g1 (0) -- 0 and consider 9:,

gs = gs g1Ib

E B] = 2n f IeIP b12 dfl

Pa[gs x(maE) E B) =

(b =gsa)

B .U.

,L"

is harmonic for each arc B CI 8E2. i6 -l b1, is therefore harmonic in a VII

for each 0S # < 2x. Ie a - all is likewise harmonic in b for each 0 S a< 2x (use gal in place of g3), and it follows that to each s

a E [0, 2x) corresponds a fi E [0, 2x) such that Ie'= a constant multiple of Because gs (0) = 0, the constant is 1, and I

b12 .

I

le'0

I-1als

min

Osa<2n lei: - als

+ I b , is likewise

Ibis

t + lals - 1 + lbls

and so gs is a topological map of each circle I a I = Z < 1 onto itself. g2 can now be adjusted so as to make gs orientation preserving on all the circles I a I = I< 1 at s s once, and using a hall, to prove that e-`a a and e-0 b t

l

IaI=IbI, 1 + lags , i. e.,

= l Ib

owe

have the same real part, it follows that, b = es" a (y = - a), i.e., gs is a rotation. g1 can now be identified as the non-euclidean motion

'.7

...

gs 1

g3 , and the problem is solved.]

7.19. Diffusions with Baowxian bitting probabilities

283

7.19. Diffusions with Brownian hitting probabilities Given a bounded d-dimensional domain D, let h (a, d b) = haD(a , d b)

be the classical distribution of harmonic mass on OD as viewed from a E D and recall that for the standard Bxowxian motion 1) h(a, db) = Pa[x(iOD) E db] (see 7.12). A diffusion on Rd U 00 is said to have Brownian hitting Probabilities if 1) holds for all bounded D C Rd; in the special case d = 1,

each diffusion in its natural scale has Baowxian hitting probabilities. Here it will be proved that the generator 0 of a d-dimensional diffusion with BRowNian hitting probabilities can be expressed as 2) u) (a) _ m(da)

m being a suitable speed measure and e" the RIESZ measure of uED((3}); see 7.5.

4'"

Given bounded D C Rd, let / E C (Rd) be = 0 on b and positive outside; then 053 Gi / = G, / is bounded below by a positive constant c on D so that (5 applied to u = G1 f/c is Z 1 on D, and an application of DYNKIN'S formula gives MOD

3)

eD (a) = Ea(maD) S Ea

f

O u dt

0

IAA

= Ea[u(z(moD))1- u(a) Ilull00 < +00 inside D. Define in = MOD, for D1 C D; then for a E D1, eD(a) = Ea (m + ma D (wm)) 4) = eD,(a) + f haD1(a, d b) eD(b) > f haD, (a, d b) eD (b), 0

and using the RIEsz decomposition [see 7.5], eD (a) is found to be the potential f GD (a, b) m(d b) * of a non-negative charge distribution m, positive on opens. Using the composition rule GD(a, b) = GD,(a, b) + f haD,(a, d6) and the resulting formula for eD, : 5)

6)

b)

a, b E D1 C D

eD, (a) = eD (a) - f haD, (a, d b) eD (b)

- f h0D,(a,

= f GD(a, b) m(db) = f GD,(a, b) m (d b),

one finds that m does not depend upon D. 0 GD is the GREEN function of D.

f GD(E, b) m(db)

7. Beovmian motion in several dimensions

284

As to 2), if (31 applied to u, E D ((b) is S - 1 on D, if u E D (@),

and if U2=U+ n u, , then O$ u2 5 0 on D for t Qt 1 10 u llo and for D, C D, DYNKIN'S formula implies -OD, I00

-A_

v (a) - f hoD1 (a, db) v (b) = -Ea f CS v dt

7)

0

0

aED, v=u, or u=. A second application of the RIEsz decomposition now permits us to express u (a) - f haD (a, d b) u(b) as the potential of a signed charge distribution e", and since non-negative potentials come from nonnegative total charges, the bounds 8a)

a f GD (a , b) m (d b) = a eD (a)

x am Df 45 u

mOD VII

S Ea f @ u dt = f haD(a, db) u(b) j u(a)

f GD(a, b) a"(db)

0

S P eD (a) = P f GD (a, b) m (db)

sup GS U D

show that a m (D) S -e" (D) S P m (D), and 2) follows at once because ® u is continuous. CAD

8b)

b.>

H. P. McKEAN, JR. and H. TANAKA [1] described the speed measures

of a wider class of motions with BROWNian hitting probabilities; in the .,,

...

present case, it can be proved that m is a speed measure if and only if each of its potentials eD = f GD dm is continuous on D and tends to 0 at the non-singular points of 8D, and each of its null sets Z is thin in the sense that, for the standard BROWNian motion,

P.[Zfl(x(t):t>o)=f]=1. .ti

Given such a measure, it is natural to hope that the associated diffusion will be the composition of the standard BROWNian motion with a stochastic clock f -1, e. g., the inverse function of the HELLINGER integral 9)

measure (s : x(s) E db, s

f

W

J

t) m(db)S

2db

Re VI'

and something close to this is true (see H. P. McKEAN, JR and H. TANAKA [1], R. BLUMENTHAL, R. GETOOR, and H. P. MCKEAN, JR [1],

1"1

and also 8.3). Because the 2-dimensional BROWNian motion is persistent, so is the motion attached to (SS in case d = 2. In d:->-' 3 dimensions it drifts * measure (s: x (s) E db, s 5 t) is not a multiple of LEBESGUE measure in dimensions d 2, i. e., local times cannot be defined.

7.19. Diffusion with BROwNian hitting probabilities

285

out to oo. P. (m00 < + oo)* = 1 if Rd can be split into A U B such that

f IbI2-dm(db) < +oo

10)

d

.ti

and B is thin at oo in the sense that, for the BRowNian motion, P. [x(t)EB i. o., It +oo] =0. 11) P. (mw < + oo) - 0 otherwise. WIENER'S test can be used to decide if 11) holds or not. Problem 1. Given a standard 2-dimensional BRowxian motion with components a and b, let t be the local time l m (2e)'1 measure (s : 0 < b (s) < e , s S t) ; the problem is to compute the speed measure e'

and the generator

C33'

of the motion D' : x' = x (f -1) (f = t + 2t)

(see T. UENO [1] for a different description of D'). [e' (d a x d b) = 2dadb on I b l> 0; also, e' (d a x0) is translation invariant and, as such, a constant multiple of the 1-dimensional LEBESGUE measi

f [i + E-' x the indicator of 0 S b< E](x,) ds

ure da. Because f = lim

e40 0

and

f /2dadb+ f /2da

R'

R' x 0

= lim f [1 + E-' x the indicator of 0 S b < e] / 2dadb, 40 R'

e' (da X 0) = 2da, and it follows that if u E D (W) and if A x B is an open rectangle meeting R' x 0, then, up to functions of class C' (A x B), - u = f GCS' 1! 2da on A x B, where G is the GREEN function of X

AxB

B. Given a E A, it is immediate that

A

l` o (2,-)-l [u (a, e) - 2 u (a, 0) + u (a, - E) ] = -1 Zm (2

f

E)-r

J 2n

AxO

= 1` 0

J (a-a')=+e' R'x0

l

= (CAS"

)'

lg (a (a a }, + -c2 t ' u (a', 0) 2da'

(5S' u(a', 0) 2da'

u) (a, 0)-

D ((J) can now be described as the class of functions u E C (R2) such

that

u' =du/2 =1 0 6-1 [u (a, e) - 2u (a, 0) + u (a, -E)]

0//R1x0 on R' X O

is of class C (R2), A being the local BROWNian generator of 7.2.3). 0' u = u' for u E D (05*).] * m0

(:: x (t -) = oo)

7. BROWNian motion in several dimensions

286

7.20. Bight-continuous paths

Consider the general motion of 7.1, but now let the sample path be merely right-continuous. (3 can be introduced as before and DYNKIN'S formula applies:

u) (a) = li mEa(m)-1[Ea[u(xm)] - u(a)]

1)

where m is the exit time inf (t: x (t) & B) from a small closed neighborhood B of a. P6(xm E db) need not be concentrated on 8B because the

path is permitted to jump out, and this is reflected in the fact that (K can be a global operator. Consider, as the simplest case, the 1-sided differential process on [0, + oo) with LEvY formula +00

2)

Eo[e-"s(t)]

= exp (_ t [ma + f (1 - e-°1) n(dl)]) `` 0+ +00

mz0,n(dl)z0' f (1 -e')n(dl)<+oo 0+

and probabilities

az0

Pa(B)=Po(x+aEB)

3) C')

(see the note on differential processes placed at the end of 1.8). Given / E C [0, + oo) and a z 0, let /6 (b) = /(a -+ b) and let G, be the GREEN operator. Because (G, f) (b) - (G, f) (a) = G, (fb - fa) (0), Cl (CS) = D (Os) n c, [0, + oo) includes G, C1 (0, + oo), i. e., C' (@) is well populated.

0 is to be computed on C' (0$). Given y > 0, u = e-y1 satisfies G, u = fl u with so belongs to D (03), and now a comparison of

(G, u) (0) and

+"0

4a)

(G3 u) (0) =lim f (e-Y' - 1) ng(dl) GLO

0

n1(dl) = E0(m.)-1 Po[x(mj E dl]

Me = inf (1: x (t) > e) and

4b)

((5i u) (0) = li+m 1-1

Eo[e-Y'

) - 1]

+00

_-my + f 0+

(e-y1-1)n(dl)

implies that (1 - e-1) n.(dl) converges as e l 0 to the measure n (dl) _ (1 - e-1) s (dl) 5) n' (0) =' M.

1>0

287

7.20. Right-continuous paths r..

But if u E CI (®), then u(1) - u (0)

1>0 A

u' (l) -

6)

1=0

=u+(0) belongs to C [0, + oo) so that +00

(CSS u) (0)

7)

ii o f [u (l) - u (0)] n.(dl) 0

- e-1) ne(dl) = f u' n' (dl) +00

+00 .`t

_ li m f u' (l) (1 +00

= m "+(0) + f [u (1) - u (0)] n (d l) , 0

and using (CSi u) (a) = ((5S ua) (0), it is seen that

(0 u) (a) = m u+(a) + f (u (b + a) - u(a)] n(db)

8)

a ? 0,

0

completing the identification of ( 3 restricted to C' (Ci) . Problem I after E. B. DYNKIN [4: 58]. Check that for the i-sided

stable process with exponent 0 < a < I and rate 1/2 (m = 0, +CO

f (1

- ey1) do = y'/2),

o+

Po [x (m,) < 1] =

f 1

sine a

r(1 - t)-m t,_I dt

n

1> e

8!1

2 ea

Eo(m.)=r(a+1) +00

[Given y > 0. f Eo[eyZ(')] di 0

+00

+00

= f e-y1 f dt PO (x(t) E dl) = 2y-' 0

0

+00

= 2r(a)-I f e-y11'- I dl, 0 +00

f dt Po(x(t) E dl) = 2r(a)-I P` dl, 0

and it follows that if f E C [0, + oo) vanishes near + oo and to the left * DYNRIN'S evaluation of E,(m.) contains an incorrect constant.

7. BROwrrian motion in several dimensions

288

of e, then (Go. /) (0) = Eo [11(x) dtJ

= 2r(a)-' f as-' J(a) da a..

= Eo [T/(x(i -m.)) dt +00

= f PO [x (m.) E d a] (Go + I) (a) 0

+00

+00

=f Po[x(m.)Eda]2r(x)-' f

(b-a)a-'/(b)db

a

0

+00

b

=2r(a)-1 f Idb f (b-a)°-'PO [x(m.)Eda], 0

0

oho

or, what interests us, b

f (b - a)°-' P. (x (m.) E d a) = b" -' or 0 according as b ? e or b < e;

now take LAPLACE transforms on both sides and solve. Coming to E. (in.), ``)

it is clear that the scaling x(t) -- e x (t/e°) preserves the measure PO (B). in, is therefore identical in law to a nil, E0 (m.) = £ ' E0(m1) , and the W.?

evaluation E0(ml) = 2/r(a + 1) follows on putting u = e-i (l ? 0) and comparing

=im

t-1 Eo[e-'()

-

"a'

((5 u) (0) and

(G. u) (0) _ -lim E.(m.)-1 Eo[e z(m`) .j0

- i]

f e-'- 1 81v

+CIO

=limE0 (m')-1 sinks a

dl

(1 - e)°

440

1

s

=f el' - l dl/Eo(ml)1'(a) I'(i - a) +00

0

_ -i/Eo(m1) ar(a).] 7.21. Riesz potentials

We know that the NEWroNian potentials

f lb-al2-de(db)

1)

e?O,dz3

are linked to the d-dimensional BxowNian motion via the formula 00

f (22r t)-d/2 g-16 -111'/29

2) 0

dt = constant x Ib - a12-d.

7.21. Ri esz potentials

289

What about the fractional-dimensional (RiEsz) potentials

fib - aIy-de(db)

3)

e

0

0<
d=f

0
d2?

Given 0 < y S 2, the FoURIER transform 4) pY(t, a, b) = (2;r) -d f ei(b-a)-ce +oo

=

t

IcIydc 2

y

(2n)-df e__2 ' rd-'dr f eilb-aircosedo st-1

p

_

(b - a).c= lb- ai rcosO

+00 _ -L ,Y

f e- 2

rd12Jd

z

o

(ib-air)dr t>0, a,bERd

is non-negative. p2 is the d-dimensional GAUSS kernel', and imitating h'1

the construction of d-dimensional WIENER measure from p=, it is possible 'L7

to construct an d-dimensional (strict) MARKOV process with rightcontinuous paths and transition probabilities p,(t, a, b) db; this is the so-called isotropic stable process with characteristic exponent y. fda is small near oo, the Given / E C(Rd) such that f = f in,

+.,

f into (2a) -4f

r.,

I:' GREEN operator Ga : f - E. f e-ai f (xi) dtl sends operator l0

Rd

(a + f I c ly) -' f d c, and introducing the 03 : f --> - j (2a)-d f i c iy / do , Rd

it is clear that (a - (J) Ga f = f ; in short, the generator of the isotropic stable process is 5)

par'

apart from the technical definition of its domain. (-A)Y12 means the

non-negative root of the non-negative operator -A in the sense of spectral resolution (FouRUER transform) over the HILBERT space L2(Rd, db); see S. BocHNER [1).

(3 is a global operator if y < 2. a b is the inner product in R4. i Here, the nuisance factor 1/2 in the GAUSS kernel is dropped. Itb/McKean, Diffusion psooesmes

19

7. Bxowxian motion in several dimensions

290

Coming to the potential kernel, +00

6)

f Y (t, a, b) d t +oo d

0

=2(2J)412 f r2YJd_1(lb-al r)dr o

2

+oo d

--2(2 70)-d/2 f r2 -YJd18-1(r)drX I b-aIY-d 0

=+oo <+00

d=1,yZ1 d=1, y<1 or d:>- 2, <2,

and this is just the kernel for the (d - y)-dimensional potentials of M. RIESZ [1]; compare W. FELLER [13].

Given compact B C Rd and y < I (d = 1) or y < 2 (d Z 2), if mB is the hitting time inf (t: i > 0, x (t) E B), then P. (mB < + oo) is the electrostatic potential, i.e., it is the greatest of the potentials

p-.f

7)

B

the electrostatic distribution 8) e(db) = l ji r-' db x Pb(mB < +o0, mB(w.) = +oo)

o

ewe

y+.,-w

and the electrostatic (d - y)-dimensional RIESZ capacity C (B) = e (B) enjoy most but not all of the properties of the NEwTONian capacity (y = 2) ; for instance, the electrostatic distribution of a solid sphere B spreads out over the whole interior and C (8B) = 0 (see problem 2 for detailed information on this point). WIENER'S test for the singular points of the electrostatic potential takes the form Pa(mB = 0) = 0 or 1 according as Z 2(d-y)" C(B,) Z oo, 9) ANN

"Z1 where a E B and B. is the meet of B and the spherical shell 2-" -' S I a - b I L1. +-1

< 2-"; the proof is the same as in the BROwxian case. Problem 1. Given a 1-sided stable process with exponent 0 < a < 1 and rate 1/2, prove that for compact B C R', P. (mB < + oo) is the (1 - a)-dimensional electrostatic potential, i.e., the greatest of the potentials

a'°

CIO

p(a) = r(a) ,1r bZa

(b

`(a)iS 1, e Z 0, e(R' - B) = 0.

Problem 2. Compute the (1 - a)-dimensional capacity

e(db)-

,f (b - a)

bZa

'II

2 C1-a(B) -a maxe(B) : e L> 0, e(R' - B) = 0, T(a)

1

8.1. Similar diffusions

291

of an interval B = (a, b].

[e(db)

= dbf(1 - b)a 2 r(1 - a) is the solution of 1

e(db)

= 1 (0 < a < 1), III

2

1'(a) J (b - a)' a

C, -,[0, 1] is the total charge e[0, 1] = 1/2r(2 - a), and

Cl-a[O,1)= 21'(2` a)

C1[a,b]=C1-a[0,l]=11

t

1=b-a.l

'V'

Problem 3. Given an isotropic d (z 2)-dimensional stable process with exponent 0 < a < 2 and E0 [e` 711(0) = e-1 jrja/2 (y E Rd) , prove

that for compact B C Rd, P. (mB < + oo) is the (d - a)-dimensional electrostatic potential, i.e., the greatest potential r (d-a p(a)

2

di22a_1r

(2)

fIb_ad_ti10160

Problem 4 after PoLYA and SZEGO [1: 39]. Prove that 1 n 2n

a-8

a

J I (1 - r2)2(2 - 2r 000

costo -}- r2)

2

r2 sinty dr dtp d8

2rr(2)r(2 2 a)

05 ES 1

and conclude that (3 - a)-dimensional capacity C3_a (B) of the 3-dimensional ball B:! b I S 1 is

=21-s,r(3

2

a)r(1 2 ")

Draw a picture of the electrostatic distribution e3-a(db) - V

2a-s(1 -

IbJ2)--12db

fbI s I

r(2 Za)r(3 Za)

for a near 0, a = 1, and a near 2 and give a probabilistic interpretation of its changing shape.

8. A general view of diffusion in several dimensions 8.1. Similar diffusions Given a (conservative) diffusion D on a space Q as described in 7.1,

its generator Ci can be expressed in terms of the hitting probabilities 19*

8. A general view of diffusion in several dimensions

292

and mean exit times 1a)

h(a,db) =hoD(a,db) =P.[x(maD)Edb, m,D<+00]

1 b)

e (a) = eD (a) = Ea (ma p)

w°.

for open D C Q via E. B. DYNKIN'S formula (0 u) (a) = Dim e (a) -1

2)

f h (a, d b) u (b) l OD

- u (a) ]

u E D ((3S) ;

J

to borrow a phrase of W. FELLER'S, h is the road map, i.e., it tells what routes the particle is permitted to travel, and e governs the speed. A diffusion is said to be similar to another if there is a homeomor-

phism between their state spaces mapping the road map of one onto the road map of the other.

Given a non-singular conservative diffusion on the open line Q = RI, its hitting probabilities can be expressed as 4)

s(b) - s(a)

and two such diffusions are similar if and only if s (Q) and s' (Q) are y;.

similar under the euclidean group, i.e., if and only if s (Q) and s' (Q) are both R', or both half-lines, or both bounded. , C ons ider, as a secon d

°'x

example, the standard 2dimensional BROwxian mo-

tion and the space-time BROwxian motion. D:mm +

Given a standard BROWN-

ian path starting at the

.%w

center of a circle y and a homeomorphism j mapping its hitting probabilities into spacetime BROwNian hitting probabilities as in the diagram, each open arc of the image y' of y under j would have positive space-time hitting probabilities. But this is absurd, so the two motions are dissimilar; as a matter of fact, in the plane, there is an infinite number of dissimilar diffusions. Using the language of road maps and speeds, the major problems of higher-dimensional diffusion can be stated as follows: Q 1) to express the basic dissimilar road maps in geometrical terms 'L7

'S7

(see 8.6 for the meaning of geometrical), Q 2) to single out from the diffusions with the same road map some especially simple one (Brownian motion) and to show how to construct 'CS

its sample paths (the meaning of simple will depend upon the geometrical picture of the road map),

8.2. 0 as differential operator

293

Q 3) to establish between two diffusions with the same road map O».

a change of speed (time substitution) sending one into the other (see 8.3 for an expression for this speed change). Q 3) is solved (see 8-3); the others are not, but a number of special .'7

cases are tractable; in the pages before us, the reader will find an informal account of the present state of the art and some rough ideas about what should be done. 8.2. 0 as differential operator

Given a diffusion on Q = Rd (d -'> 1) with standard BRowxian 'L7

hitting probabilities, the mean exit times eD = E. (maD) are the potentials f G dm of a (positive) speed measure m, independent of D (G is n ,-.

cud

the classical GREEN function of D). Each uED(0) can be expressed in the small as the potential f G (a, b) e° (d b) of a signed measure e" (d b) plus a harmonic function, and (SS u = - e"(db) 1) m(db)

C1.

(see 7.19).

T. UENO [1] discovered another case in which (3i can be expressed as a differential operator -e"(db)/m(db). Consider a persistent diffusion, i.e., suppose that 2a)

D + 0 open,

P, (MD < -}- 00)

and add the technical conditions f h(., db) /(b) E C(Q

2b)

- 8D)

/E B(8D)

OD

P.(maD,<MOD) >0

2c)

a(D,UD2, D,f1D2=fa, Q - D, U D2 connected.

Given D, and D. as in 2c) with compact boundaries 8D, and BD2, if 3)

m,=MOD,

m2=min t:x(t)EODs, t>m,)

ms = min (1: x(t) E 49D,, t > ms)

m4=min(t:x(t)E8D2,t> ma) etc., then

P. [MI < m=< m3< etc. t +oo] - 1,

and the hitting places x(ms._,) (n Z 1) and x(m,,,) (n Z 1)

294

8. A general view of diffusion in several dimensions

on 8D1 and 8D2 constitute ergodic MARKOV chains with one-step transition probabilities 4a) 4b)

db) - p, (a, db)

df)

a,rhaD,(a, d$)hoD,($, db) = p2(a, db).

A condition of K. YOSIDA is satisfied, permitting the introduction of stable distributions e1 and e2: f el (d a) p, (a, d b) = e1(d b)

5 a)

el (8D1) = 1

aD,

e. (d a) P. (a, d b) = e, (d b)

5 b)

es (8D=)

ab

and it turns out that 6)

m(db) = f e1(da) Ea[measure(t: x(t) E db, t < MOD,)] OD,

+ f e,(da) E0[measure(5: x(t) E db, t < MOD,)] OD,

is positive on opens, finite on compacts, and stable:

t > 0; A

f m(da) Pa[x(t) E db] = m(db)

...

7)

v

moreover, a second such stable measure is a constant multiple of m (see also G. MARUYAMA and H. TANAKA [2] and HAS'MINSKIi [1] for 6) and 7), and E. NELSON [1) for a different approach to stable measures). ._.

Given D C Q with mean exit time E. (MOD) < + oo, UENO proves that the GREEN measure 8)

G (a, d b) = Ea[measure (t : x (t) E d b, t < MOD)] cod

factors on D x D into the product of a Green /unction G (a, b) and the stable measure m (d b), and expresses O u as -e"(d b)/m (d b) , using 9)

f G (a, b) e" (d b)

_ - f G (a

,

b) (SS u m (d b)

mOD

_ -Ea f (0 u) (x,) dt u

= u (a) -

f h (a. d b) u (b). OD

G (a, b) should depend on the road map alone, but this is not proved

in general; it need not be symmetric (see 8.4 for an account of the potentials of such unsymmetrical GREEN functions).

8.3. Time substitutions

295

Bearing UENO's result in mind, it is plausible that each diffusion win

should have a positive (speed) measure m (d b) and GREEN functions G (a, b) !v.

...

C1.

depending upon its road map alone such that 0 u = -e" (d b)/m(d b) . e" (db) will depend upon u E D ((b) and the road map alone; see 8.5 for a probabilistic formula for it. 8.3. Time substitutions

Given two diffusions with the same road map, common GREEN functions, sample paths x and x', and generators (b3 U=- m (db)

1 a)

u=-

I b)

u E D(@)

m dab)

u E D (Csl ) ,

it is evident that there should be a time substitution i -> f -' mapping

x -> x', as conjectured in 8.1; in fact, if m' (db) = 1(b) m(db) with 0 < / E C (Q) and if t --> f -' is the classical time change with f (t) t

= f l (x1) d s, then it is a simple matter to show that x (f -') is identical 0

in law to x*. But if m' is singular to m, the above prescription loses its sense, and the HELLINGER integral 2)

f (t) _ ( measure (s : x(s) E db, s S t) m(db) Q

which appears hopeful at first glance, turns out to be intractable, though it seems that it should converge. B. VOLKONSKII [1] in the 1-dimensional case and H. P. McKEAN, JR.

and H. TANAKA [1] in the case of BROwNian hits in d Z 1 dimensions

vii

got over the obstacle by approximating i -> f -1 by suitable classical time substitutions, and R. BLUMENTHAL, R. GETOOR, and H. P. McKEAN, JR. [1] used the same method for a wide class of motions including the general diffusions.. McKEAN and TANAKA noted that, in the case of BROwxian hits, the mean hitting times e' are excessive in the sense of G. HUNT: 140, a E D 3) E.[eD(xt), t< maD] t eD(a) and defined f for t < maD as the limit of 4)

f. (t) = f c-' [eD (x (s)) -

(en (x.) , e < moD)] d s

0

for suitable e 10, the idea being that since 5)

0, u = --'[E. (u (x,), e < map) - u]

u E D (O)

8. A general view of diffusion in several dimensions

296

converges on D as e 10 to 0 it and since 0' eD = - f , it should be that 6)

o05eDdm=limo0 eDdm'=-dm'

1

and [` {t)

7)

-

J Q

measure (s : x(s) E db, s S I) 0, eo m(db) m(db)

should converge to the HELLINGER integral 2). M. G. SuR [1] has put

this idea into an elegant and useful form. Because the 1-dimensional BROwNian motion has local times, all positive measures are speed measures, but in d ? 2 dimensions the

orb

situation is not so simple; for example, in the case of standard BROWNian

hits, a speed measure has to have continuous potentials and its null sets have to be thin for the BROWNian motion in the sense that if Z2 are open and if m( ,n Zn) = 0, then n21

aEnZn

Pa[mos.j0,nt +oo] 31

nZ1

(see H. P. McKEAN, JR. and H. TANAKA [1]).

8.4. Potentials A few of the deeper properties of potentials due to G. HUNT [2 (3)] end

will now be explained; it will be seen that the standard BROWNian motion was too simple an example because the symmetry of its transition

densities obscured the significance of the backward motion described below.

Given a transient diffusion on a non-compact space Q with a positive stable measure e and continuous transition densities p(t, a, b) such that 1a)

.fe(da)p(t,a,b)

1,*

Q

Ib)

fp(t-s,a,E)p(s,E,b)e(de) Q

G (a, b)

goo

I e)

f+00p(t, a, b) dl < + o0 off the diagonal of Q X Q

0

as in the 3-dimensional Baowxian case, the backward kernel p*(t, a, b) p(t, b, a) satisfies f p*(t, a, b) e(db)

2a)

Q

f p*(t - s, a, e;) p*(s,, b) e(d) = p*(t, a, b),

2b) Q

* 1a) is almost automatic because e is stable.

8.4. Potentials

297

i.e., the p*(t, a, b) can be considered as the transition densities of a conservative backward motion starting afresh at each instant t z 0; backward because if (positive) infinite probabilities are permitted, then the stationary backward motion with probabilities 3 a)

P*[x(tl) E dbl, x(t,) E db2,

.

. ., x(t,,) E

e(db,) p*(t8 - tl, bl, by) e(dbs) ...

p*(tn - to-,, bn-1, b,,) e(dbn)

oo
P [x (tl) E d bl , x (t,) E d b=,

... , x

E d bn]

= e(db,) p(t3 - t1, bl, b2; e(dbp) ... p(tn --- to-,, bn-1, bn) e(dbn)

-oo
.,.

run backward in time. Assuming that the backward motion is a diffusion, it is easy to see that its generator (S* is dual to the generator 0 of the forward motion ; in detail, if u E D and u* E D ((s3*) vanish far out, then

fu* Jude

4)

`.t

= hoJu*(a) e(da) E-1 (J p(t, a, b) u(b) e(db)

- u(a)i

...

p* (t, b, a) u* (a) a (d a) - u* (b)

= lim 1 u (b) a (d b) e-1 SoQ

JJ

V1uQ3*u*de

= Q

A non-negative measurable function, u, is said to be excessive for the forward motion if f p (t, a, b) u e(db) T u as t i 0. Q

^oo

'`t

Given such an excessive function u, the measure e* (d b) = u e (d b) is excessive for the backward motion, i.e., f e* (da) p* (t, a, b) a (db) Q +='

.;;

t e* (db) as t t 0. Besides, if e* Z 0 is excessive for the backward motion, then u lijm f e* (d a) p* (t, a, b) is an excessive function for the forward Q

motion and e* (d b) = u e (d b) ; in brief, there is a perfect correspondence between excessive functions for the forward motion and excessive measures .SC

I".

for the backward motion. HUNT exploits this fact to prove that an excessive function u for the forward motion can be expressed as the forward potential f G (a, b) e" (d b) of a non-negative BORE!, measure e" III

.^!

plus u,o = lim f p u de (see 8.5 for additional note on a"). t t +OOQ

298

8. A general view of diffusion in several dimensions ^^,,

tan

.tip:

Given a bounded closed or open subset A of Q, the forward and backward hitting probabilities 5a)

PA = P.[mA < + -]

5b)

Pa = P? [mA < + oo]

are excessive; thus, PA is a forward potential f G (a, b) eA(db)

f

plus

e,G+

plusel n P. (m4 (wt) < + oo) (= 0 because of transience), and pa is t +00 a backward potential G (a, b) ea (db)

lim P; (mA (wC+) < + oo) (= 0).

c 1+oo

.-0

G. HUNT [2 (3) : 175] discovered the pretty fact that eA (Q) = as (Q) and called this common total charge the natural capacity of A. Here is a simple proof. Given compact A contained in bounded open B and bounded open

D)B, 6)

e.i (Q) =/ Ps (b) eA (d b)

[/4(da)G(a.

eA(db)

b)]

_ en(da) PA (a) eR(da) PD (a)

= f PB(b) eD(db) Q

f f PA(b)e.D(db)

B {A

Q

eA(da)PD(a)

where PB 4 PA on OD and eD (Q - OD) = 0 were used in the next to last step. Because ca (Q) S eA (Q) for similar reasons, eA (Q) = ed (Q) , and, letting A t B and using 7)

eA (Q)

-j eD (d a) PA (a)

tJeu(da) PB(a) = eB(Q)

* G'(a,b) =G(b,a).

AtB

8.5. Boundaries

299

I.-.

and its counterpart for e* (Q) , it is found that eB (Q) = eB (Q) for open B also. eA (Q) = ed (Q) is an alternating function of A in the sense of G. CHOQUET [see 7.8.13)].

The reader will find additional information on potentials in E. B. DYNKIN [7], G. HUNT [2 (1, 2, 3)], BEURLING and DENY [1], F. SPITZER

[2], and M. G. SUR [1].

8.5. Boundaries Consider the standard BROwxian motion on the open disc E2: I b 1< 1 stopped on 49E2, with transition probabilities

g (t, a, b) 2db = Pa[x(t) E db, t < mejp] (t, a, b) E (0, +oo) x E2 xE2,

1a)

GREEN function +c

G(a, b) =

1 b)

J

g (t, a, b) dt = 2n lg 16 - aab

I

(a, b) E Es x E2,

0

and hitting distribution (POISSON kernel) h (a, eiB) dO = Pa[x(m0z) E d0]

I c)

= lim(1 rtl - r)-1G(a,rei°)dO

I -lal' la - e'°12 dO

Ial < 1, 05 95 2x.

!'-

Given 0 S 0 S 2n, ue(a) = h(a, e`°) is a minimal harmonic function, i. C.,

2a)

OSuSu0

2 b)

u harmonic in E2

implies

u is a constant multiple of ue.

3)

On the other hand, if 0 S u is a minimal harmonic function, then it can be expressed as f ue e (d 8) (e ? 0), and e has to be concentrated 8E'

.y.

at a single point 0 E O E2, i.e., u is a constant multiple of ue i in brief, the map 0 --* ua establishes a 1 : 1 correspondence between the points of aE2 and minimal harmonic functions. aE2 is now seen to have two properties: 4a) it is not too large, i.e., the sample path tends as t t ntoEt to a single point of 8E2;

S. A general view of diffusion in several dimensions

300

VII

4b) it is not too small, i.e., it 0S u is harmonic on E2 then -off

u(a) = f h(a, ei0) e(d0)

Ian < 1, e Z 0.

aEI

of the slit disc

cad

z_4

Given a BROwNian motion on a GREENian domain Q C Rd stopped on 8Q, we cannot expect the geometrical boundary 8Q to be large enough for the POISSON representation 4b) as is clear from the simple example

Q:O
0<0<22r,

but as R. S. MARTIN [1] discovered, 4a) and 4b) can be salvaged if 8Q is properly ramified.

r'-

'L?

Consider for this purpose, the classical GREEN function G(a, b) of Q.

Given b E Q, if 8B C Q is a little spherical surface with center at b excluding a second point a E Q, then for c near 8Q, 5)

G(a, c) Z E.

f g' (t, x(ttt1B), c)dt maa E

= Ea[maB < +00, G(x(moB). C)] L- a constant multiple o l f G c) do 0B

= a constant multiple of G (b, c),

is bounded near 8Q,

Uab = G(a,

and using 6), the MARTIN boundary Q° of Q is defined as the space of

ideal points added to Q in the smallest compactification Q = Q + Q° making uab continuous on Q - (a U b) for each choice of a and b E Q. Given 0 E Q° and fixing a E Q, c) h(b, 0) = lim G(b, G(a, c)

is harmonic, but perhaps not minimal; however, the map 0 --,.

.!V

7)

0)

a',

establishes 1 : 1 correspondence between a BOREL subset Q' of Q° and the minimal harmonic functions u with u (a) = 1, and 4b) holds, i.e., if

0 < u is harmonic, then u(b) = f h (b, 0) e" (d 0), and e°

is unique.

Q.

Q' is therefore not too small and as DooB [6] proved, it is not too

P.-

large, i. e., almost all sample paths converge as t t maQ to a single Point of Q' (see also G. HUNT [3]). DOOB also found a pretty expression for e': 8)

e'(d0) =

u(b) P; [x(maQ -) E d0] h(b, 0)

8.5. Boundaries

301

where P;' (B) is the probability of the event B for the u-motion with transition probabilities P, [x(1) E db, t < an d Q ] 0.!

Pa [x (t) E d b] =

9)

u (al

u (b) ;

III

esp., taking u = 1, E dO]

h(b 9) =

10)

e'(d0) r-+

which is the analogue of the classical expression of the POISSON kernel as a hitting density, just as 7) is the analogue of its expression as normal derivative of GREEN function [see 1 c)]. t1.

DOOB's expression of e" in terms of the u-motion has applications to potentials as well; in detail, if 0 < u is excessive, i.e., if 2f g' (t, a, b) u(b) dbtu as t 4 0 and if u is not pure harmonic, then the it-motion will sometimes be killed at a time III,,. < maQ, and the RIESZ measure 11)

a"(db)=lime-1[u(b) - Eb(u(x,),e< maQ)]x2db 840

used in the expression of u as a potentialI G (a, b) e" (d b) plus a nonnegative harmonic function (see 7.5) will be 12)

a" (d b) =

u(a) P; [x (m,,,, -) Ed b, m°° < maQ] G(a, b)

inside Q.

Q* = Q° is a single point if Q = R' (d z 3) (see 8.7 for additional information on 1-point boundaries). mow

MARTIN boundaries have also been treated in connection with pop

MARKOV chains (see DooB [7], DOOB, SNELL, and WILLIAMSON [1], HUNT [3], T. WATANABE [1], and for a different boundary, W. FELLER [8], .ti

'fl

and there is no doubt that similar boundaries can be attached to any decent MARKOV process (see BISHOP and DE LEEUW [1] and D. RAY [3]). r<,

Given a conservative diffusion on a space Q with a nice boundary Q*

p°'

(neither too large nor too small), run the sample paths with so slow a clock that x (t) tends to Q* as t t m < + oo, let Z be the corresponding local generator inside Q, and pose the problem : to find all conservative generators 0 C £ , i. r., to find all boundary conditions that can be imposed upon Z. Consider, as an illustration, the standard BROwNian local generator Z*

the open unit disc Q = E2. Q' is the unit circle 0E2, and it is probable that the most general conservative diffusion operator 05 C Z can be expressed in terms of a conservative 1-dimensional differential operator

* 1u-duf2 if

uEC=(E').

302

8. A general view of diffusion in several dimensions

Sa' on 8E2 and a BOREL function 0 5 P(0) S 1: to wit, it is conjectured that 0 coincides with Z applied to the class D (0) of functions u E C (E2) such that m u has a continuous extension to E2 and .`t

p (0) ar (1-, 0) + [1 - p (0)] (C U) (1-. 0) = (Z' u') (0)

13)

05 0<2n, u' -u(1,0). -6.

Given a reflecting BROwxian motion x in the disc [--r (1- , ') 0, , if t (t) is local time at r = I for its radial (BESSEL) part and if ip is an

independent circular BROwxian, then the motions corresponding to 14a)

Or

14b)

Or

(1-, 9) = 2

(1-, 8) =

au

OF

(1 ,

e)

(1,0)

will be x eir«>

15 b)

x e$ t

...

15a)

[see problem 7.15.4]. 15 b) should be considered as a reflecting BROwNian

au motion because 14b) can be expressed as = 0, where ao means differentiation along the oblique line pointing at -45° to the outwardpointing normal.

As to the original problem, similar boundary conditions au (1 - p) m u = £1 u' should occur, and the corresponding p an + motions should be expressible in terms of a reflecting barrier motion au (an = 0), a distribution t (t, d B) of (reflecting barrier) local time on Q%

and a boundary motion associated with Z*; see A. VENTCEL [1], T. UENO [2], N. IKEDA [1], K. SATO [1], and K. SATO and H. TANAKA [1]

for some information about this. A nice problem is to explain the meaning of reflecting BROwNian motion on the (MARTIN) closure of a GREENian domain D C R3 (see J. DOOB [81 for information on this point). 8.6. Elliptic operators

Consider a nice differentiable manifold Q and, on it, an elliptic 0

differential operator £1 expressed in local coordinates as 1

2

ideij

ax£aX3

aXI

C generates a diffusion if it has HOLDER continuous coefficients and if

e = [eii] is symmetric and positive definite at each point of Q; to be

8.7. FELLER'S little boundary and tail algebras

303

0

precise, Li agrees on C2(Q) with the generator of a diffusion which is permitted to run out to the point oo before time + oo if Q is non-compact (see PocoRZELSxI [1]). K. ITO [3] proved that the desired motion satisfies the stochastic differential equation d x = ye (x) d b + /(z) dt II

2)

o

up to the time it leaves the neighborhood on which 1) holds. V e is the w

o

w

O

II

A

symmetric positive definite root of e, / = (/1, /2...., /d), b(t) is a standard d-dimensional lROwxian motion, and it is supposed that the e-1 changes according

o

w

o

II

HOLDER exponent of the coefficients = 1. Under a change of local coordinates x -+ to the rule : ..'

3)

, axe axi k,1 dekl -ax, ax;

1

.

eif

so

e,t d xx d x3

4)

induces a RIEMANNian distance on Q, and La can be expressed as the o

LAPLACE-BELTRAMI operator

d=

1

N

II

t

S)

2

2

.

T,

e!

a

ax, I

e,j

a

Ve

8x1

o

m

II

N

z

II

°

plus a first order differential operator (vector field) / corresponding to a drift (see, for example, E. NELSON [1]). 2) now simplifies to dx = d b + /(x) dl, b being the (BROWN ian) motion associated with (1/2) A. Because the differential operator (n u = -eu(db)lm(db) of 8.2 shares with elliptic operators of degree S 2 the property that 0 u S 0 at a local minimum of u E D(05), it is to be hoped that some feature of the above geometrical picture can be salvaged in a general setting. o

o

w

Besides the papers cited above, the reader can consult HAs'MINSKII [1], NELSON [2], YOSIDA [3], and 8.7 for additional information about ellipo

o

O

tic operators. Problem 1. Prove that if a differential operator Li with continuous A

.

coefficients has the property that Li is z 0 at a local minimum of m

u E C°° (Rd), then it is elliptic of degree

2 with no part of degree 0.

w

rhaD(a, db)1(b) E C(D)

m

1)

b

G

8.7. Feller's little boundary and tail algebras Given a diffusion on a space Q such that

/E B(8D),

let B be the tail algebra fl B [x (l) t Z n] modulo the ideal of negligible n 1 events. :

8. A general view of diffusion in several dimensions

304

Given B E B', P = P.(B) satisfies (a) = P. (u'm E B) = f han (a, d b) p (b)

2)

a E D,

i.e., p E C (Q), and because of

p = f e-t P. (u E B) dt = Ge p,

3)

p is a solution of

OSp51

4b)

p E D((tS)

4c)

c3p=0.

VII

4a)

On the other hand, if p is a solution of 4), then E.[p(x1) I B.] = E.(.)[p(xe-.)]

5)

p (x.) + E.(.) [ T(O p) (xe) do]

t Z s, °IN

= p(x,)

i. e., p (xe) (t ? 0) is a bounded martingale; as such, it converges to a B'

measurable limit p as t t + oo, and one gets 6a)

p(xe) = E.(P I B,) = E=ca(P)

t? 0;

in particular, if p = P. (B) (B E B'), then 6b)

:lim p (xe) =iturn P. (B I Be) = P. (B I B) +00 4 00

= the indicator of B. 'S'

G. HUNT pointed out to us that 6) expresses a 1 : 1 correspondence

between the events BE B' and the extreme points of the (compact

='°

Bpi

Cs.

convex) class of solutions of 4). W. FELLER [8] had used this correspondence to express B' as an algebra of subsets of a topological space Q' attached to Q as its boundary. Q' is what we are calling FELLER's little boundary (see J. FELDMAN [11 for its relation to the MARTIN boundary). Given a non-constant solution p of 4), it is possible to choose a < b and open A and B C Q such that p < a on A and p > b on B, and if the motion is persistent (i. e., if P. [mD < + 00] M 1 for each open D C Q),

then the sample path enters both A and B i. o. as t t + oo, and the bounded martingale p (xi) (I Z 0) cannot converge because p (xt) < a and p (x1) > b i. o.; in brief, p must be constant and the 01 law 7)

must hold.

P. (B) = 0 or l

BE B'

8.7. FELLER's little boundary and tail algebras

305

Now let us suppose that Q = Rd and that our generator 03 is an elliptic operator: 8)

i S 8x, axi J de{i

with continuous coefficients and let us pose the problem: to find conditions on the quadratic form (e , ) = e11 e + etc. such that the 01 law 7) holds, i. e., such that all the solutions of 4) are constant; in more detail, let V. and y_ be the greatest and smallest eigen-values of e at a point and let us ask what is the maximal rate of growth of y, /y- such that 9a) the motion is persistent, 9 b) all the solutions of 4) are constant, 9c) all the non-negative solutions of Z p = 0 are constant?

It is known that 9c) holds if y+/y_ is bounded. There is also the extraordinary result of S. BERNSTE[N [1] that 9b) holds if d = 2 and if y_ > 0 at each point. BERNSTEIN also gave the simple example

El=(2+4bl)a=+4baa b+aW in

which 9c) fails (p = en-b') and y. /y- - 4b4 as bt +oo; see

E. HOPF [1] for other examples connected with BERNSTEIN's result, and for more up to date information, J. MOSER [1].

ltb/McKean, Diffusion processes

20

C!'

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°°g

C+d

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app

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'CS

end

App

C°7

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r,,

List of Notations a

a LAPLACE transform variable. a BOREL algebra. B [x (t, w) : a t < b] the smallest BOREL algebra measuring x (1): a the algebra B [x, : s 1]. B, B

C (Q) C" (Q)

w00

.-.

the algebra (B : B n (m < t) E B4, t ? 0). the smallest BOREL algebra measuring all open subsets of Q. the space of bounded BOREL measurable functions t: Q -i R1. the space of bounded continuous functions / : Q - R1. !gyp

B(Q) B (Q)

fib

BM+

t < b.

the space bounded, n-times continuously differentiable functions

/:Q-.R1. C. (Q)

the space of continuous functions vanishing at oo.

C (B) d

dimension.

D

dD D D (CS) A

the NEwroxian capacity of B (see 7.8).

the space of functions / E B (Q) such that / (a f) _ /(a) if P.[m,= = 0] = I in chapters 3-6 (see 3.6), and, in chapters 7-8, a domain (= connected open set). the boundary of the domain D. a diffusion. the domain of W. the LAPLACE operator.

the uniform distribution on the surface of a ball. the unit disk. the expectation of / based on P.. E. (/) the expectation of / times the indicator function of A. E. (/, A) e(dy, a, b) = e f e an eigendifferential (see 4.11). a spectral measure (see 4.11). f(dy) the left derivative. /- (a) the right derivative. /+ _ /* (a) the BOREL measure based on the interval function /(da) do

.,a

E=

-7-

/(a,b] =/(b+)-/(a+) coo

..7

a local time integral (see 5.2). f = f (t) g = g (t, a, b) the GAUSS kernel (2rr g)_h12 exp [- (b - a)2/21]. the increasing and decreasing solutions of (5y g = a g (see 4.6). 91-92 G = G(a, b) a GREEN function (see 2.6, 4.11, and 7.4). a GREEN operator (see 3.6). G. a generator (see 3.7). 0 the classical harmonic measure of B C d D as seen from a E D (see 7.12). h (a, B) 1"'

Ah

the HAUSDORFF measure based on h (see note 2.5.2).

K. X A

kt

.C:

I = I(t)

the left shunts (see 3.4). the right shunts (see 3.4). the killing rate for K_ fl K, (see 4.8). a killing measure (see 4.3). the killing measures for shunts (see 4.8). the killing functional (see 5.6).

K_

314

List of Notations the HAUSDORFF-BESICOVITCH a-dimensional measure (see note 2.5.2).

Ls (Q, m)

the space of m-measurable functions/ with 11f IN = V f f dm < + oo. modulo null functions.

nay

Aa

:iii-1

m

a speed measure (see 4.2).

a MARKOV time (see 1.6 and 3.2).

in

the passage time to a. the entrance time into B (see problem 7.1.1). the killing time (see 2.3).

Me

ma mCO

a LEVY measure (see note 1.7.1).

P. (B)

the probabilities for paths starting at a. P.(B) as a function of a.

r-,

n

'Z1

P. (B) P = p (I. a, b) a transition probability kernel (see 4.11).

a PoissoN measure figuring in the Livy decomposition of an increasing differential process (see note 1.7.1) or a projection (see 4.11).

the state space of a diffusion. the d-dimensional EvcLtnean space. s = s (a) a scale (see 4.2). st the shunt scales (see 4.8). S' the d-dimensional unit sphere C R'+1 I time. t = t (t, a) = t (1, a, w) a local time (see 2.2 and 5.4). W a space of sample paths. w : I -. z(1) a sample path. the shifted path : x, (w;) = x,., (w) . W.* the stopped path x, (w;) = xn, (w) . W., x(1, w) = x(t) the value of the sample path w at S. ZI the set of all integers. 8 _ (I: x (t) = 0) a visiting set. (n i) the (open) intervals of CO, oo) - 8.

#<>

'ti

.Ow

.Ow

bpi

CA'

Q Rd

the smaller of a and b. the greater of a and b. the number of objects in the class C.

aAb

aVb # (C) n

for m=0, 1....,n.

m

1gsl = lg(lgl),

Ig(19.-19) (n a 3).

sup Jf Q

f fsdm. a t b

a increases to b. infinitely often.

fEt fE.

an increasing (= non-decreasing) function. a decreasing (= non-increasing) function. III

a decreases to b.

i. o.

III

ajb

(x(t) < h(t), t j 0) means that there exists l1 > 0 such that. x(t) < h(t) for .,.

0<1
.°T

(x (t) < h (1), t t +c ) means that there exists is > 0 such that x (t) < h (1) for

0

t > 12 . the empty set. an extra state (see 2.3).

f CO

the ends of the line.

..N

00

Index

additive functionals of Brownian mo-

t = local time, a = (9: x - 0) K70

d = dimension, 0 - generator,

Brownian motion, reflecting on CO, +oo) ,

tion as local time integrals

as I xI 4f

as t- - x 41 'a'

individual ergodic theorem (d = 228. (d 277 KAC's formula 54 applications of KAC'S formula 5Z 58

as x(f"I) 81 (see also local time, zeros) reflecting in a disc, application to NEUMANN problem 264 17o

190

application

(see also time substitutions)

to boundary condi-

.,.

tions for A12 273. 301

additive functional used as clock of skew product 269

skew 115

space-time 266

arc sine law (see standard 1-dimensional BROWNian motion, random

application to DIRICHLET problem for 8/at _d/2 267 KOLMOGOROV'S test 268

walk)

8pp

Are'

spherical 202 connection with projection 281

application to BRowNian local ex-

f (2r)-1 61

G3' as radial part of 4 60 passage times 62, f

,,.

,..

proof that path has no roots 6L 62, 1U Blumenthal's 01 law (d a > 25, 85, (d > 1 233 Borel-Cantelli lemmas 3 boundaries FELLER'S little 303 CPC

MARTIN 300

posing of boundary conditions on 301

Brownian motion, conditional (- tied) -, covering 219

-, elastic 4. -, Filler's 186

282

standard 1-dimensional 12

as solution of r = b + (d - 1) X

Si

individual ergodic theorem 28L arc sine laws 28, 5L 58 approximation by coin-tossing 38 CHUNG-ERDOS-SIRAO test 36

construction of P. LivY 12 construction of N. WIENER 21 excursions 7-5

excursions described in terms of tied BEssaL processes 79 0 and GREEN operators 11

M°pod5.

times 63 application to BROWNIan cursions 79

stereographic F..

backward motion 291 Bessel process 52

HOLDER condition 36

individual ergodic theorem 228 KAc's formula for additive functionals of 54 KALLIANPUR-ROBBINS law 229 KOLMOGOROV's test 31 161

LAMPERTI's renewal process based

on 8 78 law of iterated logarithm 3.3, 36 local times 63 local times, conditional, as diffusion 63

Index

316

Brownian motion, standard 1-dimensional, MARKOV times 22

MARKOVIan property of 6 22 maximum function 27, 4J moments IS non-differentiability iS passage times 25

Brownian motion, standard severaldimensional, WIENER'S test, 255 WIENER's test applications of 259

(see also boundaries, capacities, diffusion, DIRICHLET problem,

excessive functions, harmonic functions, potentials)

reflection principle of D. ANDRE 26 pap

scaling i8 time reversal 1.8

capacities, fractional-dimensional, connection with HAUSDORFF-

winding of sample path 229

e50

KALLIANPUR-ROBBINS law 277

winding, SPITZER'S law of 270

(see also standard several-dimen-

distribution of eigenvalues of 4 224 DVORETSKY-ERDdS-SPITZER

test

1-64

excessive (= superharmonic) functions 2.3 0 = J/2 234

expressed in terms of RIESz measure 24S GREEN function of (a - 4L2) u 233

GREEN function of J L / and GREENian domains 231 law of iterated logarithm 61

multiple points 261 NEwTONian electrostatic poten-

tials expressed as BROwxian hitting probabilities 241 POINCARi'S test 257

wandering out to 00 236

LAPLACE limit theorem)

Choquet's inequalities for NEWTONian capacities 251 Chung-Erdos-Sirao test for BROWNian motion 36 coincidence probabilities 158 coin-tossing (see random walk) communication 92

eon

sional BROWNIan motion)

-, standard several-dimensional 23 BEssEL process as radial part of 60 DIRICHLET problem for d 262

central limit theorem (see de MOIVRE,/1

a0.

multiple points 260 visiting each disc i. o. D6 WIENER'S test applied to tip of simple curve 257

-,

CHOguET's inequalities 251 SPITZER'S interpretation of (d > 252 (see also potentials, WIENER'S test) Riesz (fractional-diensional), connection with stable processes 2900 .:1

GREEN functions and GREENian domains 231 individual ergodic theorem 277 KAKUTANI'S test and logarithmic capacities 252

252

-, natural of G. HUNT 298 -, Newtonian 250

°b-

tions to conformal mapping and

BESICOVITCH dimension 227

-, logarithmic, and KAKUTANI's test

t,.

zeros 29

-, standard 2-dimensional, area of sample path = 0 233 covering motions with applica-

concave functions 113

relative to a linear form 122 creation and annihilation of mass 200 construction of explosions 206

explosion and extinction probabilities as solutions of nonlinear parabolic equations 209 crocodile, Littkwood's 268

de Moivre-Laplace

limit theorem, DONSKER's extension of 412 HINCIN's proof 10

proof using STIRLING's approximation 12 descriptions 85 differential processes, isotropic 288 connection with RIEsz potentials 290

0 as fractional power of A12 289 -, with increasing paths, associated

Livv formula and POISSON measure 3t 146

Index

di/fusion,

several-dimensional,

stable

mass distribution 294 time substitutions 29S

ado

differential processes, with increasing paths, BROwNian passage times

317

oo]]

as 25 (Si 286

(see also BROwNian motion, elliptic

loo

L'.

0'a

zoo

tai

'"`

33

diffusion, general 1-dimensional 83 coincidence probabilities 158 communication 22 decomposition into simple pieces

differential operators, excessive functions, skew products, spacetime motions) Dirichlet problem, for A 261 application to BESSEL hitting probabilities 62

4)0

inverse local time as 212 one-sided stable process as

harmonic measure expressed as

92

BROwxian hitting probabilities

DYNKIN's formula for Qs 99

264 LEBESGUE'S thorn 261 LITTLEWOOD'S crocodile 268

0 98

PoissoN formula 249. 300 ""'

woo

Coo

POINCARH'S test 257

'6J

general view of ($ as differential operator 105

solution in terms of BRowxian motion 262

stn

GREEN functions 154 GREEN operators 94

e'3

etc. 1g9 excursions 223

eigendifferential expansions for (bS,

WIENER'S test 262

-, for 8/a t - d/2 266

local times 174

connection with KOLMOGOROV'S

sex

MARKOV times 86 MARKOVian property of 84

test 268 solution in terms of space-time BRowNian motion 11

matching numbers 889

Dynhin's formula for 0 (d = 1) 99. I^'

S. BERNSTEIN'S

theorem 35

exit and/or entrance 19

=per

endpoints, FELLER's classification of, as 1311

entrance 19$ ergodic theorem, individual, for BROWN-

a.<

,.'

149

elliptic differential operators 302

...

ian motion (d = 11 228 (d = 277, (spherical) 28t 282 exit 10$ excessive functions, of Brownian motion, as potentials (RIEsz for0.i=

-, several-dimensional 232 backward motion 297 boundaries 222, 303 BROwNian motion as instance of

eigendi//erential expansions for CS, etc. t/3

C.-,I

.pr

""w

zeros)

286

w0'

time substitutions,

measure,

(d Z: 21

c%1

y°7

scale, shunting, shunts, speed

BROWNian motion 164

.40

shunting) 164 shunts 91 stopped diffusion 102 stopped conditional diffusion 105 subordination 176 transition densities 142 (see also endpoints, generator, killing, local time, MARKOV times, MARKOVian property,

0°0

oaf

;fib

Dvoretsky-Erd6s-Sirao lest for

sample paths, general view of (time substitutions, killing,

+'i

passage times 144 persistent diffusion 124, 134

mula) 244

as superharmonic functions 243

+^,

213

of several-dimensional diffusion 296

DYNKIN'S formula for G3 232 general view of 281 (bS as differential operator 293

excursions, of Brownian motion 75 described in terms of tied

potentials 296 similar diffusions 291

-,of non-singular diffusion 223

as potentials (RIEsz formula) 9I BESSEL processes 79

.,,

,;,

'L7

any

{9.

L'.

diffusion with BROWNian hitting probabilities 283

explosion of mass 266

318

Index

tail

and

Green functions, of BRowNian motion

Feller's classification of endpoints as exit and/or entrance 1Q8, >3o 86

250

ii.

Gauss's quadratic form for NEWtoNian potentials 253 generator,

of

general

1-dimensional

diffusion 98 DYNKIN'S formula 99

eigendifferential expansions 149 :..

eigenvalue distribution 113_4_

general view of 105

local nature of 100, 101, 134 non-singular case ill. 116, 115

sure) of several-dimensional diffusion 232 DYNKIN'S formula 232 (i expressed as differential operar°'

tor in terms of speed measure

283, 294 -, of special motions, BESSEL process

and

of au- u"2-ku)=/ with

application to K A c's formula 56

of non-singular 1-dimensional diffusion f 5Q

of several-dimensional diffusion

2"-

Green operators, of BROWNian motion

(d=1) 1L (d

)

of general 1-dimensional diffusion,

invariance of space D under 94 of several-dimensional diffusion 232

Greenian domains described in terms of BRowNian motion

Haar functions applied to P. Livv's construction of BRowNian mo-

singular case tom, 136, 142

(see also killing measure, scale, shunt, shunt scale, speed mea-

2)

electrical

probabilistic interpretation of

r'.

Gaussian distributions 4 Gauss's principle of positive total charge for NEwroxian potentials

(d=1) d u 2 --/,

x'"

Calmarino's theorem on MARKOV times

of

SFr

boundary

FO'.

little

algebras 303

tion 12 harmonic

functions

ix"

Feller's

(see

DIRICHLET

problem ford) Hausdor/f-Besicovitch dimension

numbers and measure 53 connection with fractional dimensional capacities 227 (see also local time, zeros) inverse local time (see local time)

tip

Hat's formula for Brownian functionals

...

BROWNian motion in a disc 273, 301

54

diffusions with BROWNian hitting probabilities Z83

elastic BROwNian motion on (o, + oo) 45 FELLER'S BROWNian motions 186

isotropic stable processes 282 motions with discontinuous paths 28b one-sided 2$k one-sided

applications of 57, 58 Kahutani's test for 2-dimensional BROwxian motion 252 Kallianpur-Robbins law for BROWxian motion (d = II 229, (d = 277

principle for potentials 255

Kelvin's

NEwroNian

processes

killing of sample paths, general view

stable processes 222 reflecting BROWNian motion on

non-singular case 179 of x' by V to construct elastic

differential

C,0

BROWNian motion 43

+00)

standard

',7

of f4.6

BROWNian

motion

on the shunts 196

(dam 52 234

killing measure of non-singular di f f u-

stopped BRowuian motion 243

defined in terms of hitting probabilities 107, log. 120

(d=1)

sticky BRowNian motion in a disc 264

sion, at an isolated shunt i26

Index

Kolmogorov's test for Brownian motion

times, entrance times of several-dimensional diffusion as 233 'i.

Markov

!",

killing measure of non-singular diffusion, defined in terms of conditional diffusion jig on the shunts 110, 13

319

GALMARINO'S theorem 86

of coin-tossing 6 f5.

connection with space-time

va$

of 1-dimensional BROWNian

33 rya

motion 22 Markovian property, of 1-dimensional

BROWNian motion 2¢$ MoTOO's proof of 161 Kolmogorov's 111 law 3

BRowNian motion 16. 22 .fl

of coin-tossing S. 6 of general 1-dimensional diffusion

't7

Lamperti's renewal process based on

84

78

iterated

logarithm

for

].-.

.'!

Livy's formula for differential processes ,.,

31 146

.°.

161

process 212

Livv formula for L 214 .°:

process

as

and down-crossings 4.8 c.,

and intervals of [o, + 00) - 8+ 43 60i

as density of time spent at 0 43 as (1/2)-dimensional measure on

257

Poisson formula,

for A u = 0 in a

GREENian domain 262, in a disc 299

'fl

-, of reflecting Brownian motion 42 and dim ( + 4 SQ

of 1-dimensional diffusion, Livy formula for 144 Poineart's test for BRowNian motion

maw

stable

instance of L 226

Of BROWNian motion 25 'fl

sure on $ 223 inverse local time as differential

for 8u(8t-AaY2=0 in a box ?.kz

for MARTIN boundaries 299

Poisson measure, for one-sided differential processes 31 146 for BRowNian passage times 27 LS.

as HAUSDORFF-BpSICOVITCH mea-

passage limes, of coin-tossing 7 of BESSEL process 62 133. 231

,y,

as diffusion 223

'CI

p°p

and down-crossings 222 and intervals of (0, + 00) - ,8 218

Neumann problem 264 Newtonian capacities and potentials (see capacities, potentials)

°+.

°'°

Littlewood's crocodile 268 local time, of non-singular 1-dimensional diffusion 174

one-sided

o01

motion J2

'1u y);

ago

Lebesgue's thorn 261 Levy's construction of BRowNian

(d a 2J 232

Martin boundaries 300 martingales, convergence of 52 submartingale inequality j3 matching numbers 89 Motoo's proof of KoLMOGOROV's test

°`+

the

BROwNian motion (d s 1) 33. 36, (d 2) 5i

^.I

r'1

D

for passage times of 1-dimensional diffusion 144 potentials, Newtonian, as excessive functions 244 in`

,°,

used to construct elastic BROWNian motion 4.5 -, of standard 1-dimensional Brownian .+.

opt

in terms of tied BESSEL processes f5 HOLDER condition 63

Markov times 86 BOREL algebras associated with 84

GAUSS'S principle of positive total charge 250 GAUSS'S quadratic form 253 KELVIN'S principle 255 ,.,

BROwxian passage times as 25

electrostatic potentials expressed as BROWNian hitting probabilities 241 'ti

COI

motion 3 conditional, as diffusion described

.7.

of

.,.

law

connection with smoothing of GREEN operators (d IJ 95.

Index

320

potentials, Newtonian, RIESZ formula 244

skew products 269

application to boundary conditions for 4 in a disc 273, 30t application to spinning of iso-

also NEW'oNian capacities) -, Riesz, connection with stable pro(see

cesses 288 (see also RIEsz capacities)

'-'

tropic diffusion 274 application to winding of 2dimensional BROWNian motion 270

GALNARINo's description of iso-

tropic diffusions as 271 motions, application to motions with time-dependent

space-time

probabilities 268

gyp.

Rademacher functions applied to cointossing 7 random walk (= coin-tossing) 5 approximation to BROwNian motion 38 arc sine laws 8, 4Q description in terms of RADEMACHER functions 7

MARxov times 6 MARxovian property of 5, 6

space-time BROWNian motion 266 speed measure, of non-singular 1-dimen-

sional diffusion at an isolated shunt 126

...

maximum function i2 passage times 7

defined in terms of mean exit times

107, 109. 114, 122

(see also DE MOIVRE-LAPLACE limit

i'2

sample paths, of i-dimensional diffusion

general view of iM (see also killing, shunting, stitutions)

time sub-

defined in terms of conditional diffusion 118 -, of several-dimensional diffusion, for motions with BROwxian hitting probabilities 283 stable mass distribution as 224 spinning of isotropic diffusion 274 stable mass distribution, UENO's formula 294 stable process, one-sided U BROWNian passage times as 25

:'. ono

.,,

theorem) reflection principle of D. ANDRE 26 Riesz formula (see excessive functions, potentials) Riesz capacities and potentials (see capacities, potentials)

(l3 288

-.0

scale of non-singular diffusion, defined

in terms of hitting probabilities 1Q, LO.E 112, 120

defined in terms of conditional diffusion 118 shunt 91

special inverse local times as 226 -, isotropic, and RIESZ potentials 289 3 as a fractional power of 4 282 strong law of large numbers 3 superharmonic functions (see excessive functions)

on the CANTOR set 196 't>

shunting of sample paths 193 on the CANTOR set 167 O'.

shunting with killing 196 translation on an interval of shunts 141 similar diffusions 242 singular points, of BROWNian motion

tail algebras and FELLER'S little boundaries 303 time-dependent probabilities, motions with associated space-time motions 26A

time substitutions 264 thorn, as example of BRownian singular point 259 LEBESGUE'S 261

(see DIRICHLET problem, KOL-

test,

LEBESGUE'S

thorn, POINCARH'S test, thorns, WIENER's test) of 1-dimensional diffusion (see shunts)

time

substitutions, for motions with

'"'o

shunt scale 110 >iL 134

MOGOROV'S

Lo.

discontinuous 286

time-dependent probabilities 262

-, for 1-dimensional diffusion, chain rule 176

Index

321

1-dimensional

Wiener's test, for space-time Brownian

diffusion, general view of 1614 on the shunts 195

Motion (= KOLMOGOROV'S test) 26$ -, for stable processes 29o winding of 2-dimensional Brownian

for

pa{

Q=R' 167

used to construct reflecting

L1.

motion, application of covering

motions to 280

BRowlvian motion on [0, + oo)

'.°

-, for several-dimensional diffusion 284. 245 transition densities, for non-singular diffusion t49

SPITZER's law of 270 WKB approximption, application of KAC's formula to 51

ink

81

.Ow

Q = (0, + 00) 171

'.T

time substitutions,

translation on an interval shunts of 141

zero-one law, BLUMENTHAL'S (d = 1) 25,

85, (d ? ) 233 KOLMOGOROV'S 3

e+7

%3U)

for stopped BROwNian motion ?32

'CI

zeros, of non-singular diffusion, comUeno's formula for stable mass distribution 224

parison test for dim (S)

225

dim ( ) = constant 224 t as HAUSDORFF-BESICOVITCH

4 246

Wiener's

construction

of

BROwNian

motion 21 Wiener measure 16 Wiener's test, for Brownian motion 255

applications of 257 blem 262

It61SicKean, Diffu%ion processes

LAMPERTI's renewal process based

on a 78 of reflecting Brownian motion 42

dim(3) 4S. 50 t+ as W)-dimensional measure 'in

connection with DIRICHLET pro-

measure on 3 223 of standard 1-dimensional Brownian motion 29 dim(> ) 72

I::

Weyl's distribution of eigenvalues of

on W 50

2-1

C/)

CLASSICS IN MATHEMATICS

ago

Springer-Verlag began publishing books in higher mathematics in 1920, when the series Grundlehren der mathematischen Wissenschaften, initially conceived as a series of advanced textbooks, was founded by Richard Courant. A few years later, a new series Ergebnisse der Mathematik and ihrer Grenzgebiete, survey reports of recent mathematical research, was added. Of over 400 books published in these series, many have become recognized classics and remain standard references for their subject. Springer is reissuing a selected few of these highly successful books in a new, inexpensive softcover edition to make them easily accessible to younger generations of students and researchers. ..,

o

a..

Diffusion Processes and their Sample Paths Since its first publication in 1965 in the series Grundlehren der mathematischen Wissenschaften this book has had a profound and enduring influence on research into the stochastic processes associated with diffusion phenomena. Generations of mathematicians have appreciated the clarity of the descriptions given of one- or more-dimensional diffusion processes and the mathematical insight provided into Brownian motion. Now, with its republication in the Classics in Mathematics it is hoped that a new generation will be able to enjoy the classic text of lti and INlchean.

ISB N 3-540-60629-7

11

9

83540"606291