Functional Integration
and Quantum Physics
This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks . Editors: SAMUEL EILENBERG AND HYMAN BASS
A list of recent titles in this series appears at the end of this volume.
Functional Integration and Quantum Physics
BARRY SIMON Departments of Mathematics and Physics Princeton University Princeton, N e w Jersey
A C A D E M I C PRESS
N e w York
San Francisco
A Subsidiary of Harcourt Brace Jovanovich, Publishers
London
1979
COPYRIGHT @ 1979, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART O F THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RFiTRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
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United Kingdom Edition published by ACADEMIC PRESS, JNC. (LONDON) LTD. 24/28 Oval Road, London NW1 7DX
Library of Congress Cataloging in Publication Data Simon, Barry. Functional integration and quantum physics. (Pure and applied mathematics, a series of monographs and textbooks) Based on lectures given by the author at a consortium in Lausanne, Switzerland, 1977. Bibliography: p. 1. Integration, Functional. 2. Quantum theory. I. Title. 11. Series. QA3.P8 [ QC20.7 .F85] 5 1O’.& [530.1’5] 78-22535 ISBN 0-12-644250-9
AMS (MOS) 1970 Subject Classifications: 60J65,80-00 PRINTED IN THE UNITED STATES OF AMERICA
79 80 81 82
98 76 5 4 3 2 1
Contents
Preface List of Symbols I
Introduction
1. Introduction 2. Construction of Gaussian Processes 3. Some Fundamental Tools of Probability Theory I1
32 43 48 60
Bound State Problems 8. 9. 10. 11.
IV
1 8 17
The Basic Processes
4. The Wiener Process, the Oscillator Process, and the Brownian Bridge 5 . Regularity Properties-1 6. The Feynman-Kac Formula 7. Regularity and Recurrence Properties-2 I11
vii ix
The Birman-Schwinger Kernel and Lieb's Formula Phase Space Bounds The Classical Limit Recurrence and Weak Coupling
88 93 105 114
Inequalities 12. Correlation Inequalities 13. Other Inequalities: Log Concavity, Symmetric Rearrangement, Conditioning, Hypercontractivity
119 136 V
vi
V
Contents
Magnetic Fields and Stochastic Integrals 14. It6’s Integral 15. Schrodinger Operators with Magnetic Fields 16. Introduction to Stochastic Calculus
VI
Asymptotics
17. Donsker’s Theorem 18. Laplace’s Method in Function Space 19. Introduction to the Donsker-Varadhan Theory VII
148 159 170
174 18 1 198
Other Topics
20. 21. 22. 23. 24. 25. 26.
Perturbation Theory for the Ground State Energy Dirichlet Boundaries and Decoupling Singularities in Scattering Theory Crushed Ice and the Wiener Sausage The Statistical Mechanics of Charged Particles with Positive Definite Interactions An Introduction to Euclidean Quantum Field Theory Properties of Eigenfunctions, Wave Packets, and Green’s Functions Inverse Problems and the Feynman-Kac Formula
21 1 224 23 1 245 252 258 272
References
279
Index
293
Preface
In the summer of 1977 I was invited t o lecture in the Troisieme Cycle de la Suisse Romande, a consortium of four universities in the French-speaking part of Switzerland. There was some discussion of the topic about which I might speak. Since I seem fated to be the apostle of probability to Swiss physics (see [258]), we agreed on the general topic of “path integral techniques.” I decided to limit myself to the well-defined Wiener integral rather than the often ill-defined Feynman integral. In preparing my lectures I was struck by the mathematical beauty of the material, especially some of the ideas about which I had previously been unfamiliar. I was also struck by the dearth of “expository’’ literature on the connection between Wiener integral techniques and their application to rather detailed questions in differential equations, especially those of quantum physics; it seemed that path integrals were an extremely powerful tool used as a kind of secret weapon by a small group of mathematical physicists. My purpose here is to rectify this situation. I hope not only to have made available new tools to practicing mathematical physicists but also to have opened up new areas of research to probabilists. I am pleased to be able to thank some of my colleagues who aided me in the preparation of this book. During the period of the lectures on which the book is based, I was a guest of the Physics Department of the University of Geneva. I am grateful to M. Guenin, the departmental chairman, and most especially to J . P. Eckmann for making my visit possible. The lectures were given at the EPFL in Lausanne; P. Choquard was a most gracious host there. I should like to thank the Secretariat Centrale of the University of Geneva Physics Department and Mrs. G. Anderson of the Princeton Physics Department for typing the first and second drafts, respectively, of the manuscript. I am also grateful to Y. Kannai for the hospitality of the Weizmann Institute Pure Mathematics Department where Sections 20-24 were written. vii
viii
Preface
Finally, I owe a debt to a number of people for scientific contributions: M. Donsker and M. Kac made various valuable suggestions about what topics might be included as well as offering help on technical questions; L. van Hemmen gave his permission to use an unpublished argument of his; I had valuable discussions with M. Aizenman, R. Carmona, P. Deift, J. P. Eckmann, J. Frohlich, C. Gruber, E. Lieb, A. Sokal, M. Taylor, A. Truman, and S. R. S. Varadhan; the careful reading of the complete manuscript by R. Carmona was especially valuable; finally, M. Klaus, A. Kupiainen, and K. Miller helped in the proofreading. I am glad to be able to thank all these individuals for their help.
List of Symbols
pllg'
(Eq. (9.8)) 96 b(s), b(s); Brownian motion 33.36 C,; Holder continuous functions 264 C ( K ) ;capacity 84 dh, db, dq. da; stochastic differentials 151, 154, 170 Db, Dq, D a ; measures for basic processes 38 E ( A ) , E ( / ) , E ( f ; A ) ; expectations 8 E ( f I 1 ) ; E ( , f l g l . .. . . g n ) ;conditional expectation 21 Go,,.; Dirichlet Green's function 69 h(y, k ) ; hitting probability 82 HI, 195 HD,s;Dirichlet Laplacian 69 H(a, V ) ; Schrodinger operator 159 H,(a) 161 K 7 ; Birman-Schwinger kernel 89 A, 18 p ( 4 , PA(z,B); pressure 200, 246 P,),,; Dirichlet propagator 69 P,(-x,Y ) 35 q(s); oscillator and also P ( 4 ) , process 34,57 (I,
P W' s(p); entropy S,,; Schwinger function
200 252
u , , ( X 1 , . . X , ) = ( X I , .. . , X , ) , ; Ursell function 129 y(f),W,(a.c)(b); Wiener sausage 209. 236 2 ; Brownian bridge 40 p B ;conditional measure 68 dp", a , c i , ; conditional Wiener measure 39 p);'(x; 8, 2 ) ; correlation functions 246 tu; Wiener path 38 n' ; wave operators 226 Z; grand partition function 246 ~
%:continuous functions on C[O, I ]
176
8:;'; Thomas-Fermi functional 98 :/: Schwartz space of tempered functions 12
1.1 ; Lebesque measure of A A B ; symmetric difference : : ; Wick ordering ; asymptotic series 1 1 . I/,!; uniform local norm
-
2 8 27 212 260
ix
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Functional Integration and Quantum Physics
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Introduction
1.
Introduction
It is fairly well known that one of Hilbert’s famous list of problems is that of developing an axiomatic theory of mathematical probability theory (this problem could be said to have been solved by Khintchine, Kolmogorov, and Levy), and also among the list is the “axiomatization of physics.” What is not so well known is that these are two parts of one and the same problem, namely, the sixth [287], and that the axiomatics of probability are discussed in the context of the foundations of statistical mechanics. Although Hilbert could not have known it when he formulated his problems, probability theory is also central to the foundations of quantum theory. In this book, I wish to describe a very different interface between probability and mathematical physics, namely, the use of certain notions of integration in function spaces as technical tools in quantum physics. Although Nelson [190, 1911 has proposed some connection between these notions and foundational questions, we shall deal solely with their use to answer a variety of questions in conventional quantum theory ; some typical problems which we shall solve using functional integration are the following:
(I) Consider apotential V in threedimensionsand let N ( V ) be thenumber V (i.e., the dimension of its spectral projection for of bound states of - A (-00, 0)). In a semiclassical approximation, this is expected to be (271)-~ I {(p, x) Ip2 + V ( x ) < 0) I (the (271)-~comes from the fact that h = 2m = 1 so h = 271);i.e.,
+
1
2
I.
Introduction
Throughout this book, I {. . .} I denotes the Lebesgue measure of {. . .}. It is clear that, in general, N ( V ) can be zero even though Nsc(V ) is quite large: for a shallow square well will not bind states, so if V is a sum of lo6 such wells, each lo6 light years apart, N ( V )will be zero but Nsc( V )will be very large. On the other hand, one might hope that N ( V ) cannot be large without N J V ) being about as large; i.e., one might hope to prove that
for some suitable universal constant c (see Section 9). (2) Let En@) be the nth eigenvalue of -d’/dxZ + x’ + bx4.Since E,(O) = (2n + l), we have that E 3 ( 0 ) - E 2 ( 0 ) = E,(O) - E,(O). A moment’s reflection convinces us that the left-hand side of this equality should increase faster with than the right-hand side. How do we prove that
a
E3(B)
- E 2 ( / j ) 2 E2(b) - E,(B)
(see Section 12)? We note that the more general conjecture that “ E , ( b ) is convex in n” is open. (3) Fix a positive continuous function V on R 3 and a, a C’-function on R3 with values in R3.Then (see, e.g., [ 132, 215, 229, 2421) H(a)
=
(-iV
-
a)’
+ I/
is essentially self-adjoint on C,7(R3). H(a) is, of course, the Hamiltonian of a particle in a magnetic field V x a and electric field -VV. Think of fixing V and varying a. Let E(a) = inf spec(H(a)). In [250], Simon showed that E(a) 2 E ( 0 ) by the following elementary argument: Let us compute
+
V(u*u) = [(V - ia)u]*u u*[(V - ia)u] = 2 Re(u*(V - iu)u)
Thus, using V(u*u)
=
2 Iu I V I u 1
IVlull I [u-’(Re[u*(V - ia)u]l I I(V - ia)u(
Squaring and integrating over x we find
so that E(a) 2 E ( 0 ) by the variational principle. (1 . l ) says that the energy of spinless Bosons goes co so fast up when a magnetic field is turned on. Let V go to infinity as x --f
3
I. Introduction
that Z(a) = Tr(eFBH‘”’)is finite. How can one prove the finite temperature analog of (1.1):
Z(a) I Z(0) (see Section 15)? (4) Fix V and W, even functions on ( - co,co) with d W/dx 2 0 for x 2 0. Suppose that -d2/dxZ V and -d2/dxZ V W both have normalized eigenfunctions, R, and R, + , respectively, at the bottom of their spectrum. Since W is “attractive,” adding it should “pull the ground state in”; i.e., one expects that for any a > 0
+
,
/;alR“+w(x)12dx
+ +
2 /~~IRY(X)l2dX
The difficulty with proving this lies in the facts that R is only given implicitly, that the energy shift must be taken into account, and finally that the normalization condition must be taken into account (see Section 12). (5) The Hamiltonian of a hydrogen atom in a constant magnetic field (O,O, cB) is (units with 2p = h = lel = 1):
It is not hard to show that H commutes with L, = i[y(d/dx) - x(d/dy)], and one expects that the ground state Ic/ of H has m = 0; i.e., L z $ = 0. The usual proof [217] that ground states of systems without statistics have 1 = m = 0 (when rotationally invariant) breaks down since eCfHis no longer positivity preserving; indeed, it is not even reality preserving (see Section 12). ( 6 ) Let V be a positive potential with compact support but also with some rather severe singularities ( e g , r - ‘ with c( large). One’s intuition is that particles should just “bounce off,” so that the basic existence and completeness questions of scattering theory for the pair ( - A , - A + V ) should be solvable. However, since V and - V have a very different status, the usual perturbation methods [199], will not be applicable (see Section 21). (7) One can ask to what extent the ground state (lowest eigenfunction), Ic/(x), of -d2/dx2 V(x) mirrors properties of V . Suppose that V is even Then modulo technical hypotheses we shall prove (see and $(x) = Sections 12 and 13) that V’ 2 0 (respectively, V” 2 0, V”’ 2 0) on (0, co) implies thatf” 2 0 (respectively,.f” 2 0,j’”’2 0) on (0, 00). We note that the analog for four derivatives is false; see the example at the end of Section 12. (8) Consider the ground state energy of H : in the Born-Oppenheimer approximation; i.e., for R E [0, xi), let
+
H ( R ) = - A - 1 x l - l - I X - (R,O,O)I-’
4
I.
Introduction
as an operator on L’(R3). Let e ( R ) = infspec(H(R)). Then E(R) = e ( R ) R-’ is the Born-Oppenheimer energy curve. It is reasonable to suppose that e(R) is monotone increasing as R increases but how does one prove this? (See Sections 12 and 13.)
+
It is not true that these problems all require functional integration for their solution (although, at the present moment, some of them have only been solved with such methods), but they all share the property of being problems with “obvious” answers and with elegant, conceptually “simple” solutions in terms of the tools we shall develop here. Once the reader has understood these methods and solutions, he will probably have little trouble giving a “wordby-word translation” into a solution that never makes mention of functional integration but rather exploits the Trotter product formula (Theorems 1.1 and 1.2 below) and the fact that eta is an integral operator with a positive kernel. That is, there is a sense, somewhat analogous to the sense in which the Riemann integral is a systematized limit of sums, in which the FeynmanKac formula is a systematic expression of the Trotter product formula and positivity of etA.In part, the point of functional integration is a less cumbersome notation, but there is a larger point: like any other successful language, its existence tends to lead us to different and very special ways of thinking.
*** Basic to a mathematical elucidation of path integration in quantum theory is Trotter’s extension [279] to infinite dimensions of a result of Lie. Nelson [l89] has isolated a special case (which is the one we mainly need) with an especially simple proof: Theorem 1.1 (special case of Trotter’s product formula [279]) Let A and B be self-adjoint operators on a separable Hilbert space so that A B, defined on D ( A ) n D(B), is self-adjoint. Then
+
(1.3)
If, moreover, A and B are bounded from below, then
+
Remark The theorem remains true if A B is merely supposed to be essentially self-adjoint (see [279] or [34]) but the proof below does not extend to this case.
1.
5
Introduction
Proof ([l89])
w
Let S, = e i f ' A f R1.; ) , = eitA,
=
eirB,U , = V,
w,and let
$t = S, $ for some $ in 2,the underlying Hilbert space. Then
II (St
-
:;il
q J $ I1 = 1 U:/n(Sf/n uf,n)s:,;J'$ -
5 n SUP
II (Sf/,-
U,in)$s
l
II
(1.5)
05S5f
+ B)4 as s 10 and - I)+ = K(iB4) + y[s~'(K. 1) iB]4 + s - ' ( V , - I)4 + iB4 + 0 + i A 4
Let 4 E D(A) n D(B). Then . S - ' ( S , ~ l)$
s-'(Us
+
i(A
-
-
so lim [nll(Sti. fl+
U,,,)4 Ill
+
0;
each
4 E D ( 4 D(B)
(1.6)
00
+
+
Let D denote D(A) n D(B) with the norm Il(A B)$Ij 11411 = 1114111. By hypothesis, D is a Banach space. By the above calculations, {n(Stin- Ut,,)} is a family of bounded operators from D to -Y?with sup,{ lln(St,n - U!,,J$Il} < co for each 4. As a result, the uniform boundedness principle implies that IIn(Sr/n - Ut/J4Il 5
~lll$lll
(1.7)
for some C . (1.7) implies that the limit in (1.6) is uniform over compact subsets of D. Let $ E D . Then s + $$is a continuous map from [0, t ] into D,so that {$,I0 5 s 5 t } is compact in D. Thus the right-hand side of (1.5) goes to zero as n -,co. The proof of (1.4) is similar. I Recently, Kato [154] has found the ultimate version of (1.4), which we will give without proof, primarily as a "cultural aside." Recall [152, 2141 that there is a one-one correspondence between positive self-adjoint operators and positive closed quadratic forms given by u(4, $) = (A'"$, A'"$). Let us extend the notion of positive self-adjoint operator: Definition Let A be an operator on a subspace D of X' which is symmetric and positive and which is self-adjoint as an operator on D (which may not be X'). We call A a generalized positive operator and set e - r Aequal to zero on D' and define it in the obvious way on D. (Note: e - O A may not equal one.)
With this definition, the above one-one correspondence is between positive closed quadratic forms (not necessarily densily defined) and generalized positive operators. The point is that if u and h are closed quadratic forms,
6
I.
Introduction
+
a b, defined on D(a) n D(b), is always a closed quadratic form, but it may not be densily defined, even if a and b are. Given generalized positive operators A and B,with associated forms a and b, we define A B to be the generalized operator associated to a b.
+
+
For any two generalized positive operators Theorem 2.2 (Kato [154]) A and B,s-lim,+m (e-rA/ne-fB/n)" exists and equals e - r ( A + Bfor ) any t > 0. Remark 1. An amusing and illuminating example of Kato's theorem is the following: Let U and W be two closed subspaces of 2 and let P, Q be the corresponding self-adjoint projections. Let A have domain U with A = 0 on U so that e - f A= P for any t. Similarly define B so that e - f B= Q for any t . Then Kato's theorem says that (PQ)" converges strongly to the projection onto U n W, a well-known result, but not one previously realized to be a case of a Trotter-type product formula. 2. For the extension to more than two operators, see Kato-Masuda [1%a]. Following Nelson [I891 we can use the Trotter formula to give sense to Feynman's path integral [83,84] : Let V be a smooth potential on R3.Then e - i f H= s-lim(e-i'Ho'"e-ir"'")", where H , = -*A and H = H , + V. Since - i f H o is an operator with integral kernel (see, e.g., [215]):
( I X 21"l')
K,(x, y ; t ) = ( 2 7 ~ i t ) -exp ~ / ~i
~
we see that (e-irHo/ne-ir"in)n has an integral kernel K(")(x,y ; t), where K(")(xo,x,; t ) = Nfl-'
s
exp(iS(x,, x,, . . . ,x,; t))dx, . . . d x n P l
where N , is the "normalization factor," (271it/n)~"'~, and -1
n i= 1
which is an approximation to the action
1.
7
Introduction
for a polygonal path going through xJ at time j t / n and linearly in between (the+ in S is m/2 corresponding to the 3in H , being 1/2m; i.e., m = 1 is chosen). Thus, in the limit, we obtain the formal expression
K ( x , y ; t)
=
s
exp(iS(w)) “dw”
(1.8)
for the integral kernel for e - j t H .In (1.8), the integral is over all paths between x and y in time t. A particularly beautiful aspect of (1.8) is that if we reinsert the h’s then exp(iS) becomes exp(iS/h) and a formal application of stationary phase in the semiclassical h -+ 0 limit gives the principle of stationary action of classical mechanics. See the end of Section 18. The “do”in (1.8) is certainly not a positive measure because of the normalization and indeed, general arguments [29] imply that it cannot be chosen as a signed measure even if one attempts to absorb the free action into it. (By extending the notion of measure, various attempts at defining the righthand side of (1.8) have been made; see, e.g., C. Dewitt-Morette [47-50a], Albeverio-Hoegh-Krohn [2] and Truman [280a-f]. These methods, while useful computationally and for formal heuristics, have not thus far turned out to be analytically powerful; we will not discuss them further.) Kac made the fundamental discovery that when one follows the above procedure for e-rH,e-‘“, e-IHoone can make sense out of the combined quantity exp( -$ cb2 ds) “do”and get a finite positive measure dp(w); indeed, this had essentially been done already by Wiener [286]. The resulting analog of (1.8) is now known as the Feynman-Kac formula; see Section 6. Of course, using the Wiener-Kac approach we obtain information aliout e P t Hand not e - i t Hthe , fundamental object of quantum dynamics. However, if we want to study eigenfunctions, it is hard to claim that e - j t His any more basic than e - l H . Indeed, it is often easier to study the ground state using e-rH since ’
inf spec(H)
=
-
’
lim t - In($,
e-lH$)
(1.9)
I+ m
for any vector $ whose spectral measure dp$ obeys dp,(a, a + E ) > 0, where a = inf spec(H). Moreover, if a = inf spec(H) is a nondegenerate eigenvalue, H q = aq and ($, q ) > 0, then (1.10) (1.9) is easy to prove and (1.10) follows by noting that e - f ( H - “ ) converges strongly to the projection (7, .)q as t + co. In fact, the reader of the recent literature in both constructive and “particle-theoretic” field theory could well conclude that we are living in an imaginary time era.
8
I.
Introduction
2. Construction of Gaussian Processes
We begin with some probabilistic notions; see Breiman [ 2 3 ] or Feller [82] for further discussion. A probability measure space is a triple ( X , 4,p ) of a set X , a cr-field 4 of subsets of X , and a positive measure p on X with p ( X ) = 1. (Such a measure is called a probability measure.) A real-valued measurable function on X is called a random variable. Given a random variable,f, the measure d p s on ( - co, co) defined by p s ( A ) = p ( f ‘ [ A ] ) is called the probability distribution o f j ’(to be distinguished from the “distrir ] which is common in the literature and bution function,” F ( t ) = ps( - a, which we shall not use). Given n random variables f l , . . . , f“ we define .fl 0 . . . 0.L: x R” by ( . f I 0. . . O f ” ) ( X ) = (fI(X), f*(x), . . . sn(x)) and = p ( [ f l 0 . . . Of;1]-’[A]),a the joint probability distribution ps,,.,,,fn(A) probability measure on R“. We use E ( A ) for p ( A ) , E(f) for f d p , and E ( f ; A ) to denote the integral offover the set A . There is a reason for introducing the term “random variable” for the equivalent “measurable function,” namely, an implied change of viewpoint. For suppose that one has a “random function on ( - co, co),” i.e., for each t E R a random variablef, (with some measurability in t ) ; a functional analyst would most naturally think about f,( .) for each t or perhaps the function f,(x) of two variables. The probabilist’s language leads to the consideration of the functionsf: (x) for each x. As is usual, one does not distinguish random variables which are equal almost everywhere. Typically, one goes even further and, at least formally, A and B, are called removes the points of X from consideration: two sets in F, equivalent if and only if p ( A A B) = 0 where A A B = (A\B) u (B\A). The equivalence classes are called events and the family of events, S/9,, inherits the notions of intersection and union. A random variable is, more properly, a map (f I ) from 9, the Bore1 subsets of R, to 9/9, preserving countable unions and intersections and ps is the composition off- :,%’ -+ S/ 9,and p : F / 9 , -+ [0, co). An isomorphism of two probability measure spaces ( X , 4,p ) and ( X ’ , F, p‘) is a map T : 4/9,+ 9’/9,. which is a bijection respecting countable unions and intersections and with p‘ T = p . Random variablesf‘on X andJ’ on X’ are said to correspond under T if and . pointed out the need for the above general only if T f’- = ( j ’ ) - ’ Having abstract framework, we will usually be colloquial and talk about points, about random variables as functions, etc. Let 9 be an index set for a family of random variables. If I c 9 is a set with n = # ( I ) < co, we have a joint probability distribution p I on R“ for {,fa},, I . The measures p, are consistent in the sense that if I c I‘, then p I can be +
9
0
0
’
9
2. Gaussian Processes
obtained from p I , by “integrating out” the variables in l‘\l; i.e., if { f l , .. . f,} = { . f u ) g c l and if,, . . . , f m } = i f z } , , l . ( m 2 n), then p / ( A ) = p J A x Rm-”). A fundamental result in probability theory is: Theorem 2.1 (Kolmogorov’s theorem) Let 9 be a countable set and let a probability measure p I on R’“) be given for each finite set I c 9 so that the family of pi's is consistent. Then, there is a probability measure space ( X , F, p ) and random variables { f u } u s , aso that p , is the joint probability distribution of Moreover, this space is unique in the sense that if ( X ’ , F, p’) and { f h } u e , Ialso have these properties and if 9(and respectively, 9’) is the smallest cr-field with respect to which thef, (respectively,fk) are measurable, then there is an isomorphism of the probability measure spaces under which eachf, corresponds tofh. Proof The existence and uniqueness aspects are quite distinct. To prove existence we will take X = k qwhere k! = R u {a}is the one-point compactification of R. Since X is compact, we can use the Reisz-Markov theorem to construct Baire measures (this finesses the proof of countable additivity, or more properly, places it on the shoulders of the proof of the ReiszMarkov theorem; see, e.g., Berberian [ 1 11 for this proof). Let C,,,(X) denote the family of continuous functions of finitely many coordinates {x,},~~(as I runs through all finite subsets) and let f(f’) =
Sf(.,)
&/(xu)
iff is a function of {x,},,~. / is a well-defined positive linear functional on C,,,(X) because of the consistency conditions and clearly I/(f)l S II .f II . By the Stone-Weierstrass theorem, C,,,(X) is dense in C ( X ) , so !extends uniquely to a positive linear functional on C ( X ) . Therefore, by the ReiszMarkov theorem, there is a Bake probability measure, p , on X with ~
K f )= S . f ( X ) W ) Definefu(x) = x, if xu # co and zero if x u = co.Then clearly the d p , are the joint distribution of the { f u } a c l . This proves existence. To prove uniqueness, we first show that the condition that the,f,’s generate F implies that the bounded functions of finitely many ,f,’s are dense in L 2 ( X ,d p ) . For let Y? be the closure of these functions in L2 and let xa in 2 be a characteristic function of some set A. Then we claim that there are A , whose characteristic functions, x,,,depend on only finitely many fa’s with xn % x A . For let g, 5 x A and by passing to a subsequence, suppose that g, -+ x A pointwise almost everywhere. Let A , = { X I + < g,(x) < 2). Then
10
I.
Introduction
4 xa pointwise almost everywhere and so in L z by the dominated convergence theorem. Since X, + x A , the set of A’s with x A in X is closed under finite intersections since this is true of the generating sets. Clearly it is closed under complements and it is closed under countable unions by the monotone convergence theorem. Thus, the set of A with xA in A? is a a-field, Thus A? = L 2 ( X ,d p ) . and so it is 8. Given two models ( X , 9, p), {f,}and (X’, B‘, p’), {I:}, define U :L 2 ( X ,dp) + L2(X’,d p ’ ) by U[F(f,)] = F(J:), which is well defined and unitary and extends to L2 by the above density result. It is not hard to see that U takes characteristic functions into characteristic functions (use X, + zA) so that T defined by U ( x A )= x ~ (defines ~ ) a map of F/9,,to F’/Y,,,. This T is easily seen to be an isomorphism under whichf, corresponds tofh. I
xn
( X , 9, p ) is called a model for the p,. If we take .a = (0, 1,2,. . .}, then by the above R’ can be taken as a model. (In the above, k’ was used a priori, but one notes that p{x(some xu = co} I ~ , p { x ( x=, co} = 0.) It is often useful to know if some nice subset of R” has measure one. Two particularly useful subsets are a and 3’ defined as follows: for m E Z,
a 9’
=
nm
Urn
a, with the Frechet topology induced by the 11. ,/I and i)‘ = .9 , is the topological dual of a if x E $9‘ is associated to the linear functional
LAY)= C x n Y n
The cylinder sets o-field on a’ is the smallest a-field with respect to which the functionals x H L,(y) are measurable for each y E d. It is easy to see that this is identical to the a-field generated by viewing 6‘ as a subset of R’ with its natural a-field. Given a probability measure, p, on a’ with the cylinder set a-field, we can define the function 0 on 6 by (2.1)
called theFourier transformof p. Often this is called the characteristic function of p . To avoid confusion with the characteristic function of a set, we only use the former name in this book. 0 clearly has the following three properties [(c) follows from the dominated convergence theorem and the continuity of O(0) = 1.
0 is positive definite ; i.e., given z I , . . . ,z, E rC and y I , . . . ,ynE 6 , ” zizjo(yi- y j ) 2 0
1
i. j = I
0 is continuous when a is given its Frechet topology.
11
2. Gaussian Processes
The following infinite-dimensional analog of Bochner’s theorem (see [215] for a proof of Bochner’s theorem) is a special case of a theorem of Minlos [187]. The beautiful proof we use we learned from van Hemmen [283] (although the literature on Minlos’ theorem is so large, it may well be older): Theorem 2.2 (Minlos’ theorem for ,J’) A necessary and sufficient condition for a function @ on 9 to be the Fourier transform of a probability measure on il’ is that it obey (a)-(c).
Proof Necessity has already been discussed, so suppose that (a)-(c) hold. By Bochner’s theorem, for any finite I , there is a measure pf on R’ with @((Ya)ae 1) =
J
It
exp(ix . Y) dp,(x) RI
By the uniqueness part of Bochner’s theorem, the p’s are consistent, so by our proof of Kolmogorov’s theorem, there is a measure p on R” so that (2.1) holds for all y E il with y, nonzero for only finitely many a’s. Each 9, and thus d is measurable in R”. If we show that p ( d ) = 1, then p may be restricted to 6‘ and (2.1) will extend to all of 3. The proof is thus reduced to using (c) to show that p ( d ) = 1. Given E, we can find rn and 6 so that Ilyll, I 6 implies that I@(y) - 1 I I E. We claim that Re@(y) 2 1 - I: - 26-211yjli
(2.2)
for all y e il. For (2.2) holds if llylli I b 2 , since 11 - @(y)l I E in that case, and it holds if Ilylli > d2, since Re @(y) 2 - 1 for all y [we use here the fact that condition (b) implies that J@(y)lI @(O)]. Fix a sequence {q,,} and for LI and N let do,., be the measure on R N + ’:
Notice that (for all integrals over R N +I ) (2.3a)
erx‘ydo,, N ( y )=
exp
(2.3b)
12
I.
Introduction
Integrating (2.2) with respect to d G , . N and using (2.3), we find that exp( - !!
i:
2n=O
[Wf
q,,xi) dp 2 1 - c
+
c1
2F2ct
5
(1
+ nz)"q,
(2.4)
n=O
qn(l n2)" Choose qn so that monotone convergence theorem :
Now take obtain
-
= K < co and take N
+ CQ,using
the
to zero, using the monotone convergence theorem again to
Choosing qn = (1
+ n2)-"-
I,
we see that
p(4') 2 p ( L m - , )2 1
- E
so that, if we take E to zero, we see that p(ci') = 1.
I
The above theorem only depends on the structure of ci and 6' as a particular topological vector space and its dual. But 6 is topologically isomorphic to 9 ( R ) under the Hermite expansionfE ,Y H {.f,} withf, = 1Q,,(x)f(x) dx where cb,,(x) = (2"n!)- I"( - 1 ) " ~ - 1'4e"'z (d/dx)ne-*', the nth harmonic oscillator wave function (see [240] or [214, Appendix to Section V.31) and also to .Y(R") for each v. As a result, Theorem 2.2 immediately extends to the following. Definition A cylinder set measure on .!Y"(R") is a measure on the a-field generated by the functions T + T(Q,)as Q, runs through all of 9'(Rv).
Theorem 2.3 (Minlos' theorem for 9") A necessary and sufficient condition for a function @(.) on cY(Rv)to be the Fourier transform of a cylinder set probability measure on 9 ' ' ( [ w v )
@(4>= Jexp(iT(4)) M T ) is that @(O) = 1, @ be positive definite, and @ be continuous in the Frechet topology on 9'(Rv).
13
2. Gaussian Processes
Remurk The above proof shows a little more; namely, if @ is continuous in the norm associated to d, then p is concentrated on any set of the form (c > O),
For example, these methods imply that if w(t) is a Wiener path and d, E C r , then &o is in the domain of l d / d x [ ' / 2 - e .This is just short of implying continuity (any function in ) d / d ~ I l ~ is' +continuous), ~ so we will need a subtler argument to get continuity (see Section 5).
*** In the above, we considered countably many random variables, but much of the discussion is applicable to arbitrary families, e.g., families indexed by t E (a, b) c R ("stochastic processes"). In particular, Kolomogorov's theorem, in theform we give i t (and its proof) extends to arbitrary families. We can use by for X ; however, the use of R' does not extend since {x I x, = co for some a } is no longer even measurable ! There is one subtlety ("the problem of versions") associated with this extension that we shall discuss now and henceforth generally brush under the rug. The subtlety can be illustrated in the following trivial example: p ) be [0, 11 with its Bore1 sets and Lebesgue Example Let ( X , 9, measure. For each t E [0, 11, let y(t) be the random variable on X which is identically one and let q"(t) be defined by
q"(t)(x) = 1, =
0,
t - x nonrational t - x rational
Then for each fixed t , q and q" are equal almost everywhere so that, from the Kolmogorov theorem point of view, q and q" are identical families. But notice that for every EX, the map t ~ q ( t ) ( xis) continuous while t ~ q " ( t ) ( x ) is discontinuous ! This example illustrates dramatically that if {q(t)lt E [O,
111
is a family of random variables on ( X , 9, p) and 9is the minimal o-field, then {x I t + q(t)(x)is continuous} may not be measurable. This example is especially disturbing because one of the most celebrated results in the development of the Wiener process is that the "paths," q(t)(x) are continuous in t for almost every x. One can, for example, define the Wiener process for t 5 1 a priori on b[('*' I with the Baire field, in which case it does
14
I.
Introduction
not strictly make sense to say that t + q(t) is pointwise almost everywhere continuous. (It does make sense to say that t -+q ( t ) is continuous in L2norm and this will be trivial to prove.) There are a variety of ways around this problem: (a) (The one we will use) Suppose we show that for almost every x E X , there is a C , so that
I&)
- q(s)l 5 C,lt
- sI6
(2.5)
for some fixed 6 and all rational t, s in [0, 11. Since only countably many t and s are involved, (2.5) is “version independent.” Choose any explicit X for the process and consider the q(t) for t rational and let X , = {x E X ( ( 2 . 5 ) holds}. For x E X , , q ( t ) ( x ) defined for t rational extends to a unique continuous function g(t)(x) for all t E [0, 11 and thereby, we can define random variables g(t) on X , and extend them to be zero on X\X,. Clearly t + @ ( t ) ( x ) is continuous for each X E X . Moreover, the joint distributions of ( q ( t l ) ,. .. , q(t,,)) agree with those for (cj(t,), . . . , ij(t,,)). This is obvious for ti rational and extends to all t by the pointwise continuity of ij and the L2 continuity of q. We thus have a “version of q” with continuous “paths.” This point of view is further discussed in [39]. (b) One directly constructs a probability measure on the space of continuous functions and defines this to be the Wiener process; the above philosophy is not then directly relevant. One disadvantage of this method is that a detailed proof of countable additivity can get somewhat more involved than in our discussion. This procedure is used in [14]; see also Section 17. (c) Use Minlos’ theorem to construct a measure on Y ’ ( R ) so that the . . . , T(f,) agrees with that of j q ( t ) f , ( t )dt, . . . , joint distribution of T(f,), j q(t)f,(t) dt. Then find in 9” a measurable family, F , of distributions equal (as distributions) to continuous functions with p(Y’\F) = 0. The Levy [168]-Ciesielski [35] proof (described, e.g., in [183]) can be interpreted in this way. (d) Use the compact model of [r;Q[O,ll but use the Borel field rather than the Baire field (see, e.g., [214] for the distinction) in which case the Holder continuous functions are Borel measurable. The construction we used in Theorem 2.2 only defines a Baire measure which will have many Borel extensions, but precisely one regular Borel extension. In this extension, the Holder continuous functions have measure one. This is a point of view advocated by Nelson [188]. While the words leading to the final result are different in the above arguments, the end definition of Wiener measure as a measure on the space of continuous functions is the same.
***
15
2. Gaussian Processes
A particularly important class of abstract processes are the Gaussian processes. It is easiest to describe them by using the Fourier transforms of the joint probability distributions; i.e.,
m 2 n ) if One reason for this is that d p f l , , , , , f mis consistent with d p f r , , , , . f(rn and only if C f ,,...,f n ( t l , . . ., t n )
=
Cfl ..... f,,,(tl,. . ., t n , O , O ,
. . .,O).
We callfa Gaussian random variable of variance a if and only if C,(t)
=
exp( -+at2)
Equivalently, dpf(x) = (2na)- ' I 2 exp( - + x 2 / a ) dx
(a # 0)
(a
= G(x)dx
=
0)
We callf,, . . . ,,fn jointly Gaussian with covariance { a i j }( a j j= a j j )if and only if
a j jis only dependent o n 5 andjj, since
a r. j. = JL.f,dP If { a j j } sli < j 5 n is a nonsingular matrix with inverse b, then dpf,,...,-,(x) = (2n)-""(det u)-'"exp(-fC
bjjxjxj)
(2.6')
Given an n x n real symmetric matrix aij, it will be the covariance of some jointly Gaussian random variables if and only if a is positive semidefinite. For, in that case, (2.6) defines a positive definite function and so, by Bochner's theorem, a measure via Fourier transformation. Occasionally (but not here), one discusses Gaussian random variables with mean mi and covariance aij in which case those described above are Gaussian random variables of mean zero. The right-hand side of (2.6) is then replaced by uijtiti + i C rnjtj) exp( -f
C
16
I. Introduction
Throughout this book, the phrase “fand g are Gaussian random variables” will be used to indicate they are jointly Gaussian with mean zero.
Theorem 2 3 A Let X be a separable r e d Hilbert space. Then there exists p) and for each u E % a random variable, a probability measure space ( X , 9, +(u), so that u H $(u) is linear and so that for any u l , . . . , u, E 2,(4(ul), . . . , 4(u,)) are jointly Gaussian with covariance ( u i , u j ) ((., .) = inner product on 2).The same kind of uniqueness as occurs in the Kolmogorov theorem holds here. Remark
We will call {~(ZI)}the Gaussian process with covariance (., .).
Proof Pick u l , . . . , u,, . . . an orthonormal basis for X . Let c(t) be defined for any sequence t , , . . . ,t , , . . . eventually zero, by c(t) = exp(
-+ C ti’)
By the above remarks on when exp( -f C a i j t i t j )is positive definite and on p) consistency, we can apply Kolmogorov’s theorem to construct ( X , 8, and 41,. . . , 4,, . . . ,jointly Gaussian with covariance d,, (this is somewhat circumlocutory, one can just take X = R”, 4i= x i , and m
p = @(2n)- ‘ I 2 exp( - + x i ) dx, n= 1
xy=
x?=
directly). Now given a finite sum u = cqui, set 4(u) = I ui4i. Then I + ( r ) 1’ d p = /I I/ 2 , so, by continuity, 4 extends from finite sums to a map from &‘ to L 2 ( X ,dp). It is easy to use continuity to see that I‘
J
eiQ(4
4l
=
exP(-311~l12>
so the 4’s are jointly Gaussian with the proper covariance. Uniqueness of the process restricted to finite sums of the ui follows from the uniqueness part of Kolmogorov’s theorem and general uniqueness follows from the L2 continuity deduced above. I Corollary 2.4 Let c(t, s) be a jointly continuous real-valued function on K x K where K is a separable topological space. Suppose that for any t , , . . . , t, E K , c ( t i , t j ) is a positive semidefinite matrix. Then, there exists an p ) and a random variable q(t) for essentially unique measure space ( X , 9, each t E K so that the q(t) are jointly Gaussian with covariance c.
Proof For each t E K , introduce a formal symbol 6, and consider the vector space of finite sums aidrt.Define an inner product on this space by
1
(Caidt,, C b j d , , ) = C z i b j c ( t i , s j )
3.
17
Probability Theory
By hypothesis, this is a positive semidefinite inner product, so by the usual procedure ofquotientingby null vectors and completing we can form a Hilbert space, X . The separability of K and continuity of c imply that X is separable. Let 4 be the Gaussian process with covariance (., .), afid let q(t) = 4(&). I Warning:q ( t ) is not, in general, linear in t. We want to give one final result in the abstract theory of Gaussian processes, a result due to Feldman [80a] and Hajek [121a]. It is essentially equivalent to a result of Shale [237] about the implementability of Bogoliubov (= symplectic) automorphisms in the theory of the free Bose field. Most field theorists are unaware of the work of Feldman-Hajek and most probabilists of the work of Shale! Theovem2.5 Let X be a real Hilbert space and let A be a bounded positive invertible operator on A?. A necessary and sufficient condition for } there to exist a single measure space ( X , F)with functions { 4 ( ~ )and two mutually absolutely continuous probability measures dp and dv so that the 4(u) are jointly Gaussian on ( X , 8, p) with covariance (., . ) x and v) with covariance (., A . ) , is that A - 1 be jointly Gaussian on ( X , 9, Hilbert-Schmidt. A proof may be found, for example, in [258]. It is quite easy to prove; indeed, we will essentially prove it as Lemma 18.6. The useful direction is that A - 1 Hilbert-Schmidt implies mutual absolute continuity. The idea is to choose a basis for X with A$, = u,$,. Then formally d p = @ ( 2 ~ ) l-i 2 exp( -+xi) dx, and dv = @ (2naJ ' I 2 exp( -~xi/sc,) dx,. The condition (u, - 1)' < 00 can be used to show that
1
n N
u,- 'I4 exp( -*(a;
I
- 1)~:)
n= 1
converges in L 2 ( X , d p ) . Of course, the square of the limit turns out to be dvldp. There is an extensive probabilistic literature on properties of Gaussian processes; see, e.g., [61a, 82b, 82c, 85a, 180, 195al.
3. Some Fundamental Tools of Probability Theory
In this section, we present a number of related topics: Borel-Cantelli lemmas, the notions of independence and conditional expectation, Doob's martingale inequality, Levy's maximal inequality, the individual ergodic
~ ~ ~
18
I. Introduction
theorem (without proof), and finally some calculations with Gaussian processes related to the above. Definition We define
Let A, be a sequence of sets in a probability measure space.
i.e., x E 6A , if and only if x lies in infinitely many A,,%. One reason for the name is that if A , = { x l f . ( x ) > A} for some random variables f’,, then { x l ( l i m f ’ , ) ( x ) > A} c l i m A , c {xl(limf’,)(x) 2 A}. Theorem 3.1 (first Borel-Cantelli lemma) with E ( A , ) < co.Then E ( E A,) = 0.
1,
Let A , be a sequence of sets
Proof Let x, be the characteristic function of A,. Then x E lim A , if and ;m(x) < 00 almost everyonly if x,(x) = co,so we need only show that where. But, by Fubini’s theorem,
c,
1
A fortiori,
1,1, < co a.e. I
The classical second Borel-Cantelli lemma is for independent A, (see below for the definition). The following result is a special case of a theorem of Erdos-Rennyi [78]. Some related results can be found in [14,223]. Theorem3.2 (second Borel-Cantelli lemma) Let A , be a sequence of E(A,) ,, = 00. Suppose that there is a matrix a,,defining a bounded sets with I operator on / z so that IE(A, n A,)
Then E ( E A,)
=
-
E(A,)E(A,)I I anmE(A,)112E(Am)112
(3.1)
1.
Proof’ Let c be the norm of a and let f’, characteristic function of A , . Then
=
X,
-
E(A,) where X, is the
3.
19
Probability Theory
Thus, for any a 2 0 and N so that
I
&An) 2 a,
c;=l
-cr=l
since on the set where xn I a, we have that &An) 2 0. Fixing a and taking N -+ 00, we see that
since
I
E(A,)
= m.
fn
2 -u
+
Hence
As an example of the power of these two results, we consider the following situation. Letf'be a random variable and let dv be its probability distribution. =: I d v ( x , ) on R" (conBy an independent sequence of copies of / w e mean @ structed a la Kolmogorov), which is to be thought of as a model of "sampling f."Byf, we mean the nth coordinate function. Suppose thatfis unbounded with nonpathological behavior at m and let a, be defined by v(u,,, .o>= l/n
(3.2)
One would expect to need about n trials to get a value o f f in the interval (a,,, m); i.e., one would expect that fifJu, = 1 (which should be thought of as a fluctuation result since we will also have 6f n / ( - a,) = 1 if d11is an even measure). Theorem 3.3 Let fn be an independent sequence of copies of a random variab1e.f: Let b,, be a sequence of numbers so that for each 6 > 0:
(3.3a) (3.3b) Then
lim f,/b, = 1 with probability one.
20
I.
Introduction
Proof Let A’’ = { X I f,(x) 2 (1 +_ 6)b,). By (3.3a) and the first BorelCantelli lemma, almost every x is in only finitely many A:’; i.e., 6f,/b, < 1 + 6 with probability one. By the second Borel-Cantelli lemma, (3.3b), and the fact (take anm= d,,,,,) that E(A,-’ n A,’) = E(An-b)E(A;6) (n # m ) (we have a product measure), almost every x is in infinitely many A,’; i.e., kfn/hn2 1 - 6 with probability one. Taking 6 to zero through a countable set, we obtain the result. I
Remark
(3.3b) is trivial for the u, of (3.2) and “normally,” (3.3a) will hold.
Example dv. Then
Letf(x) have the probability distribution (271)- ‘ i 2 e - x 2 ’ 2dx
(27r-1/2u-1(1+
=
exp(-a2/2) I v(u, m) I (271)-1’2u-1exp(-u2/2) (3.4)
for a > 0. This may be seen as follows ([lSS, p. 41):
Let b, = (2 In ,)Ii2. Then (3.4) shows that (3.3) holds. Thus for independent Gaussian trials, 6f,/(2 In , ) I i 2 = 1 with probability one. The celebrated “law of the iterated logarithm” and some other limit theorems we prove in Section 7 are only one step beyond this simple example. An alternative to McKean’s inequality (3.4) is the following inequality of Fernique [ S ~ C :]
+
( 2 7 ~ - ’ / ~ ( u I ) - ’ exp(-u2/2) I v(u, co) I $ ( 2 7 ~ - ” ~ (+u I ) - ’ exp(-u2/2) (3.4‘) (3.4’) follows from
( r 2 0) g’(t)
I exp( -it2)5 $g‘(t)
’
where g ( t ) = - ( t + 1)- exp(-$t2), for we can integrate this inequality from t to infinity. By elementary calculus this last inequality is equivalent to
2I (t2
+ t + l)/(t +
1)2
= q(t) I 1
3.
21
Probability Theory
which is easy to check since q’(t) has exactly one zero in (0, co) at t = 1 and q(0) = q(co) = 1, q(l) = 2. (3.4’) is clearly sharper than (3.4) for u small but not quite as sharp for a large.
*** Definition n random variables .fl, . . . , f, are called independent if and only if their joint distribution tlv,l,,,,,,m is the product measure dv,, @ . . . @ dv,,
n events A , , . .. , A, are called independent if their characteristic functions are independent random variables.
For two events, A and B, their characteristic functions have a joint probability distribution supported at the four points (0, 0), (0,1). ( L O ) , and (1, I), and it is easy to see that independence means p(A n B ) = p ( A ) p ( B ) .Equation (3.1) can be interpreted as A, and A, are “asymptotically independent.” Suppose that one has the special situation where the underlying probability measure d p can be realized as d p f 0 dv where dpu, is the distribution forf, and ,f becomes the first coordinate function in this realization. Clearly functions g of the second variable are independent off, and given an arbitrary random variable h(x, y), the variation off h(x, y) dv(y) as .x varies will be some kind of indication of how far h is from being independent off. The conditional expectation we will now discuss is the analog of integrating over y ” in cases where d p is not a product measure. “
Definition Let ( X , F,p ) be a probability measure space and let C be a Let Lg be the (closed) subspace of functions in a-field contained in 9. L 2 ( X ,9, d p ) which are C-measurable and let P , be the orthogonal projection from L2 to L;. ForfE L2(X,9, d p ) , P,,fis called the conditional expectation off’ with respect to C written E ( f 1 C ) . If gl, . . . , q,, are random variables, E ( f l g , , . . . , g,) denotes the conditional expectation offwith respect to the a-field generated by gl, . . . , g,l. Remarks 1. Since E( 1 I C) = 1 and E ( f I C) 2 0 iff 2 O(this follows from 11 f 11 1 SO E(-I C) Remark 2 below), it follows that forfE L n L 2 , II E ( f I C)ll1 I extends to L’. The second remark also shows that E ( f l C) 2 E(gIC) if
’
f
2 9.
2. E ( f I C) is determined by two conditions: That E ( f I C) is C-measurable and that
22
I.
Introduction
for any C-measurable y. (3.5) implies, in particular, that
if h is C-measurable
E ( f h l C ) = h E ( f IC),
(3.6)
3. If y is the characteristic function of a set B, then
so that E ( f Ig)(x) is the expectation off with the additional information that we know g(x) and nothing else. More generally E ( f I C)(x) can be interpreted as the expectation offknowing the value of all C-measurable 9’s. .f is independent of y if and only if E(eiaf Ig) is a constant
Proposition 3.4 for each real CI.
Proof Iff is independent of g, then E ( F ( f ) l g ) is constant for any F . Conversely if E(eiafIg) is constant for all CI, then the constant is necessarily E(eiaS)and ,qeia.r+
for any CI and
ill!?) =
,c[E(eia.r I y ) e i l l ~ ]
E(eid)E(eibg)
fl so, taking Fourier transforms, pf,s= pf 0 pg. ***
Definition Let , f l , ... , ,fi be a sequence of random variables. We call them a martingale (respectively, submartingale) if and only if for each m, E(l.f*l) < and
(respectively, E ( f i I f 1 , . . . ,.fntRemarks
2
.fm-
, (pointwise)).
1. It follows by induction that for j < m, E(.fm If
l3
. . fj) = ,fj ‘ 9
(resp-7 2 .rj>
(3.7)
2. If X , , . . . , X , are random variables which have mean zero and are X i is a martingale. Then Doob‘s inequality independent, then f, = (below) can be interpreted as saying that a certain gambling strategy will not do any better than the strategy ofjust waiting n steps; see Feller [82].
x{=
3.
23
Probability Theory
Theorem 3.5 (Doob's inequality [61]) Then for each I > 0 and n:
Let {f,}
be a submartingale.
(3.8)
Proof Let x j be the characteristic function of the set A j of points where be the characteristic function of the set f1, . . . , f,- < I andf, 2 A. Let A with max,,j,n f, 2 A. Since the Aj's are disjoint sets with union A , we have that n
2 AE(A,J
IE(A) =
j= 1 n
1
I EUiXj) j = 1 n
2
I E(E(.f;,I .fi . . . ,fj.)xj) 9
j= I
n 2
1 E(fnX,j) = E(.fnX) IE ( f ' , f ) j= I
In the above, we usedf, 2 A in the first inequality, the submartingale relation (3.7) in the second inequality, the fact that x j isf;, . . . , &-measurable and (3.5) in the next equality, and the calculation E(fnx) I E ( f z ~ I) E ( f ; ) in the final step. I The exciting thing about the above inequality is that it extends to continuously indexed processes:
Definition Suppose that { 4 , } r G , u .ish 1a family of random variables. We say it is a martingale (respectively, submartingale) if and only if
(respectively, 2 q , ) for all a I ~1 5 r 5 b. Notice that a fortiori, one has that
24
I.
Introduction
if s1 I s z . . . Is, It [check (3.5)!] so that for s1 I . . . s, I. . . the sequence f, = qs, is a (sub)martingale. As a result,
Taking the mesh of the points si to zero we have the following. Corollury 3.6 Let {qt}fG1u,bl be a submartingale with continuous sample paths (i.e., t + q1 is almost everywhere continuous; this is “version dependent ”). Then
Remark Doob’s inequality is just one of a number of interesting developments in the theory of martingales described in Doob’s book [61]. For example using
E(F 2 1) I1 - ’ E ( f , ; F 2 A)
with F = maxoSjsnfj, an inequality proven as an intermediate step above, one can show that E(IFlP)’’P I p(p
-
l)-lE(lfny)”p
as follows:
where q-’
=
1 - p - ’ , and we used Holder’s inequality in the last step.
***
3.
25
Probability Theory
There is another powerful inequality which has a proof similar to that of Doob's inequality: Theorem 3.6.5 (Levy's maximal inequality) Let X I , . . . , X , be n random variables with values in some vector space R". Suppose that the joint probability distribution for X , , . . . ,X , is invariant under any change of sign X i + q X i (each E i = k 1). Let S j = X I + . . . + X j . Then for any A > 0,
(
1
E max I S j [ 2 A I 2E(IS,J 2 A)
If v
=
I<;<,
1, then, in addition
2E(S, 2 E(maxSj 2 A ) I Proof
(3.8a)
A)
(3.8b)
We prove (3.8a); (3.8b) is similar. Let A j be the set with IS, I < A = A;. Let TJ = X I ... X j -
u
A, . . . , ISj- I < A, lS,l 2 A, and let xj+l- . . . - X , and notice that
+
+
sj = is, + f T j Thus {ISjI2AIc {ISnl2A)u{lTjl2A)
so E ( A j ) = E ( A , ; lS;l 2
A)
+ E(Aj;
< H A , ; lS,,l 2 A) = 2 E ( A i ; IS,I 2 A)
2
A)
where we use the invariance under changing the sign of X j + . . . , X , . Since the A j are disjoint:
E(A)
=
C E ( A ; )I 2 1 E ( A ; ; IS,] 2 1)2 2E(IS,I 2
4
which is (3.8a). For further discussion of the role of reflections in probability theory, see [85a, pp. 21-29] and [198a, Chapter 51.
*** Definition A map T on a probability measure space is called ergodic if and only if T is measure preserving and T [ A ] = A for A E 9 implies p ( A ) = Oor 1.
26
I.
Introduction
Examples 1. ("Kolmogorov 01 law") Let (Y, B,, v) be a probability measure space and let X be the two-sided infinite product of Y's with dp = @Corn dv. Let ( T Y )=~y,- Then T is measure preserving and ergodic. For given A with T [ A ] = A and given E, we can find n and B only depending . . . , y, so that p(A A B ) I E (such B's have characteristic functions on L2-dense in the characteristic functions of all measurable sets; see the proof of Kolmogorov's theorem). Let B' = T'""[B]. Then B' is independent of B so p ( B n B') = P ( B ) ~Since . A
A ( B n B') c ( A A B ) u ( A A B')
we see that p(A A ( B n B')) I 2r:
so that
lAB2 - p(B)I 5 3E thus ldA>2- P ( 4 I
4r:
Since E is arbitrary, , u ( A ) ~= p(A). 2. Suppose that T is measure preserving on ( X , 9, p ) and that for each [a, b] c Iw, we have a subalgebra B [ u , b ] of -9 so that: generate 9, and ,F, c .FJif I c J , (i) the (.F[a,b]} (ii) if a < b < c < d , then
.f(c.- b)
I d B n C ) - p(B)pL(C)I
for all B E B [ a . h ] and C E .F,r,dl withf(f) (iii> T I B [ a , b ] l = B[a+ l , b + 11.
+0
as t
-+
co,
Then, as in Example 1, T is ergodic: For given A, we find B E .Fl-n,,l so that p(A A B ) 5 c. Then, if B' = T 2 " + k [ B ]we , have that Ip(B n B') - p(B)I
Taking k
--f
00, we
< 3c
so that
Ip(B)' - p(B)I 5 38
+ f(k)
find that ~ P ( B-) p(B)I ~ I 3 ~so, Ip(A)* - p(A)I I 4~.
In two places, we will need the following result which we state without proof; see Halmos [1223 or Shields [238] for further discussion. Theorem 3.7 (Birkhoff ergodic theorem) Let T be a map on a probability measure space which is measure preserving. Letf E L ' ( X ,dp). Then for almost every x, the limit 1 n- I lim C f(T'x) f g(x) ~
n-tm
j=o
3.
27
Probability Theory
exists and i g ( x ) d p constant f ( x ) dp.
S
= if(x)dp.
If moreover T is ergodic, then g is the
*** Next, we turn to the applicability of some of the above ideas to Gaussian random variables. First, we have the following trivial proposition. Proposition 3.8 Two Gaussian random variablesf and g are independent if and only if their covariance ( f , g ) = f g dp is zero. Proof The Fourier transform of their joint distribution is M ( t , s ) = E ( e i t f + i S g= ) exp(-$/ltf =
+ sgll:)
W r , O)M(O, 4 exp( - W ,9))
is a product if and only if (L g)
=
0.
1
Slightly more subtle is the following.
Theorem 3.9 Let 2 be a real Hilbert space and let 4(.) be the Gaussian be a closed subspace of 2,P the orprocess with covariance (.;) Let thogonal projection onto A, and C the 0-algebra generated by {4(u)Iu E .A!}. Then for any w E 2 :
,
E(ei4(w) 1 1 ) , = c,14(Pw)e
- li2
I1 w l l Z e l / 2 ll Pw 112
(3.9)
Remark (3.9) is connected to Mehler's formula; see [258].
Proof Since the right-hand side (r.h.s.) of (3.9) is clearly Z,,-measurable, it is enough to check that
s
Fei4(w) dp =
s
F(r.h.s. of (3.9))dp
(3.10)
for all C,-measurable F . By a limiting argument, we only need to check (3.10) for F = G(+(u,), . . . , ~ ( I I , , ) )with G E Y ( R " ) and, therefore, using the Fourier transform, for F = ?i zfi4csi)= ei4(")with u = t l u , E ,A'.But by a direct calculation J e14(0)ei4(w) d p = exp( -$llu ~ 1 1 ' ) . Since (u, w ) = (u, Pw) for u E A', (3.10) holds. 1
+
(3.9) suggests we single out the object .ei4b(l').= ei4(i))e+1/211vlIz
(3.11)
28
I.
Introduction
called the Wick-ordered exponential; more generally we define .eub(Io. - eab‘u’e-
l/z~~lllJllz
for any a E @. Then (3.9) and its analog for general a read E( .ea4(w): IC,,)
=
(3.9‘)
:@d(PW):
See [258] for further discussion of Wick ordering for Gaussian processes. To apply Theorem 3.2 to Gaussian processes, we will need the following theorem. Let
Theorem3.20
4(.) be the Gaussian process with covariance
(. , .) for some real Hilbert space X‘. Let A, A’ be two subspaces of 2 and let P , Q be the corresponding orthogonal projections and C,, C,K the
corresponding u-algebras. Then, for any C,-measurablefand
C,-measurable
g, both in L2, we have that
I E ( f g ) - E ( f ) E ( g )I 5 I/ PQII I1 f /I 2 Ilg II 2 with 11.
Proof
(3.12)
the L2-norm. Let 11
=
g - E(g). Then
I~(fg> - E(f)E(dl = IE(f4l =
l~(.fml~dv))l
5 Ilfllzll~@l~.k)Ilz
Since Ilh 11 < IIy 11 2, it suffices to show that for C ,-measurable h with E(h) = 0, we have that IIE(hIC.v)Il2 5 IIPQIIllhllz
(3.13)
By a limiting argument, we may suppose that -Ais finite dimensional. Let A = PQP and let wl,. . . , w, be an orthonormal basis for A of eigenvectors for A , Aw, = a,w,. Notice that sup, a, = IIAIl = llPQ112.By a limiting argument, we can suppose that h
=
s
H(x):exp[i(x,4(w,)
+ ... + x,~(w,)]:d“x
for H E Y ( R ” ) .Then using (3.9’) and the fact that E(:eab(w): :ebb(”):)= eao(w,u),
s R ( ~ ) H ( y ) e (d”x ~ ~d“y ~) ~ H ( x ) H ( ~ ) ~ ( d”x ’ , ” d”y ~”
3.
29
Probability Theory
Now, expand the exponential in each of the above expressions and note that each term is positive, e.g.,
py’H(x)H(p) d”x d”y
=
1s
xDH(x) d”x
/2
Moreover, in comparing the two series, all terms except for the ( j H ( x ) Px)’ term for the second are smaller than for the first by a factor always bounded by sup, a,. But since E(h) = 0, J H ( x ) d“x = 0, so (3.13) is proven. I Remurks 1. The above shows that to get equality in (3.12) and (3.13), we should take h so that j x’H(.x) is only nonzero for = (1, 0, 0, . . .) (if sup ai = a l ) ;i.e., H ( x ) = - d h ( x ) / J x , , or h = $(wI). With this choice one easily sees that the constants in (3.12) and (3.13) are best possible. 2. This result is more transparent if one first develops the r(.)functor, see [258].
*** Finally, we want to do some explicit calculation with Gaussian processes. We have already seen that S e x P ( ? 4 ( m L o = exp(+.’(.f, f ) )
(3.14)
In general, one can explicitly do one-dimensional Gaussian integrals for functions which are exponentials of quadratic polynomials so one should be able to do the same for Gaussian processes. Let A be a trace class operator on 2.Then A has a canonical expansion [214]: A =
1
un(.fn,
.)gn
n
where.f,, gl, are orthonormal and
cnall < co,a, 2 0. We define (3.15)
1, the sum converges in L 1 .If A 2 0, then gn = f n since J I d(.fJ4(gn)I d p o I and (4, A d ) 2 0.
Let d p o be the Gaussian process over a real Hilbert space, Theorem 3.11 .x‘ with covariance (., .). Let A be a positive trace class operator. Then:
(3.16)
30
1.
is the Gaussian process with covariance (., (1
Introduction
+ A)-’.)
(c) =
Remark
(exp[$cr2(f’, (1
+ A ) - ‘f)])[det
(1
+ A)]-”’
(3.17)
For A trace class
+ A ) = n (1 + &,(A)) m
det(1
n= 1
where A, are the eigenvalues of A . (See [259] for further discussion.) Proof’ (a) Choose coordinates x, corresponding to 4(.f,) so that d p , = @ (27c-”’ exp( -$xi) dx,. Let F , = exp( a,xf) so that F , 1exp[ -%$, A$)]. Then
-*cr=
N
JF. d p o
=
+ c(,)x~]dx, = n (1 + N
IJJ(2n)-l/2
expC-31
afl)-1/2
n= I
n= 1
Taking N + CCI and using the monotone convergence theorem, (3.16) results. (b) By the proof of (a), dv = lim[F, dp,/J’ F , dpo] is obviously Gaussian with xkxIdv = (1 mi)- Idkl. (c) Follows from (a), (b), and (3.14). I
+
Theorem 3.1 1 can be extended in two ways. First, A 2 0 is not necessary. All that is needed is 1 + A > 0. Moreover, with an elementary subtraction, one can extend to A Hilbert-Schmidt. Namely, define
Since [4(f’,)’ - 1][4(fm)’ - 13 dpo = 26,,, the sum converges in L2(dp,) so long as a: < co. Moreover, a simple extension of (a) shows that
1
n [(I + m
[exp[-+:(S,
A$):] d p ,
=
an)e-anl”2
n= 1
= [det2(l + A ) ] (See [259] for discussion of det, .)
3.
Probability Theory
If one defines
31
The Basic Processes
4. The Wiener Process, the Oscillator Process, and the Brownian Bridge
In this section, we define the three basic Gaussian processes whose perturbations will concern us in the remainder of this book. Intuitively (and even rigorously in a strong sense, see Section 17), the Wiener process, b(t),is the limit of elementary random walks. An elementary random walk is defined as follows: Let v be the measure { - 1, 1) with v( - 1) = v(1) = and let dv, on X ,”= { - 1, l } . Let y , be the nth coordinate function and dp = X, = y,. The family of random variables {X,}:= is called a random walk. Since E(y,y,) = 6,, we have E ( X ; ) = n so that the variables 8, = n-”’X, at least have a chance of having a limit. In fact the well-known De Moivre-Laplace limit theorem (special case of the central limit theorem) says that 8, approaches a Gaussian:
4,
pm=
Theorem 4.1 (central limit theorem) Let {y,},“, be a family of independent, identically distributed random variables with E ( y , ) = 0, E(y;) = 1. Let
- = n-11’ x, C Ym m= fl
1
Then 8, approaches a Gaussian random variable X , with variance one in the sense, that for any continuous, bounded function on R, E(f(8,))
32
-+
(2n)-’/’
4.
33
Basic Definitions
'as an interProof By a simple limiting argument (going throughfe 9 mediate step), it suffices to prove that E(eiu'*#)+ e - u z i 2for all real CI. But E(e'"2n) = [F(./&)]"
where F(B) = E(eiaY). Since E(y,)
=
(F(/j)-
as
0, E ( y ; )
1)[(-2
+
=
1, we have that
-12
0 (by using the dominated convergence theorem) so that
by the compound interest formula.
I
n-"2X[,,1 (where [ a ] = integral part of a). Intuitively, b(t) is Thus, each b(t) should be Gaussian of variance t. To find the covariance, we note that X , - X , and X , are independent if m < n. Thus, we expect that b(t) - b(s) should be independent of b(s) for s < t ; i.e.,
E(b(s)(b(r)- b(s)N = 0 so E(b(t)b(s))= s for s < t. Therefore, we define the following. Definition The Wiener process (or Brownian motion) is the family {b(t)},s t of Gaussian random variables with covariance E(b(t)b(s))= min(s, t).
Of course, to be able to make this definition, we need the following.
Lemma 4.2
Let 0 I s1 I s2 I .. . I s, and let zl, . . . , z , E C. Then
CJ,'= sizj min(si, s j ) 2 0. Proof
Note that (with so
is obviously positive.
=
0)
I
Remark This lemma can also be proven by noting that b(t) really is the limit of n - l i 2 X L nas t 1 far as joint probability distributions of finitely many b's are concerned. Notice that E((b(t)- b ( s ) ) 2 )= t
-
s
(t >
S)
(4.1)
34
11.
The Basic Processes
Moreover, by writing b(t) = b ( s ) + (b(t) - b(s))and using the independence of b(t) - b(s) from { b ( ~ ) } ~ , , , , ,we , see that E(b(t)I (b(U):o
suss)
=
b(s)
(t
’s)
(4.2)
so that b(t) is a martingale. An important property of Brownian motion is that it continually starts afresh; i.e., for each fixed s, the process b(t) = b(t s) - b(s) for t > 0 has the same joint distributions as b and is independent of b(s);i.e., at any given time a particle following the paths b(t) stops and except for the addition of b(s) follows the exact same paths as a particle beginning at s = 0. Later (Theorem 7.9) we will prove an even stronger version of this property of starting afresh. There are a large number of results known characterizing Brownian motion in terms offairly weak conditions. Typical is the following result, which we will not use and which we state without proof. (It is Theorem 5.1 of [SS].)
+
Theorem 4.3 If x(t) is a process for 0 I t with a continuous version so that for 0 I sI t
E M t ) - x(~~l~x(u)lo,,,,~
=
0
E((x(t) - x(s)>’l {x(u)lo,u,s)
=
t -s
Then x ( t ) is Brownian motion. Remarks 1. The proof begins by noticing the above conditions, say, that x(t) and x’(t) - t are martingales. We will see later (Section 7) that more generally certain special polynomials (:x(t)” :) are martingales. 2. An interesting realization of Brownian motion comes from the fact that if xs is the characteristic function of [0, s], then (xs, ~ ~ =) min(s, ~ 2 t). Thus if 0 is the Gaussian process associated to L’(0, a),then b(s) = @(x,) is Brownian motion. 0 is often called white noise and the formal relation db/ds = O(s) is often expressed by saying that “the derivative of Brownian motion is white noise.” A second main Gaussian process we will consider is the following.
Definition The oscillator process is the family {q(t)}- < I of Gaussian random variables with covariance E(q(t)q(s)) = iexp( - If - sl).
To be sure this is a legitimate definition, we note the following.
4.
35
Basic Definitions
Lemma 4.4
Let sl, . . . , s, be real and z l , . . . , z, E @. Then
c Zizj(hxp(n
ISi - S j l ) )
20
i.i=l
Proof By direct calculation of the Fourier transform of e-Ifl and the Fourier inversion formula,
so that n
1 Zizj(+exp(- Isi - s j l ) ) = (27r-l
j.i=l
which is clearly positive.
I
From a probabilistic point of view, the oscillator process is “natural” as the only “invariant Gaussian Markov process” (up to changes of scale); see Corollary 4.11. Our interest comes from the fact that it is a “path integral for the harmonic oscillator” as we shall see. Often, the process we have called the oscillator process is called the Ornstein-Uhlenbeck velocity process” since Uhlenbeck and Ornstein [281] introduced a process x ( t ) with differentiable paths so that dxldt = q. We defer the definition of the third major Gaussian process, the Brownian bridge. Our interest in the Wiener process comes from the fact (responsible for its invention by Wiener [286]) that it is intimately connected with the semigroup e-IHo where H , = -+d2/dxz. Given the connections of b(t) with random walks, of random walks with diffusion, of diffusion with H o via the diffusion equation, this should not be surprising. e P t H ohas the integral kernel : “
Since P,(x, .) is in L z and varies continuously in L z as x is varied, P,(x, y ) f ( y ) dy is a continuous function for any f~ L2. Using this continuity, we define ( e - ‘ H . f ) ( x ) for every x (a priori it is only defined almost everywhere).
36
s1
II. The Basic Processes
Theorem 4.5 Let f,,. . . , < ... < s,. Then
fn-
E L"(R),
fn E L2 n L"(R),
and let 0 <
E(f,(b(s,)) * . . J,(b(s,))) = (e-f1Hofie-'2H"j2 . . . e-'nHOf)(0)
(4.4)
where t l = s,, t 2 = s2 - s,, . . . , t , = s, - s,-,, where f, is viewed as an element of the Hilbert space L 2 and wheref,, . . . ,fn- are viewed as bounded multiplication operators on L2.
,
Proof It clearly suffices to show that the joint probability distribution of (b(s,), . . . ,b(s,)) is Prl(O, x1)Pi2(xI,~
. . Pt,(xn-
2 ) .
1,
-4d"x
(4.5)
But since b(s,), b(s2)- b(s,), . . . , b(s,) - b(s,- ,) are mutually independent Gaussian random variables of variance t , , t 2 , . . . , t,, their joint distribution is Pt,(O, .YI)P,~(O,~
. . Pr,(O, y,) d " ~
2 ) .
so that (4.5) holds since the Jacobian of the change of variables y, = x,, = x2 - x l , y 3 = x j - x 2 , . . . is 1. I
y2
It is easy to generalize Theorem 4.5 to several dimensions:
Definition vdimensional Brownian motion, b(t) ( t 2 0), is the family of R"-valued "random variables" whose v-components are v independent Brownian motions; i.e., (bj(t)),<'; j i v are Gaussian random variables with covariance
E(bj(t)bk(s))= aj, min(t, s) An identical calculation to that in Theorem 4.5 proves the following theorem. Theorem4.6
Let H o
=
-fA on L2(Rv). Let f,, .. ., fn-, EL"(R"),
f n L2~ n L m and let 0 < s1 < . . , < s,. Then E(fl(b(s1)). . . f,(b(s,)>)
where t l
=
sl, t 2
=
=
(e-'lHofi . . . e-'nH"f.)(0)
(4.4)
s2 - s,, . . . , t, = s, - s,-,.
TO find the analog of (4.4) for the oscillator process, we let
L0
- - L 22 d /dx2 -
+f
~ '- f
and Ro(x) = n- 114e-(112)x2 so that LORo= 0 and IR, l2 dx = 1. Notice that xR, is also an eigenvalue of Lo:L,(xR,) = (xR,) and that x2 I Ro l2 dx = f.
4.
37
Basic Definitions
Let fo, . . . , f f lE L"(R) and let - co < so < . . . s, < co.
Theorem 4.7 Then
E(fo(q(so)).. . L(q(Sfl))>= ( 0 0 ,f o e - ' l L O f i . . . e-'nLYnQo)
(4.6)
where ti = si -
Proof Fix t l , . . . , t,. We first claim that there is a Gaussian pr6bability measure, G, on R"+ so that r.h.s. of (4.6) =
s
f o ( x o ) .. . f,(x,,) dG(x)
(4.7)
For, using the Trotter product formula, )m li.m f ll. . . f , Q , ) r.h.s. of (4.6) = lim (Qo, f o ( e - ' l H o i m l e - ' i W m+ m
with W ( x ) = +(x' - 1). For each fixed m = (ml, . . . ,m,,), this is of the form JfO(x0).
. L ( X n ) d ~ m ( xY, >
where G, is a Gaussian measure in x and auxiliary variables y obtained by putting together the explicit Gaussian kernel of e-'Ho,the Gaussian SZ,, and the Gaussian in e-'W. The partial integral J,, dG,(x, y ) of Gaussians is again a Gaussian and a limit of Gaussians is a Gaussian, so (4.7) holds. To find the covariance of the dG in (4.7), we note that for i < j ,
I
x i x j d G = (xQo, e - c s ~ - s ~x?,) ) L o = *e-(sJ-sC)
since Lo(xSZo) = ( x n , ) and (xR,,xR,) = i.This shows that dG is just the joint probability distribution of q(so), . . . , q(s,). I
where dG is the Gaussian measure on R2 with covariance matrix
II. The Basic Processes
38 Thus, inverting this matrix and using (2.6) dG(x,y) = z - l ( l
-
e-2')-''2exp[-(1
-
+
e-2r)-1(x2 y2 - 2e-'xy)]dxdy
or Q,(x, y)
=
~-'/~( 1e - 2 r ) - 1 / 2 x exp{ -[1
-
e~~']-l[(+ x ~y2)+(1 + e-2')
-
2e-'xy])
This is known as Mehler's formula and this proof is essentially due to Doob [61].
*** The formulas (4.4) and (4.4) clearly distinguish the point zero; to restore the translation invariance of H o , we introduce some measures which are not normalized (although they are still positive). DeJinition Let dx be Lebesgue measure on R" and let ( B , 9#,Db) be the measure space for v-dimensional Brownian motion. By Wiener measure, we mean the measure d p o on R" x B given by dx @ Db. We let w ( t ) --= x + b(t). Remark We use Db, Dq for the measures associated with b and q to avoid confusion with the stochastic differentials db of Chapter V.
Let dpo be Wiener measure. Let fo, . . . , ,f.E Lm(Rv)with Theorem 4.8 .f;,fn E L2. Let 0 I so < s1 . . ' s , - < s,. Then Jfo(m(so)) . . . ffl(o(sn>> dpo(o) =
(f0,e-tlHofi . . . e-'nHy,)
(4.8)
where ti = si - siProof Suppose first that so = 0. Then writing o = x (4.4) to do the b integration, we see that
1.h.s. of (4.8) =
+b
and using
s
fo(x)(e-'lHyl . . * e-"'""Jfl)(x)dx
which is the right-hand side of (4.8). If so > 0, we let g R be the characteristic function of the ball of radius R in R". Then, by the dominated convergence theorem 1.h.s. Of (4.8) = lim R+U2
s
gR(w(o))fo(~(So))
' *
. f n ( O ( S f l ) ) dpo(C0)
4.
39
Basic Definitions
while the right-hand side is
sincefOe-SoHogR converges tof, in L z by a simple application of the monotone convergence theorem. Thus, the general case of (4.8) follows from the special case with so = 0. 1 Remarks 1. For reasons of symmetry and also for emphasizing the analogy with the oscillator process, one can consider R" x B x B with measure d x @ Db 0 Db, with b , (respectively, b,) the first (respectively, second) Brownian coordinate function. One then defines, o(t)= x b,(t) for t 2 0 and x b,( - t) for t I 0. (4.8) then holds for all t's and the joint (nonprobability) distribution of (~(s,), . . . , o(s,)) is time translation invariant. 2. If we prove some "translation invariant" statement like continuity for the Brownian paths with probability one, then, by Fubini's theorem, it automatically holds for o on a set whose complement has p,-measure zero.
+
+
The next special measure which concerns us is formally just h(o(0) - a)S(o(r) - b)dp, which is not a probability measure, but rather one of mass Pr(a,b)
fly=
1
=
Pr(a;, bi).
Definition
Fix a, c E R"and t > 0. For 0 5 s I t define random variables for (o(sl), . . . ,o(s,)) (0 < s1 < s2 I . . . I s, I t )
o(t)with the joint distribution
where t , = s,, t , = s2 - sl,. . ., r , , + , = t - s, and P,(x, y) The consistency and normalization conditions follow from J P t k Y>Ps(Y,2 ) d'Y
=
=
6(x
-
y).
P t + s ( X , 2)
be the corresponding probability measure and let us define Let dv,, conditional Wiener measme by d ~ oa.,c: r(W) = pr(a3 c) dva. c ; t
40
II. The Basic Processes
Using the explicit form of the distribution of finitely many o(s)'s one sees that (4.9)
so long as f is measurable with respect to the o-algebra generated by {o(s)IO I s I t } . Similarly, (4.10)
for the Wiener process. (4.9) and (4.10) have an interpretation in terms of conditional expectations. For example, (4.10) says that
~ ( f ( b ~b(t))(a> ) =
Jf(o)dP0,o. a ; t
where E ( . I b(t))(a) means the value of E(. . .) at those points with b(t) = a. [More precisely, it is defined for almost every a with respect to dv(a), the joint distribution of b(t), by
E(*g(b(t)))= Jg(a)E(. I b(t))(a) M a ) for all measurable g.] We will occasionally use the symbol E ( . I o(0) = a, o(t)= c)
for expectations with respect to dva,c:f.There is a useful way of representing all the dv,, c : processes at once. Definition The Brownian bridge, { ~ ( ( s ) } ~is ~the ~ ~Gaussian process with covariance E(a(s)a(t)) = s(l - r)
(0 I s I t I 1)
Rather than directly showing that the required covariance is positive definite, we note that E(s) = b(s) - sb(1)
is a family of Gaussian random variables with (s I t ) E(E(s)E(t)) = s
+ sr
-
2st = s(l - t )
4.
41
Basic Definitions
Moreover, E(E(s)b(l)) ( O I S I 1)
=
0 so that, by Proposition 3.8, one can write
b(s) = ~ ( s )+ sb(1)
(independent sum)
(4.11)
where means that the distributions on both sides are the same. Now let a(s) be v-independent copies of ~ ( s ) .Using (4.1 1)and the definition of o,one immediately sees that (4.12) for the dv,, ;, ,process. Thus, (4.9) becomes P,(a, c) #a d"c Dcr (4.13)
*** We want to close this section with a brief discussion of the Markov property as it applies to the processes and results just discussed. We emphasize that these notions will not be used in the remainder of this book. See Dynkin [67] for further discussions of Markov processes. Definition A stochastic process { ~ ( t ) } , , , , , is called Markovian if and only if for a I u I t , the conditional expectation E ( f ( x ( t ) ) l{~(s)},,,,,) is measurable with respect to x(u). Colloquially, this says that the future depends on the past only through the present. Theorem 4.9 Let { ~ ( t ) } , , , , , be a Gaussian stochastic process with (pointwise) strictly positive covariance C(t, s). Then {x(t)} is Markovian if and only if C(t, s) = f(t)g(s) ( t > s) for suitable functionsfand g.
Proof Let t 2 u 2 s. We first show that the process is Markovian if and only if for such triples: C(t, s)C(u, u ) = C(t, u)C(u, s)
(4.14)
For the Markov property is equivalent to the condition that E ( : P ' r )I:Ix(s)),,,,,>
is x(u)-measurable. By (3.9), this is equivalent to the fact that, in the notation of Corollary 2.4, the projection of 6, onto the span of {Ss}o,scu in the C( .. .)inner product is just a multiple of S,, and this is equivalent to the fact that
42
II. T h e Basic Processes
6, - E,(6) is orthogonal to 6, in the C-inner product, where E,(6,) = C(u, t)C(u, u)-'6,,. Thus, the Markov property is equivalent to (4.14). Now, clearly if C(t, s) = f(t)g(s) (t 2 s), then (4.14) holds. Conversely, let (4.14) hold and pick some u and let (t 2 u)
C(t, t>C(u,u)lC(u,
(t I u) (s 2 u )
t)
(s
I u)
Using (4.14) and considering the three cases t 2 s 2 u, t 2 u 2 s, and u 2 t 2 s, one easily sees that C(t, s) = f ( t ) g ( s ) for t 2 s. I Remarks
1. The formulas forf(t) and g(s) can be guessed by supposing
g(u) = 1 and that C(t, s) = f(t)g(s).
2. Some kind of restriction on C is needed to deduce C(t, s) = f(t)g(s) from (4.14). For example, take C(t, u ) = d,,,. 3. A sufficient condition for C to be strictly positive, is that it be jointly continuous and C(t, t ) > 0 for all t. For C is then uniformly continuous so C(t, s) > i f o r It - sI < 6. Given any t < s find t < s1 < . . . < sk < s with Is1 - t I < 6, IS^ - S I I < 6 , . .., I S - ski < 6. By (4.14)
is strictly positive. Corollury 4.20
{b(t)},?,
and {q(t)}- m c f < are Markov processes.
Corollury 4.22 Let {x(t)} be a Gaussian stochastic process which is time invariant [i.e., the joint distribution of x(tl), . . . , x(t,) agrees with that of x(t tJ, . . . ,x(t tn)l7continuous in quadratic mean [E(x(t)x(s)) continuous], and Markovian. Then for suitable a, /3 > 0, x(t) = a@t) where q is the oscillator process.
+
+
Proof By the time invariance and symmetry of C(t, u), we have that C(t, u ) = h(lt - u l ) for a suitable continuous function h. The Markov property implies (4.14) and thus h(t)h(O) = h(t - u)h(u) for t > u > 0. This and continuity imply that h(t) = h(O)e-YI"
The Schwarz inequality implies y 2 0. Take p
=
y and
c1
=
(2h(0))1'2.I
5.
43
Regularity Properties-I
There is a close connection between formulas (4.4) and (4.6) and the Markov property. Given any stochastic process, { x ( t ) } we can define the spaces L: = L2(R, dv,) where v, is the probability distribution of x ( t ) and for t 5 s define U ( t , s):Lf
-+ L:
by the formula
U ( t , s)[f(x(s))]
=
E(f(x(s))lx(t)).
t I s, The Markov property says that for u I E(f(x(s))
I X(t))
so, since E(E(.I x(t), x(u)) I x(u)) Markov processes U(u, t ) U ( t , s)
=
=
I
= E(.f(x(s)) x(t), x(u))
E ( .Ix(P)
for any process, we see that for
U ( u , s)
(u I t 5 s)
In the case of the oscillator process where the L: are isomorphic and U ( t ,s) = U ( s - t ) , we see that U ( t ,s) = ep(s-r)Bfor some generator B of a contraction semigroup; this is clearly why (4.6) holds. (4.4) is more complicated but similar. Moreover, one can go quite easily from (4.4), (4.6) to the Markov property. For example, (4.6) implies that for t 2 u ; JJ3f(q(t))I
{ds)J,,d
=
Q,(du))- l(e-(f-“”Y-Q 0 du))
which is clearly q(u)-measurable. In this way, one easily sees that the P(+),processes to be constructed in Section 6 are Markovian. Notice that the above expectation has the form (eC‘Ef) on L2(R, dv) where dv = Qidx and B =
sr, 1L, a,.
5. Regu Ia r it y P rope rt ies- 1
We will deduce the existence of continuous versions for the Wiener and oscillator processes from the following beautiful result (“Kolmogorov’s lemma ”) : Theorem 5.1
Let { x ( t ) J a S f lbe h a stochastic process obeying E(lx(t
+ h)
-
x(t)lP) I
Klhl’+‘
for some K , some r < p , and all t , h with a I t
-
x(s)l I C(x)lt -
+ h Ib. Fix 0 < a < v / p . Sl3
(5.1)
44
II.
The Basic Processes
for all dyadic rational t and s where C ( x ) is finite almost everywhere. In particular, x has a Holder continuous version [i.e., one can find x(t) defined for all points with the correct joint distributions and for which (5.1) holds for all t and s]. Proof We prove (5.1); the version question is then solved via method (a) of Section 2 (p. 14). Without loss, suppose that a = 0, b = 1. Let e = r - up. Then E(lx(t
+ h)
-
x(t)l 2 Ihl") IIhl-""E(lx(t <
+ h)
-
x(t)l">]
Klhll+&
so that
Thus
f
n = l 'ki= 'O E
(lx(T)
I);(.
-
2 2-")
< co
so, by the first Borel-Cantelli lemma, there exists a random integer v(x), almost everywhere finite so that
Now let n 2 v(x) and let t be a dyadic rational in [k/2", ( k t = k/2" + yi/2"+'; each yi = 0 or 1. Then
c;=l
+ 1)/2"]. Write (5.2a)
where d
=
(1 - 2-')-l. Similarly (5.2b)
Now let t and h be dyadic rationals with h I 2-"(x).Take n with 2-O-l I h < 2-"andksothatk/2"+' st < k 1/2"+'.Thenk 1/2"+' I t +h < k 3/2n+1.It follows that
+
+
Ix(t
+
+ h) - x(t)l 5 (2d + 1)2-(n+1)aI (2d + 1)h"
Since, for fixed x this holds for all sufficiently small h, we have that (5.1) holds. I
5.
45
Regularity Properties-I
Theorem 5.2 The Wiener process, v-dimensional Brownian motion, and the oscillator process all have Holder continuous paths of any order CI < +. Proof We consider the Wiener case. The v-dimensional result is then immediate. The oscillator case is similar. For any Gaussian variable X :
E(lX1”) = C,E(IX12)”’2
(5.3)
(5.3) follows by scaling once we know that the expectation is finite which is trivial. Froin (5.3) and the fact that h(r + h ) - h(t)has variance h, we see that E(lb(t
+ h) - b(t)lP) = C,hpi2
We can, for fixed p > 2, choose ct arbitrarily close to 5 - p - so by taking p large, we can obtain CI’S arbitrarily close to 5. I By pushing the ideas of the proof of Theorem 5.1 to their extreme (see [183, Section 1.63) one can obtain Levy’s precise law for the local smoothness [167] :
Theorem 5.3
With probability one
for the Wiener process. In particular, the Wiener process is (everywhere) Holder continuous of order on a set of measure zero. Remark
We will see below that b(t) and q ( t ) have the same local behavior.
Levy’s law only assures us that b(t) is not Holder continuous of order
f at some point and not that it fails to be Holder continuous of order f at all points [indeed, we shall see in Section 7 that the behavior at a fixed point s differs from the behavior over all s in (0, 1); explicitly
with probability one]. The following is a slight generalization of the proof that Dvoretsky et al. [63] gave of the celebrated result of Payley, Wiener and Zygmund [199] that b(t) is everywhere nondifferentiable; the result below is also in [199].
46
II. The Basic Processes
Theorem 5.4 Fix CI > $. Then with probability one, b(t) is nowhere Holder continuous of order cc; i.e., for almost every b, inf OblSl
[-limIb(t + h )
- b(h)l lhl-"
h+O
1
=
(5.4)
co
In particular, b(t) is nowhere differentiable.
Proof Fix an integer k with k(a - 3) > 1. Suppose that b(t) is a path with the left-hand side of (5.4) finite. Then, there is a t , a C , and an ho with Ib(t
+ h)
-
b(t)l 5 Clhl"
for all h with I h 1 I ho. Pick m so that (k + l)/m < h,. Given n let i so that i/n, (i l)/n, . . . , (i k)/n E [t, t h,] if n > m. Then
+
+
+
for j = 1, . . . ,k (since I b(s) - b(u)I I C [ I s - t 1" inequality). Thus
+ I u - t I"]
=
[tn]
+1
by the triangle
{ bI b somewhere Holder continuous of order a}
< Dn-"forj
=
i
+ 1, . . . , i + k
I
It thus suffices to show that
1, . . . , i + k
1)
=O
This follows from
since the b(.in) - b((j - l)/n) are independent and distributed identically to b(l/n). Now by scaling E(bJlb(l/n)lI Dn-")
= E(bllb(l)l I = O(n-("-"2')
Dn-n+1/21
since the distribution of b(1) is continuous. Since k(a - $) > 1 , (5.5) holds.
I
5.
47
Regularity Properties-I
Finally, we turn to showing that q ( t ) and b(t) have the same local behavior. We give two distinct proofs of this fact. The first relies on an elementary observation that q(t) A e-'b(e2')/$
(5.6)
where means that both sides have the same joint distributions. To prove (5.6), we need only check covariances since both sides are Gaussian. For t > s, we have that 1 --f Ze e - s min(e2', e2') = $e"-'
From (5.6) we immediately conclude that Theorems 5.3 and 5.4 extend to q(t). Expressions like (5.6) occur with functions of b on both sides. Since q( - t ) I q(t) trivially, (5.6) suggests that b(t) and b(l/t) are related. In fact b(l/t)
since min(l/t, l/s)
= (ts)-'
(5.7)
t-'b(t)
min(t, s). More trivially b(at) A a'I2b(t)
(5.8)
Notice that (5.7) and (5.8) extend to v-dimensional Brownian motion since they hold for each component. The second way of seeing that b(t)and q(t) have the same local behavior is to apply Theorem 2.5: Let Db (respectively, D q ) denote the measure on paths to Brownian motion (respectively, the oscillator process). Let C @ b ) denote the a-algebra generated by { o ( t ) l a < t < b}. Then for 0 < a < b < co, Db 1 C(o,b)and Dq 1 C @ b )are mutually absolutely continuous. Theovem5.5
co(t) associated
Proof Pick c > b and on [a, b] consider the four processes with covariances : Km(r,s) K,(t, s)
=
=
+e-''-"l fe- I-f--sI - ge2'
- l)-l{es+-f
- e"--f-
&S
+
e2ce-r-s}
A,(t, s) = min(t, s) ~ , ( t s) , = min(t, s) - c - ' t s
The positive definiteness of K , and A , follows if one notices K , (respectively, A,) is the kernel of ( B + 1)- ' (respectively, B - ') where B is the operator - d 2 / d x 2 on L2(0,c ) with u(0) = u(c) = 0 boundary conditions. We will prove the mutually absolute continuity of the pairs ( K , , A,), ( A , , Am), ( K c ,K,) by application of Theorem 2.5 from which the result then follows.
48
II.
The Basic Processes
Let (f,g)A,, = (f, B-'g)LZ, (f, 9)K.c = (f, ( B + 1)-'g)L2, and let denote the corresponding Hilbert space, etc. Then
(1,g)A,c -
(f?
g)K,c =
( f ?
(
0
,
.)K,C
also
cg)A,c
+
where C = B1l2(B-' - ( B l)-')B'12 = ( B + 1)-'. For C to be HilbertSchmidt on (., . ) A , c , it is necessary and sufficient that C be Hilbert-Schmidt on L2 and this is obvious since its eigenvalues go as n - 2 . It is clear that 1 - C = B(B + 1)-' is invertible as a map on (., .)K,c since B is invertible. This shows that K , and A , are equivalent. Next we note that for q E (., . ) A , , (q, q ) A , c = (11, 111.4, m - c where 6, has norm ( 6 b , 6*)'/' = b'12. Thus
'(113
(119
q)A,c =
(119
(1 -
6 b ) A , m(6br
V I A , a,
Q)v)A, m
where Q is rank 1 (and so Hilbert-Schmidt) and IlQll = bc-' < 1. It follows that A , and A , are equivalent. The proof of the equivalence of K , and K , is similar. The difference is rank 2 with 6, and 6, both involved. Remarks 1. The above proof was suggested by work of Guerra et al. [120, 1211 on the P(+)2 field theory. In the language of those papers, K , is the free field of mass rn = 1, K , has added Dirichlet boundary conditions at 0 and c, A , is the free field mass 0 with a Dirichlet condition at 0, and A, has an extra Dirichlet condition at c. The three equivalences above have analogs in the two-dimensional case but, e.g., A , and K , are not equivalent in four or more dimensions since ( - A D + 1)-' is not Hilbert-Schmidt if -AD is the Laplacian with Dirichlet conditions on the boundary of some bounded set. We emphasize that in this paragraph "dimension" refers to the dimension of the parameter t (see Section 24 for discussion of multidimensional t's) and not to the number of components of b. 2. For c = 1, A , is the covariance of the Brownian bridge. Thus we have also shown that Db and Da restricted to times in [0, d ] with d < 1 are mutually absolutely continuous.
6. The Feynrnan-Kac Formula
Our main goal in this section is to give a number of variants of a basic V ( x ) or formula relating e - r Hto path integrals when H is of the form H o Lo V ( x ) . The basic formula is illustrated by the following result for which
+
+
6.
49
The Feynman-Kac Formula
we now give two independent proofs. Third and fourth proofs appear in Sections 14 and 16. Theorem 6.1 Let V E Cm(R"), the continuous functions vanishing at a,and let dpo be the v-dimensional Wiener measure. Let H = H o + V as a self-adjoint operator on D(Ho). Then
Remark
Since V(w(s)) is almost everywhere (in w ) continuous,
yo V ( o ( s ) ) d scan be taken as a Riemann integral and so is o-measurable as a limit of Riemann sums. First proof
By the Trotter product formula:
(1,e-IHg) = Iim (,f, (e-tHo'ne-t"'n)ng) n-.m
so by Theorem 4.8: ( f , e-IHg) = lim n+ m
i-
f ( o ( O ) ) g ( w ( t ) exp )
Since w is almost everywhere continuous,
as n -+ 00 for almost every w. Moreover, the integrand in (6.2) is dominated by If(o(O))IIg(w(t))Iexp(tll Vll,) which is L' since
Thus, by the dominated convergence theorem, (6.1) holds.
Second proof For fixed V and t , the right-hand side of (6.1) clearly obeys (by Holder's inequality)
I (r.h.s. of (6.1)) I 5 e''i""m II f )I 119 II since
50
II.
The Basic Processes
- [)(w(s)j
ds] du (6.3)
Thus we can find an operator A , so that r.h.s. of (6.1) = ( f , A , g ) We want to show that A Next notice that
=
exp[ - fV(oj(n)) 0 dn]
e-'".
=
1 - J:V(w(u)) exp[
(6.3) can be proven by integrating the perfect differential in the integral on the right-hand side. Multiply both sides of (6.3) by f(w(O))g(o(t))and integrate with respect to d p o to find
( f , A,g) = ( f , e - ' H o g )
- fa,, du 0
with
=
JE(f(W9) I &), u I s > V 4 u ) )exp
=
(f, e-UHol/At-,,g)
where E(.lw(s),u I s) stands for the L2-projection onto those functions measurable with respect to {o(s)lu I s} (this remark is needed since dy, is not a probability measure). In the above, we use ~(f(s(O))Iw(s), u
I s)
=
(e-""of)(w(u))
(which follows from Theorem 4.8), and we use the translation covariance of d p o . We have thus proven that A,
=
e - f H o- J ~ e - " H o V A f - , , d u
and from this one can conclude that A, = e - t H ;e.g., one can iterate the above equation and obtain an expansion for A , which is identical to the DysonPhillips expansion for e - f H .I Remark Formula (6.1) is a variant of a result of Kac [138] (see also [139]) who was trying to understand Feynman [83]. Both proofs appear to be well known to probabilists. In the mathematical physics literature, the first proof appeared in [189] and the second in [120].
6.
51
The Feynman-Kac Formula
Once we have (6.1) for smooth V's it is easy to extend it to a wide class of V's (note that f E L:,,([W" \\G) means JA I .j' I d"x < co,for any compact subset of R"\G): Theorem 6.2 Let V be a potential so that V+ E L:,,(R"\G) where G is a closed set of measure zero and V- obeys Q(H,) =I Q(V-) and
I(+?
V- $11 5
449 HI341 + m4>4)
for some CI < 1 and all 4 E Q(H,). Let H Q(H,) n Q( V ) . Then (6.1) holds.
=
Ho
+ V as a sum of forms on
Remark Since V and Q are measurable, V ( o ( s ) )is a measurable function of s, so modulo the question of convergence of the integral fo V(w(s))ds = g(co) can be defined for each w as a Lebesgue integral. That the integral converges or diverges to m almost everywhere and that g is measurable as a function of (11 follows from the proof below which establishes that it is a pointwise limit of a sequence of measurable functions.
+
Proof Suppose first that V E L a and let V , = h,(V * j n ) where J,(x) = n"4(xn) and h, = 4 ( x / n ) with 4 E C z , 0 I 4 I 1, 4 ( x ) dx = 1, and 4(O) = 1. As n -+ 00, V,(x) + V ( x ) for x # K , some set of measure zero. For each fixed t , { w l w ( t ) E K }has measure zero since the distribution of o ( t ) is just Lebesgue measure. Thus by Fubini's theorem with respect to dp, @ dt, (((0, t ) I w ( t ) E K } has measure zero. Tt follows, again by Fubini's theorem, that for almost every w, { t I co(t) E K } has Lebesgue measure zero. Thus, by the dominated convergence theorem, V,(w(s))ds + V ( o ( s ) )ds almost everywhere in u.Thus as n + x, the right-hand side of(6.l)converges. Since H , + V , H , + V in strong resolvent sense (since they converge on a common core; see [152,2143 for discussion of strong resolvent convergence), the left-hand side converges. This establishes (6.1) for V E L". Now, let V be an arbitrary function of the type allowed in the theorem and let
yo
yo
--f
Then, first taking n + co and then m -+ x;,both sides of (6.1) converge, the left-hand side converges by application of monotone convergence theorems for forms [152,221,253,254] and the right-hand side by monotone convergence theorems for integrals. I
Remark If one interprets e-IH in the proper way for nondensely defined closed forms [154,245] then the condition V+ E L:,,,(R\G) can be droppedit was only needed to assure that H was densely defined.
52
II. The Basic Processes
By mimicking the above proofs, one easily shows that the following is true.
Theorem 6.3 Let V be a potential so that V+ E L:,,,(R) and so that Vobeys Q(Lo)3 Q(V-) V-
I(49
4 )I
4 4 , Lo 4) + P(4?4)
for some CI < 1 and all ~ E Q ( L ,Let ). L Q(Lo) Q(V). Then
for allf, g E L’(R, Corollary
=
Lo
+ V as a sum of forms on
Qidx).
Under the above hypotheses: = - lim
inf spec(L)
t - In
t-+m
Proof In case L has an eigenvalue E (with LR = E n ) at the bottom of its spectrum, the proof is immediate from (1.9) since (Q, a,) > 0 (for R, is pointwise strictly positive and R is pointwise positive, see [217]). In general, we proceed as follows: Choosingf = g = 1 in (6.4) we see that 7
limt-’ In[.] I-infspec(L)
On the other hand, pickf and g withf’Q,, Schwarz inequality and (6.4)
g2Qo E L’(R, dx). Then, by the
so that lim{t-’ ln(fQ,, e-tLgQ,)} 5 -+inf spec ( L ) + + h t - ’ In[.] Varying over allfand g we see that -+infspec(L) I +limt-lln[.] completing the proof.
I
Remark The above “general case” proof is patterned after a field theoretic argument in Seiler-Simon [235].
53
6. The Feynman-Kac Formula
This completes the general discussion of Feynman-Kac formulas; in the remainder of this section, we will present some complementary material: , ~ ; proof of Mehler's Feynman-Kac formula for Db and d ~ ~ , . ,a ~second formula ;definition of P(4)l-processes, and the use of Feynman-Kac formulas to do calculations with Wiener measure.
*** Formulas (4.9) and (6.1) imply that kernel
e-lH
is an integral operator with
(6.6) can only be interpreted a priori as holding almost everywhere in a, b. The right-hand side is defined for each a and b but the left-hand side is only defined almost everywhere. We want to investigate when e-lH(a, b) and the right-hand side of (6.6) are continuous so one can interpret (6.6) pointwise. We begin by studying when the related formula:
is valid. We first study all operators on C( R"), the bounded continuous functions on R". The operator (e-'Hi"f)(x> = JP1(X, Y ) f ( Y ) dY
is easily seen to define a continuous semigroup on C(R"). Moreover (4.4) holds iffl, . . . , f . lie in C(R") and k, is replaced by for the proof of V Theorem 4.5 goes through without change. If I/ E C(R"), then i?, defined on D(fi,) is the generator of an exponentially bounded semigroup on C(R"), so using the Trotter product formula for semigroups on C(R"), we conclude that (6.7) holds (the convergence is now in 11.11 and thus pointwise). Moreover if J E L2 n C(R"), then the two definitions of e - f H fagree (this follows, e g , from the equality of the Trotter approximations). Thus the following theorem is proven.
no,
+
+
Theorem 6.4 If I/ E C(R"), and fi = A, I/ as an operator sum on C(R"), then (6.7) is valid forf E C(R"). In particular, iff E L2 n C(R"), then
e-lHf(in L2-sense)is continuous and (6.7) holds for the continuous representative.
54
II. The Basic Processes
This result ignores the fact that e-fH has a tendency to be smoothing; indeed Ran(e-IH) c Cm(H)and under fairly general hypotheses ([l50, 2431 and Section 25), C"(H) comists of continuous functions. When v I 3, D(H,) is already contained in C(R") and this makes proofs easier; for this reason we state the following for v I3. Using the machinery of [243], an extension exists for v 2 4. Theorem 6.5 Let v I 3. Let V E L2(R") + Lm(R") and let H = H , + V as an operator sum on D(H,). Then e-"ff'is continuous for each t > 0 and f~ L2, and (6.7) holds for the continuous representative of e-rHf.
Proof Let j, be an approximate identity as in the proof of Theorem 6.2. Letj, * I/ = V,, so that (6.7) holds forf E C(Rv) n L2,and H , = H o + V, by Theorem 6.4. Now {e-fHn}and e-IH are uniformly bounded from L2 to D(H,) and so to D(H,). Since D(H,) is continuously embedded in C(R"), the maps ePfHn and e - f Hare equi-continuous from L2 to C(R"). It follows that (6.7) forf E C(R"), and H , implies the result for allf E L2 and H , since the left-hand side is L2-contiguous inf by the above and the right-hand side is continuous by Holder's inequality. If we take n --* co, the right-hand side of (6.7) converges by Holder's inequality and the dominated convergence theorem. Thus, we need only prove that for fixedf E L2, eCtH:f converges to e - f H f in C(R"). This follows if we show that ( H , l)[e-lH"f] converges to ( H , 1)[eCfHf]. But since I-f, converges to H in a strong resolvent sense, we know that ( H , + I) (e-"',f) .+ ( H + I)e-lHf. By a simple argument V,(e-lH,f) -, V(e-'*f). Now use H , = H , - V,. I
+
+
We remark, that in any event, (6.1) implies that under the hypotheses of Theorem 6.2 (e-lHf)(x)
=
Jexp( - J:V(x
+ b(s))ds
)
.f(x
+ h(t))Db
(6.8)
almost everywhere in x. Theorem 6.6 Let I/ E C(R"). Then e-IH has an integral kernel eCfH(a,b) jointly continuous in a, b and t > 0, and (6.6) holds for all a, b, t > 0.
Proof
This is a simple exercise in the use of (4.12)-(4.13). Namely, let
a be the Brownian bridge and note that since V E C(R"),
6.
55
The Feynman-Kac Formula
is jointly continuous in a, b, t by the almost everywhere continuity of a(s) and the dominated convergence theorem. But by (4.12) r.h.s. of (6.6) = Q(a, b, t)(2nt)-"i2exp( - la - bI2/2t)
(6.9)
This proves the result. [ Remark In Section 25, we shall show that e-rH(a, b)ELm(RzV)for a very general class of potentials I/.
*** The Feynman-Kac formula also provides another proof of Mehler's By the Feynman-Kac formulas (and a formula for the integral kernel of ePfLo. slight extension of Theorem 6.6)
where g(s) = a(1 - s) + bs and c( is the Brownian bridge. Now, as noted in the proof of Theorem 5.5, the covariance, s(1 - u) (s I u), of the Brownian bridge is the integral kernel ofthe inverse of - d 2 / d s 2 with vanishing boundary conditions at zero and one. This operator has eigenvalues (nn)' with eigenfunctions
4,(s) If An = (nn)-
=
4 sin(nns)
(n
=
1,2, . . .>
', we can write a,
a(s)
C Axn4n(s)
n=l
with the xn's Gaussian with covariance
a,,
(for we need only check that E(CCAnxn+n(s>ICC A n z x m + m ( ~ ) I ) = C Af4n(s>+n(u) = ~ ( 1- u ) for s I u)- A direct calculation shows that g(s) = g,4n(s) with
56
II. The Basic Processes
Thus
with D x =
n:=l ( 2 ~ ) - "exp(~ - + x i )
dx,. Using (3.18), we find that
(e-'La)(a, b) = (2nt)- "2e-"2f(t)exp[ - ) A ( t ) ( a 2
+ b2)
-
B(t)(ah)]
where
n
f(t) =
(1
+ A,'")
3'"
+ c 2A,'t(l + PA,')-' m
A(t) = t -
n= 1 m
B(t) =
t-'
+ C(-1)"+'2A,2t(l + t 2 A y n= 1
The identification with Mehler's formula now follows from the formulas :
n(1 + m
sinh x
=
x
A,'x2)
n= 1
m
(sinhx)-' = x - l
+ C(-1)""2A,2x(l + x2A,2)-' n= 1
cosh x sinh x
=
x-
+
m
2A,'x(1
+ x2Ai)-
n= 1
(These formulas are the Weierstress-Hadamard factorization of sinh x and the Mittag-Leffler expansions of (sinh x ) - ' and coth x . ) If one deals directly with Green's functions for - d 2 / d x 2 and does not discretize, one can avoid these arcane formulas at the cost of doing slightly more involved integrals.
*** We will use P as a generic symbol for a potential obeying the hypothesis of Theorem 6.3 with the additional property that E ( P ) = inf spec(&,
+ P)
is a simple eigenvalue with an associated strictly positive eigenvector R,. Most reasonable V's will obey this extra condition (see, e g , [ 2 1 7 ] ) and, in
57
6 . The Feynrnan-Kac Formula
particular, P can be any polynomial which is bounded from below. Set L = Lo P - E(P).
+
Definition The P($),- process is the stochastic process with joint distribution of q(tl), . . . , q(t,) ( t l < . . . < r,,): ~ , ( x , ) ~ , ( x n ) ( e ~ ~ ~ l i )x( xz l>, .. . ( e - s n - l L )(xn-
1,
xn)
where (e-")(u, h ) is the integral kernel of e - s i (which is certainly defined as a measure) and si = t i + - t i . dv, denotes the corresponding measure. Consistency of the measure follows from e - u L e - b L = - ( a + b ) i and e-%, = Q,. Thus, the oscillator process is just the P(4),-process for P = 0 and the definition is just made so that a suitable version of (4.6) holds for Lo + P - E(P). There are two versions of Feynman-Kac relevant to the P(4),-process. The first is the following. I
.
Theorem 6.7 Then the measure dv, restricted to Let g p = Q,(Q,)Cla,b],the a-algebra generated by { q ( f ) } o < l < b is absolutely continuous with respect to the oscillator measure, Dy, and lf1'P(4)
Proof
r
C [ u , b]
= F [ a . b](q;
1 C [ u , b]
(6.10a)
One must show that
for any function G, of {q(t)la I t I b}. When G = h,(q(u))h,(q(b)) for suitable h , , h,, this is just Theorem 6.3, and for G = nl=I hi(g(ti)),it follows from a simple extension of Theorem 6.3. Since every CI,,,I-measurable G is a limit of sums of such products, the result is proven. I Since we have just seen that all P(4),-processes are locally mutually absolutely continuous, one can ask naturally about global absolute conalthough one tinuity; for convenience we think of all processes on @)'can realize most of them on Y ' (R) or some other similar space. If P , - P , is nonconstant, then dv,, and dv,, are Theorem 6.8 ([223]) mutually singular as measures on (k)'- o'. m ) .
58
It.
The Basic Processes
Proof We first claim that dv,, and dv,, are distinct on functions of q(0). For the distribution of q(0)is !2$dx which therefore determines R, and so P - E ( P ) = +Q, '(n;)- +x2 + 3. Next we claim that every P(+),-process restricted to {q(n)lnan integer} is ergodic with respect to the map U : q(i) -+ q(i 1); see Example 2 preceding Theorem 3.7. Now pick some set A c R with dv,,(q(O) E A ) = a , # a2 = dv,,(q(O) E A). Then, by Theorem 3.7,
+
almost everywhere with respect to dv,,. Thus, we have disjoint sets which support the dv,,. 1 We will see later how to distinguish the dv,, more explicitly. As a final result on P(+),-processes note: Theorem 6.9
For any G
/ G dv,
=
lim Z,' / G exp( - /:f'(y(s))
d s ) Dq
(6.1
T+ m
where (6.1
Proof ItsufficestoconsiderGoftheformnr=, f;(q(ti))witht, < ... < t,. In that case for 7'with - T < t 1 < t, < T :
where L = Lo + P and so = T + t l , s, = tz - t,, ..., s , - ~= t, - t n + l , s, = T - t,. (1.10)completes the proof. 1 Equations (6.11)give the study of the P(+),-process something of a statistical mechanical flavor; see Sections 12 and 19.
*** In these notes, we will be primarily interested in the direction of the Feynman-Kac formula that says one can study e-IHusing Wiener integrals. However, one can turn this analysis around and use e - t Hto study Brownian
6.
59
The Feynman-Kac Formula
motion. The following example (suggested to me by Donsker) illustrates this nicely. For each t > 0, consider the function of the Brownian path b, given by ~ , ( b= ) t - ' 1 {s 10 I s I t ; b(s) 2 011
(6.12)
where 1.1 is Lebesgue measure; i.e., U,(b) is the fraction of time that the path is to the right ofzero.Think of U , as random variables. Since { t - 1i2b(st)}o,,, has the same joint distributions as { b ( ~ ) }the ~ U,'s ~ ~ all , ~have the same probability distribution dp(x) which we want to compute. Let V be the function which is zero (respectively, one) for x < 0 (respectively, x 2 0). Clearly,
so by the Feynman-Kac formula in the form (6.7) and Theorem 6.5: JeFtui'')fif(b(r))Db =
s
e-'H(x,O)f(x) d x
+
where H = - H d 2 / d x 2 ) V. On both sides of the last expression, we can choosef, E L2 converging monotonically upwards to one. Since e-'H(x,0) and e - ' " ~ ( ~are ) positive, we see that Se-'. d p ( y ) = Je-'"(x, 0) dx
Multiplying both sides by e-'" and integrating dt (using Fubini's theorem to interchange orders), we find that /(y
+ a)-'
dp(y) = J(H
+ a)-'(x,
0)dx
(6.13)
+
Now, one can easily compute ( H a)- ' ( x , 0) = g ( x ; a) explicitly since it satisfies -$g" Vg ag = 6 with the boundary conditions g --+ 0 at f co. The net result is
+
+
$/(A + ,/ Thus, aintegrating ).on the right-hand side
with a(a) = of (6.13), we see that
60
11.
The Basic Processes
Now F(a) is analytic in the plane cut from - 1 to 0 with F ( a ) -+ 0 at infinity. The Cauchy integral formula thus implies that 0
~ ( a=)
71-1
J- (x
- a)-’
Im(F(x
+ io)>dx
1
nl
since F is pure imaginary for - 1 < x < 0. Thus, we have proven the following. The random variable U , of (6.12) has probability
Theorem6.10 distribution
i0,&, n-’(x(l
d&) =
; dp(x) Since 1 law. 7.
=
-
x>)-’’~dx,
2n-’ Arc sin
0<x < 1 otherwise
this result is often called the Arcsin
Regularity and Recurrence Properties-2
In this section, we continue our study of the regularity of Brownian paths and study certain global properties. The first result we will prove is Khintchine’s [157] famous law of the iterated logarithm: Theorem 7.1
For one-dimensional Brownian motion
with probability one where log,(y)
=
ln(1n y).
Before proving this result, we state a number of results which follow from (7.1). Since b(t + s) - b(s) A b(t) and since b and q and any P(4),-process are locally absolutely continuous (Theorems 5.5 and 6.7): Theorem 7.2 For one-dimensional Brownian motion, for any P($)lprocess and any fixed s:
with probability one.
7.
61
Regularity and Recurrence Properties-2
+
The result for b(s - r) - b(s) follows from that for b(s t ) - b(s) if we notice that b(2s - t ) - b(2s) = b(t)has b(t) A b(t) for 0 I t I 2s. The equivalences (5.6) and (5.7) immediately give the following. Theorem 7.3
For one-dimensional Brownian motion -
lim
*+m
h(t) [2t log,(t)]”2
(7.2)
=
with probability one. Theorem 7.4
For the oscillator process:
with probability one.
To get (7.3), we use the fact that In(2t)/ln(t) -, 1 as t -+ a.(7.2) and (7.3) show explicitly that Db 1 X(a, and Dq X ( a , are mutually singular measures for each a > 0. One can also distinguish various kinds of P(+)lprocesses by similar means; e.g., if P ( x ) = x Z m ,then o[i)
r
(with urna nonzero constant) for almost every P(+)l-path. See [223] where “higher order oscillations” are discussed ; e.g., for the oscillator process an explicitf(t) is exhibited with
lim f ( t ) [ q ( t ) - (In t)’/2]= 1 (earlier references to this subject may also be found in [223]). Now b(t) = -b(t) and 4 ( t ) = - q ( t ) . Thus the following theorem holds. Theorem 7.5 In (7.1), (7.2), and (7.3) b (respectively, q ) may be replaced by -b (respectively, -4). Equivalently, 6and = 1” may be replaced by ljm and “ = - 1.” “
These results are another indication of the roughness of Brownian motion. They imply that b(t) = 0 for infinitely many t’s near zero, since b(t)is positive and negative infinitely often. In fact, one can show that, with probability one, {tlb(t)= O} is a nonempty perfect set with Hausdorff dimension (see [136]), and, in particular, it is uncountable. In multidimensions, one has the following.
62
II. The Basic Processes
Theorem 7.6
For v-dimensional Brownian motion
with probability one.
Proof
Let {en};= be a dense set of vectors on the unit sphere. By (7.1)
for each n with probability one. Since Ib(t)l 2 I b(t)-e,I, the lim in (7.4) is greater than or equal to one. On the other hand, if the 6were some a > 1, then by a compactness argument limb(t)-e, 2 +(1 + a ) for some en. This proves (7.4). The proof of (7.5) is similar. I Remarks 1. One curious feature of the v-dimensional case is the following: In one dimension, the set of limit points of b(t)/[2t logz(t-')]''2 as t 10 is clearly [- 1, 11 since b is continuous and we know that the lim (h) is 1 ( - 1). One can ask about the limit points in v-dimensions. Since the components are independent, one would naively expect that the set of limit points is just the v-fold product of [ - 1,1]. But this cannot be since (7.4) implies that the limit points lie in the unit ball! In fact the entire unit ball occurs. 2. The law of the iterated logarithm has been proven for many Gaussian processes; see, e.g., [196a]. We now turn to the proof of (7.1). The key to understanding why (7.1) holds comes from (7.3) and its comparison to the exam le following Theorem 3.3: If the q ( t ) were independent, then lim y(n)/& would be one. The point is that the q(n) are "almost independent" so that this is still true. But q(t)for t near n is not significantly different from q(n). The "easy" part of the proof of Theorem 7.1 just follows this intuition; note that by (5.6) and (5.9), 4(n) and b(e-'") are related:
Lemma 7.7
For one-dimensional Brownian motion
& b(e-")/[2e-"log n]1/2 n+m
=
1
7. Regularity and Recurrence Properties-
63
2
with probability one and, in particular, the equal to one with probability one. Proqf
e-
In-mIi2
lim in
(7.1) is greater than or
Let x, = e+""b(e-"). Then, the x , are Gaussian with covariance . Let A , be an event depending on x,. Then, by (3.12):
IE( A,
n A,) - E ( A , ) E ( A , ) I I e- ~ n - m ~ ~ 2 ~ ( ~ , , ) " 2 ~ ( ~ m ) l i Z
Using the strong Borel-Cantelli lemma, Theorem 3.2, one can now mimic the proof of Theorem 3.3 and the example following to see that E x , / (2 In n)l" = 1 which completes the proof. I The hard part of proving (7.1) involves showing that sampling the points = e-" does not result in a smaller lim than sampling all the times. We will give two different proofs. The first gets control with martingale inequalities:
t,
First proofoflim I 1 (following [183]) In the inner product (hi,6,) = min(t, s), the projection of 6, onto [6,ls I u] with u < t is 6,. It follows from (3.9) that
E(:eab('):Ib(s); sI u ) = :cab(“): i.e :eab(r).. = eab(i)-azii2 is a positive martingale. Thus by Doob's inequality (3.;)
Li[
TI 1
E max b(s) - - > fi
=
Li 1 > eZP
E max
< e - "PEE( :eab(t)..) - e-zP
Let h ( t ) = [2t log,(t-')]''2 and choose 0 < 0 < 1 and 0 < 6 < 1. Let = On-' and choose a, = (1 6)0-nh(O"), fi, = h(0")/2 so anfin= (1 6) log, 0-" and e-*"On= exp(-(1 6)[ln n log, O-']) = C H - I - ~ . Thus by the first Borel-Cantelli lemma, for n 2 N ( b ) (with N ( b ) < 00 almost everywhere), t,
+
+
[
+
+
";"I
max h(s) - - I fin SSI,
Let t I
and suppose that r,+ b(t) I max b(s) I fi, sst,
+ 21 a, 0"- ' ~
64
II.
The Basic Processes
since h(t) is monotone increasing for t I e - e . Thus
Letting 6 1 0 and 9 1, the result follows.
I
Remark The above implies that :b(t)": is a martingale where :x":= (d":e"x:/da")l,,o for Gaussian x; see [258] for discussion of :x":.
The second proof depends on the following remarkable fact whose proof we defer. =
2E(b(t) 2
A)
A
20
(7.6)
osssi
Second proofoflim I 1 (following [86]) By (7.6) and (3.4)
Pick 6 > 0 and then BE(O, 1) with (1 + 6)29 > 1, t , (1 6 ) h ( ~ , + ~ ) / &where h(t) = [2t l ~ g , ( t - ' ) ] ' ~so ~ that
+
x, and thus since (1
+ 6)'9
=
(1
=
On, and x,
=
+ 6)[29 Inn + c]'"
> 1, -x 3 2
5 cn-",
a
>1
Xn
Let A , = { b ( t ) > (1 + 6)h(t)Isome t E [t,+ t,]}. Then b E A , and n so large that h is monotone on [0, t,] implies suposisinb(t) 2 (1 6)h(t,+ 2 x,&, so by the first Borel-Cantelli lemma, b $ A , for all n sufficiently large; i.e., with probability one
+
Taking 6 LO, the result is proven.
I
Notice that the full power of (7.6) was not used; all that was needed was osssr
7.
Regularity and Recurrence Properties-
65
2
This is just Levy’s inequality, Theorem 3.6.5, in this situation: the hypotheses of that theorem are applicable once one discretizes time since a product of Gaussians is clearly invariant under sign changes. More significantly, Levy’s inequality is applicable to n-dimensional Brownian motion yielding
1
(
E max I b(s)l 2 A I2E( I b(t)l 2 A) OSSSf
(7.6‘)
for n-dimensional Brownian motion. Of course the fact that equality holds in (7.6) is interesting. We turn now to proving (7.6). A second proof will be given below following Lemma 7.10. To understand why it should be true, consider an elementary random walk X,. The analog is ( N , k positive integers)
(
E max X , 2 k OSnSN
1
=
2E(X, > k)
+ E(X,
=
k)
(7.7)
(7.7) comes from the fact that if max05,5N X , 2 k , it is equally likely that also X , > k or also X, < k, since we can reflect the path about the first n where X , = k. This idea is easy to make rigorous. One lets A j = { X I X j = k , X o , . . . ,X j p l < k } and notes that E ( ( X , - X j ) > 0, A j ) = E ( ( X , - X j ) < 0, A j ) since A j is independent of X , over j
-
X j , and X ,
X , > k , max X , 2 k OSnSN
1
=
(
-
X j is even. Thus, summing
E X , < k , max X , 2 k OSnSN
which easily yields (7.7). The intuition is the same for Brownian motion but the fact that t is continuous makes it difficult to find analogs of A j . The key to finding a suitable way is to discretize. It is useful to introduce the following.
Definition
is the o-field generated by { b(s)I s I t } ;Bm=
uf
<
Bt.
A stopping time is a function T with values in [0, a]of the Definition ~ so that Brownian path with the property that {blz(b) < t } E B and E(T < 00) > 0.
66
II. The Basic Processes
Roughly speaking, a stopping time is a function given by " z ( b ) is the smallest time, t , so that . . . has occurred before t." Examples 1. Fix A > 0. Let r(b) = inf{t I1 Ib(t) 2 A}. If maxO.,,, b(t) < A, set z(b) = 1. Then, for t < 1:
u.
is obviously in B t . As a result {blz(b) < t } = {blz(b)I t - l/n} is in g t .Thus z is a stopping time. 2. z(b) = inf{t Ib(t) = 0, I b(s)l 2 1 for some s E [0, t ] } . As above, this "first return time" is a stopping time. 3. Given any stopping time z, define its discretization 7(")by
if (k
r(")(b)= k/2" Then, for t E (k/2", (k
-
1)/2" I z(b) < k/2"
+ 1)/2"]:
{blz'"'(b) < t } = {blz(b)< k/2"} E Bt Thus, t(") is a stopping time. Notice also, that {blz'"'(b) = k/2"}
=
{blT(b)< k/2"}\{bl~(b) < (k - 1)/2"}
is 21ki,,-measurable. We can now prove (7.6). Theorem 7.8
For Brownian motion and A 2 0,
Proof Since b(s) A tli2b(st-l),we can suppose that t = 1. Let z be the stopping time of Example 1 above and let z(") be its discretization modified so that z(")(b)= 1 if z(b) 2 (2" - 1)/2". LetfE C(R). We claim that
E(f(b( 1) - b(T("))))= E(f(b(T'"') - b(1))) For introducing the symbol
(7.8)
7.
67
Regularity and Recurrence Properties-2
we have that
: : :
= k = O E(f(h(1) -
);
b(k));.(")=
=k = O
E(l(b(1) - b($)))E(+)
=k = O
E(f(b(;)
=
-
b(l)))E(t(")
);
=
=
$)
E(f(b(z'"') - b(1)))
where we have used the facts that b(1) - b(k/2") is independent of B k / 2 " , that (t(") = k/2") E Bk/2n and that b( 1) - b(k/2") is even. As n + 00, T(") 5 z so b(t(")) -+ b(t) by the continuity of paths. Thus, by the dominated convergence theorem : E ( f ( b ( 1 ) - b(r)))= E ( f ( b ( 7 ) - b(1)))
or
+
E(f(b(1) - A); t < 1) f ( O ) E ( t = 1) = E ( f ( A - b(1)); t < 1) + f(O)E(t
=
1)
The continuity of the distribution of b(1) then allows us to letfbe the characThus, teristic function of (0, a).
E(b(1) > A; 5 < 1) = E(b(1) < A;
t
< 1)
Since E(b(1)
=
A;
t
< 1) + E(t
=
1, b(1)
=
A)
=
E(b(1) = A)
=
0
we have E(T < 1) = 2E(b(l) > A;
T
< 1) = 2E(b(l) > A) I
Stopping times will be useful in our study of recurrence; especially useful is Theorem 7.9 due to Dynkin [67] and Hunt [130] which says that "Brownian motion starts afresh at stopping times." As a preliminary, we need the following. Definition
Let z be a stopping time. Then
68
II. The Basic Processes
Examples
1.
t
is BT+-measurableas is b(t) if we set b(z) = co when
z = co. Also j;, f(s)F(b(s))ds is Br+-measurable.
2. If Br+=
t o is
a fixed positive number, then t(b)
nszlo Bsexplaining the notation.
Definition by
= t o is a stopping time and
Let B be an event with E(B) > 0. Then the measure pB given PB(A) = p ( B
A)/p(B)
is called the B conditional measure. Given n random variables fi, . . . ,fn, their joint distribution with respect to p B is called their joint distribution conditional on B. Let t be a stopping time for Brownian Theorem 7.9 (Dynkin-Hunt) motion. Let 6(t) b(t z) - b(t). Then, conditional on t < co, b”(t) has
=
+
identical joint distribution to b(t). Moreover, the b”(t) are independent, conditional on z < co, of B r + . Proof
The theorem asserts that E ( F ( i ( t ) ) ;t <
02) =
E(F(b(t)))E(z < co)
and for B E Br+: E(F(b”(t));B n (z < c o ) ) ~ ( < r
00)
=
E(B n (t < . o ) ) ~ ( ~ ( b ” (tt )<) ;co)
This pair of statements is clearly implied by
E(F(b(t));(t < co) n B ) = E(F(b(t)))E((t< co) n B ) It suffices to prove (7.9) when F(b(t)) = G(b(ti),. . ., N t n ) )
with G E C(R”). Define g(t) = G(b(t,
+ r ) - b(t),. . . ,b(t, + t ) - b(t))
Then ~ ( ~ ( b ” ((t t )< ) ; 0 0 ) n B ) = E(g(t); (t < 02) n B ) =
lim E(g(t(”));(t(”) < co) n B ) n-. m
(7.9)
7.
69
Regularity and Recurrence Properties-2
where (7.10) follows from the facts that (t(") =
k/2") n B
= ((7
< k/2") n B)\(t < ( k
-
1)/2") E Bki2"
since BELB~+and t is a stopping time and that b(t and identically distributed to b(r). independent of Bk,2n
+ k/2")
-
b(k/2") is
I
*** We next want to study the question of recurrence for Brownian paths. In v = 1 dimension, we know that Km b(t) = 00 while h b ( t ) = -03. It follows that, with probability one, b(t) returns infinitely often to any given set, since it must sweep through the whole real line infinitely often. We want to examine the analogous question in detail when v 2 2. We will give two presentations of the basic facts, one after Lemma 7.21 using stopping times and harmonic functions; the major tool in our first approach will be the use of Dirichlet operators: Definition Let S be a closed subset of R'. Consider the quadratic form with form domain C;(Rv\S) and form
(f, HD;Sf)
=
a
The form closure of this form defines a self-adjoint operator HD: on L2(Rv\S). We let PD,,(x, y ; t ) and GD;s(x, y; u ) denote the kernels of e-tHD:sand (HDis u)- (A priori these are only distributions, but using elliptic regularity and the form of their equation, one shows they are C" for x # y in R'\S, and of course, they are positive; see, e.g., [ l 2 l , 2171.) We let GD;s(x, Y; 1) G D : ~ ( xY,). For S's with sufficiently large interiors, we expect that (xs is the characteristic function of S)
+
(H,
+
PxS
+ z)-'
--*
(HD;s
+ z)-'
(7.1 1)
strongly as p -+ 03, for all z 4 [0, 00). We call S regular if P D ; s ( ~y;, t ) is continuous on all of R' x R' (set equal to zero if either x or y E S) and if (7.1 1) converges pointwise for the integral kernels. When S is a nice set like the exterior of a ball, it is easy to see that H D : s is the classical Dirichlet operator and that S is regular. Lemma 7.20 Let S be a regular closed set so that the Lebesgue measure of as is zero and 0 4 S. Then
E(b(s) 4 S""10 I s I t )
=
(7.12)
70
II. The Basic Processes
Proof Let x be the characteristic function of S'"'. Then yoX(b(s)) ds > 0 if and only if b(s) E S'"' for some s E (0, t ] since paths are continuous. Let gn be a sequence in CF with gn(x)/l 1 for each x. Then 1.h.s. of (7.12) = lim E(g,(b(t)); b(s) $ S'"', 0 I s I t ) n+m
n+m I - m
=
lirn lim (exp[ - t ( H o
+ Ax)]g,)(O)
n-tm I + m
=
lim /pD;s(o, Y; t)gn(Y) dy n+
=
00
r.h.s. of (7.12)
In the first two and last steps above, we use the monotone convergence theorem. In the third step, we use Theorem 6.5 and we use regularity in the fourth step. I Example We can now give an alternate proof of (7.6). For let S c R . Then, by the method of images:
=
[A, 00)
Thus, by (7.12): max b(s) I A 05s5z
11 s_, =
PD:S(x, 0; t ) dx I
=
[P(x, 0; t ) - P(2A - x,0;t ) ] dx
=
E(b(t) I A) - E(21 - b(t) I
=
E(b(t) IA) - E(b(t) I
=
E ( - A I b(t) I
-
A)
2)
A) = 1 - 2E(b(t) 2 A)
which is (7.6). Theorem 7.21 Let S be a ball of radius ro about a point y with lyl R > ro. Then, for v-dimensional Brownian motion:
E(b(t) E S I some t ) =
1,
{(ro/R)v-2,
v = 1,2 v23
=
7.
71
2
Regularity and Recurrence Properties-
Remarks I. The fact that for R fixed, and ro small, this “hitting probability” is O(r;-’> not O ( r ; - l ) (= cross-section) is another expression of the “wigglyness” of Brownian paths; see Section 22. 2. An alternative proof of this theorem is given in the Aside following Lemma 7.21. That proof uses hitting probability ideas in place of Green’s functions. P r m f Let p(s) = E(b(r) $ S 10 I t I s). Since p(s) as s -+ co,we have that E(b(t) $ S I all t ) = lim a10
=
lim a10
= lim
+ E(b(t) $ S [all
t),
som sRs
ae-aSp(s)ds ~omae-usPD;s(O, x ; s) dx ds
J
110
U G ~ , ; ~X(; Oa ), dx R’
Now, by scaling So using translation covariance, we see that E(b(t)$ S I all t )
=
lim a10
s
GD;s ( o ) ( ~ , y,) dx
(7.13)
where ya = al/’y and S(a) is the sphere of radius a1/2roabout 0. If we replace S with the empty set, we see for comparison that 1
=
s
lim Go(x, ya)dx a10
(7.14)
For v = 1, one easily sees that the right-hand side of (7.13) goes pointwise to zero (use the explicit formula for G,) and so the integral goes to zero since GD I Go. For v 2 2, GD and Go have spherical harmonic expansions, and
k,
G . . . (x, Y) = g . . . (1x1, Ivl)
where dR, is the measure on the sphere and So(r, r’) = r - 1/2(v- 1)(r‘) - 1/2(v- ” Av (r < ) & ( r > ) where r < = min(r, r’); r , normalized solution of -u”
=
max(r, r’) and A , (respectively, B,) is a suitably
+ $r-2(\,
-
l)(v
-
3)u
+ 2u = O
72
II. The Basic Processes
which is regular at Y = 0 (respectively, O(e-'J2) at a). If gD;ocomes from the Green's function with a Dirichlet condition on the sphere of radius a, then
Thus,
with y
=
ro/R.Thus, dividing (7.13) by (7.14) (7.15)
-
--
If v 2 3, then A , x112("-l)and B , x ' " ( ~ - " for ) x small, so the limit in (7.15) is Y " - ~ . If v = 2, then A , x ' / ~and B, xli2 In x, so the limit in (7.15) is limxlo(lnx)/ln(yx) = 1. I
-
Let fn be a sequence of bounded functions Example (Friedman [87]) on R" with supp f. c {x 11x1 I l/n}. Friedman noted that for v 2 4, - A + f n -+ - A in strong resolvent sense irrespective of how fast )I f 11 a, grows! For C& = {g which are in Corn, g = 0 near 0) is a core for - A if v 24andforg~C:~: II[(-A =
+ f. + i)-' lI(-A
+ + fn
+
(-A i)-'](-A i>-'f.gll = 0
-
+ i)gII
for n large. Since {(-A + i)g} is dense by the core statement, the strong 3 by explicit example, resolvent result is true. This assertion is false for v I but Friedman did prove it for v = 2, 3 if the,f,'s are positive. His proof is expressed quite nicely in terms of path integrals: the key fact is that when v 2 2 for fixed T, Q = {olo(t) = 0, some t E [0, T ] } has Wiener measure zero. For v 2 3, this follows easily from Theorem 7.1 1; for v = 2 a different argument is available: see the Aside following Lemma 7.21. If o $ Q, then by the continuity of paths, info,,,, Io(s)I > 0, so for n sufficiently large 1; f.(w(s))ds = 0. It follows that for any g, h E L2,
7.
Regularity and Recurrence Properties-
73
2
for almost every w. Since h 2 0, we can apply the dominated convergence theorem to conclude that (g, exp( - T( - A
+ h)W)
-+
(9, exp(TA)h)
as n
--f
03
this implies the strong resolvent convergence. Theorem 7.11 and stopping time arguments are what is needed to complete the study of recurrence. As a preliminary, we note that if b(t) is a copy of v-dimensional Brownian motion on a space with some additional independent coordinates, x, and if R(x) is a random rotation on R" depending only on x,then R(x)%(t) is also a copy of Brownian motion. This follows since for each fixed x,Ep(f(R(x)%(t))) = E(f(b(t))) so by independence Ed;.(f(R(x)~(t>>>= W(b(t))) As an example, let z be a stopping time which is everywhere finite and let R,(b) be a rotation with R,(b)b(r) = (I b(r)l, 0, . . . , 0). Then R,(b)(b(t + z) b(z)) is a copy of Brownian motion.
Theorem 7.22 that v 2 3. Then
Let b be v-dimensional Brownian motion and suppose lim 1 b(t)l
=
co
f+cci
with probability one. In particular, for any bounded S, {tIb(t)ES} is a bounded set.
Proof
Define stopping times z, =
7,
by
inf{s 1 1 b(s) I
=
2")
Then z, < cc everywhere by (7.5). Since R,,(b(t + 7,) - b(z,)) = %,(t) is a copy of Brownian motion, Ib(z,)l = 2", and b(s) E {x 11x1 < r } for some s > z , i f a n d o n l y l f % , ( t ) ~ { x ) J x + ( 2 " , 0. ., . , O ) I < r } f o r s o m e t > O , we see that for n > In r/ln 2 E( I b(s) I < r I some s >
7,) =
(r2-n)v-2
Thus, by the first Borel-Cantelli lemma, there is, for almost every b, an n(b) so that Ib(s)l 2 r for all s 2 z,*,(b). Thus, since each z, < 03, l&Ib(s)I 2 r. Since r is arbitrary, the theorem is proven. 1 Theorem 7.13 Let b be v-dimensional Brownian motion and suppose that v I 2. Then for any nonempty open set S
I { t I b(t)E S i I =
74
II. The Basic Processes
for almost every b where 1.1 indicates Lebesgue measure. In particular, the above set is unbounded and for any a E R' bIb(t)
-
a1 = 0
t-+W
Proof It suffices to take S a ball, say of radius r about xo. Let T = inf{t I I b(t) - xo I I r / 2 } . Then z is a stopping time, so it suffices to show the result for all r and xo = 0, since then I{tllb(r t ) - b(t)l I r/2}1 = co. Finally, by scaling we can suppose that r = 1 also. Define stopping times t l , al,T ~ a2, , . . . inductively by
+
T~
= inf{t Ilb(t)l = 2)
a, = inf{t > T , I I b(t)l = t } T , = inf{t > a,-l Ilb(t)l = 2) It is not hard to see that each stopping time is almost everywhere finite: For z, < co given 6,- < 00 on account of (7.5) and a, < 00 since 6(t)= R,,(b(t + z), - b(t,)) is a Brownian motion and an < 00 is equivalent to
E(~(~)E{xIIx+(~,o)I
11};somet)= 1
which follows from Theorem 7.11. Using the Dynkin-Hunt theorem, {a, - T,} and {z, - a,- 1 } are two mutually independent families of identically distributed independent random variables. Therefore, T, = I{tllb(t)l I
1;T,
< t < T,+l)l
are strictly positive random variables which are independent and identically distributed. Let S , = min(T,, 1). Since T, > 0, E(S,) > 0. Clearly E(I S, I) < co, E(lS,lz) < co. Thus, by the strong law of large numbers (see Lemma 7.14 below),
with probability one and, in particular: m
c s j = C o j=1
with probability one. Since
the result is proven.
I
In the above, we needed the following lemma.
7.
75
2
Regularity and Recurrence Properties-
Lemma 7.14 (strong law of large numbers) Let {X,,}:= be a family of independent, identically distributed random variables and suppose that
with probability one. Proof By replacing X j by X j - E ( X j ) , we can suppose E ( X j ) = 0. Notice that in this case, if Y is the sum of any kX's, then E(Y2) = kE(X?)
In particular, if
=
(7.16)
Cy=lX i , then E(12-"YpI')
=
2-"E(X:)
so that E(12-"Y',lI 2
E)
I &-'2-"E(X:)
Thus, by the first Borel-Cantelli lemma, \ 2 - " Y',, I < E , eventually; i.e., 2-"Yz, = 0. Now let Z y ) = 2-"+' X i t z m - Then l.
( Z p + E((2-"+'X1)') > (Zj" 1)2 so ( Z y ) ) 2is a submartingale. Doob's inequality (3.8), thus, implies that E((Z?))' I (Zj"!1)2)
=
IZY'1' 2
max 0 5j 5 Z n -
I
E*
1
I E-'E((Z$,-,)*) =
4~-'2-"E(x:)
Thus, by the first Borel-Cantelli lemma (again), IZp)I < E for n 2 N o , 5 m < 2". Then
j = I r . . . , 2 " - l . Let 2"-
m-llYrnl
for n sufficiently large.
I
2-"fllY2"-11+ ~z:!zn-l~1 2 E
1
Remarks 1. The law of large numbers also follows from the Birkhoff ergodic theorem (Theorem 3.7) and the Kolmogorov 01 law (Example 1 before Theorem 3.7). 2. One can dispense with the condition E( I X , 1)' < co;see [lSS].
As a final set ofresults involving recurrence, we want to note that while onedimensional Brownian paths spend an infinite time in any given bounded set,
76
II. The Basic Processes
there is a significant probability that they will take a long time to reach the set. Explicitly, we want to show that for the stopping times used in the proof of Theorem 7.13, E(T, - T,,-’) = oc. This fact prevents an amusing paradox from taking place. While I { t I I b(t)l 5 1) I is infinite if v = 1 or 2, it is fairly clear that t - ’ I {sllb(s)l I 1; s I t } I should go to zero as t -+ GO for the Brownian path should forget where it began; i.e., limt-l~{sllb(s)lI l;s1t}~=limt~’({s(b(s)~[n,n+2];s~t}( for any n. If E(T, - T,were finite, then we would have T , / n + E(z, - rl) - T,, we would also have by the strong law of large numbers. Since T, I T,+ IT;. + E(T,)/E(T, - T ~ in ) contraTJn -+ E(Tl) so that T;:’ diction to intuition, which says that t - ’ I . I + 0. Fortunately, E(T, - 2,- ’) = co,so there is no problem. Proposition 7.15
In one dimension, tli2E(b(s) I 110 I sI t ) + (2/x)’”
ast+co. Proof As already noted in the Example following Lemma 7.10, Equation (7.6) is equivalent to =
E( - 1 Ib(t) I 1)
=
(2nt)- l l z
osssr 1
which proves the result since exp( -x2/2t)
J-l e x p ( g ) d x
-+
1 pointwise as t
+
co. 1
Actually, one can explicitly compute the probability distribution for the stopping time which gives the first time that b hits one: Proposition 7.15‘
The stopping time T =
inf{slb(s) = l }
has the distribution (271t3)-‘1’ exp( -(2t)-’)dt
In particular, as t
+
co,
t’I2E(b(s) < 110 < s I t ) = t”’E(T > t ) + (2/Z)”*
7.
77
Regularity and Recurrence Properties-2
First proof ([272]) f(t)
Clearly
= E(t > t ) = E
so by (7.6): f ( t ) = E(lb(t)l I 1)
Thusfis differentiable and
which establishes the probability distribution for t . Second proof ([183])
Introduce stopping times TN," by if ( k - 1)2-" I t < k2-";
k I N
We will first prove that for y > 0, E(:exp ~ b ( t N , ~ = ) : 1)
(7.17)
For E(:exp yb(tN,"):)
=
E(:exp y b ( ~ , + ~ , ~z) < : ;( N
+ E(:exp yb(N2-"):;
t 2
-
1)/2")
( N - 1)/2")
But b(Nz-") - b((N - 1)2-") is independent of t 2 ( N - 1)/2" since 2 ( N - 1)/2" is measurable with respect to B ( N1),2m. Thus
T
E(:exp yb(N2-"):; t 2 ( N - 1)/2") = E(:exp yb((N - 1)2-"):; t 2 ( N - 1)/2")) i.e., we have shown that E(:exp yb(tN,,):)
=
E(:exp y b ( t N -
I,m):)
78
II.
The Basic Processes
so (7.17) holds by taking N to one where b(z) = b(1/2"), and the result is evident. Now, set N = 2"M with M an integer and take m 30. Pointwise b(z,,,) + b(fM)where fM = min(z, M ) . Moreover -+
:exp(yb(z,,,)): I max exp(yb(s)) O<s<M
=
(
)=
exp y max b(s) O<s<M
F(b)
But, by (7.6)
=
1
+ 2y
SI
eYAE(b(M)> A)dA < co
Thus, by the dominated convergence theorem, we can conclude that
E( :exp(yb(f, )):)
=
1
Now z < co, by the law of the iterated logarithm so b(fM)-+ b(z) = 1 as M + 00. Moreover, by the definition of z, b(fM) < 1, so :exp(yb(f,)): < ey and thus we can take M + co and conclude that E(:exp(yb(z)):) = 1 But b(z) = 1, SO E(exp( -+y2z))
=
(7.18)
exp( - y)
As in the inversion procedure used to reach Theorem 6.10, one can go from this to E((a + z)-') and from there to the claimed distribution. Remarks 1. Similarly, one finds that the distribution for first hitting a > 0 is (27ct3)- "'u exp( -a2/2t) dt, and that for IT,the first time that Ib(s)l = 1, one has that E(exp( - h'o))
=
[cosh(y)] -
'
2. We were especially careful about the use of the dominated convergence theorem, because one can consider the stopping time y~ =
min{s I b(s)
Clearly 4 > z so since b(q)
=
=
b(z)
1; b(t) 2 2, some t < s} =
1,
E(:exp(yb(q)):) < E(:exp(yb(z)):) = 1
7.
79
Regularity and Recurrence Properties-2
But the proof above goes through most of the way! In fact E(:exp(yb(rj,)):) = 1. But as M + 00, :exp(yb(f,)): is no longer bounded by an L'-function invalidating the use of the dominated convergence theorem. One can explicitly compute E(:exp(yb(q)):) since 4 has the same distribution as the stopping time p = inf{slb(s) = 3}, since we can reflect the path about b = 2 for times past the first time the path hits two. Thus, E(:exp(yb(q)):) = eYE(exp( - + y 2 q ) ) = eYE(exp(- )y2p)) = e-2YE(:exp(yb(p)):)= e - 2 Y
as above. 3. The argument in the second proof can be abstracted to the following. Theorem 7.15A Let f ( t ) be a (sub-respectively, super-) martingale for Brownian motion with ~ u p , , ~ I f (b)( t , < co.Then, for any stopping time, t: E ( f ( t ) ) = E ( f ( 0 ) ) ( 2; respectively, r ) As in the above proof, one can relax the supremum requirement by additional
argument and conditions on t. 4. One can directly obtain that E ( t ; t < t ) diverges like t'" from (7.18) which implies that
E ( t exp( - ut))
d
exp( - (2u)'l2)
=
- -
=
( 2 ~ ) 1'2 - exp( - ( 2 a ) ' 9
da
Now use a Tauberian theorem (see Section 10) to obtain limt+mt ' / 2 E ( ~> t ) . Corollary 7.16
The stopping times G, t of Theorem 7.13 obey E(a, -
t,,)=
co
Proof Let 6(t)= R,,(b(t + t,,)- b(t,)) Then G, - T,, 2 inf(sJb,(s)I -3) so g(t)
1
(
= E(o, - t,,2 t ) 2 E max b(s) I 3 05S5l
+
= + ( 2 / 7 ~ ) ~ / ~ t -o(t-1/2) '/~
As a result
E(o, -
t,,)= - s,"t
dg
=
Iomg(t)d t =
03
80
II. The Basic Processes
The reason that it may take very long for b(s) to reach one is not that the path gets “trapped” in ( - 1, l), but rather that it makes excursions far to the left: Proposition 7.17
1
(
E maxIb(s)I I 1
(4
o<ssi
where
=
E(b(t)EA) - E(b(t)EB)
(7.19)
1 ) ~ ( 3 , 5 ) ~ ( - 3 ,- 5 ) ~ . . .
A = (-1,
and B
=
(1, 3) u (-1,
1
-3) u (5, 7) u . . .
(2k + l ) - ~ ( - l ) k e - n 2 ( 2 k + ~ ) 2 r(7.20) /8 4 “ E( - a I b(s) s 1 10 I s I t ) = O(e-‘“‘“’) (c) where ~ ( a = ) z 2 / ( 2 ( a+ 1)’).
(b)
(
E max Ib(s)l i 1 ocsct
=-
=k=1
Remarks 1. (7.19) is good for t small and (7.20) for t large. 2. (c) verifies our remark, that for large times, b(s) can only stay below one by going far to the left, indeed, it must go a distance O(t”’) to the left.
Proof
By Lemma 7.10, 1
E ( - u I b(s) i 110 I s i t) =
i.
P ~ . ( x 0; , t)dx
where PD:.is the propagator for times Laplacian with Dirichlet conditions by the method of images and at one and -a. (7.19) comes from writing (7.20) by expanding PD:a in its explicit eigenfunction expansion and noting that this expansion is uniformly convergent so that it can be used for P(x, 0; t ) and the sum and integral can be interchanged. (c) comes from the same kind of eigenfunction expansion noting that the lowest eigenvalue of -$ d2/dxZon L’( -a, 1) with Dirichlet conditions is zZ/(2(a 1)2). I
+
One can ask how I { t I I b(t) I 5 1 ;t 5 T } I diverges as T + 00 when v = 1,2 (we have already remarked it cannot be as fast as T). We will answer the weaker question concerning how E( I { s I I b(s) I i 1 ; s t} I) diverges as t + co. Proposition 7.18
For v
=
1
7.
Regularity and Recurrence Properties-2
and for v
=
81
2: lim(lnt)-'E(I{sIIb(s)) I 1 ; s It } l ) = d t-a2
for suitable nonzero c, d. Proof
Let x be the characteristic function of [ - 1, 13. Then, for each b 1;s ({sllb(s)l I
Thus, by a Feynman-Kac formula:
Jo
Thus q'(t) 2 0 and as CI 1 0 : ~ o m e - ~ ~ q ' ( t= ) d( t( H ,
+
for explicit c l , dl depending on the explicit small CI behavior of the integral kernel for ( H , a)- '. The v = I result follows from the Tauberian theorem (Theorem 10.3) and the v = 2 result from an extension of that theorem. I
+
Remark
The same method shows that in v 2 3 dimensions, E(I{slb(s)ES)I) <
for any bounded S. This implies that almost everywhere I {sl b(s) E S}I < co, which is not quite as strong as the fact that I b(s) I + 00 almost everywhere.
*** Finally we should like to discuss self-intersection of Brownian paths. In 1940, Levy [1691 proved that almost every two-dimensional Brownian path had a self-intersection, i.e., a pair t , s so that b(t) = b(s); and Kakutani [145] proved that this could not happen if v 2 5. The cases v = 3, 4 were settled in [63] ;the question of n-fold points with n 2 3 was settled in [64,66]. These results are summarized in the following. Definition A continuous curve {b(t)lO 5 t < cx)} is said to have an n-fold point if and only if there exist r , , . . . ,t , distinct with b(t,) = . . . = b(r,).
82
II. The Basic Processes
Theorem 7.19
(a) (b) (c) (d)
Let b be v-dimensional Brownian motion. Then
If v I 3, almost every b(t) has infinitely many double paints, If v 2 4, almost every b(t) has no double points, If v I2, almost every b(t) has infinitely many n-fold points for each n, If v = 3, almost every b(t) has no triple points.
We will give only the proof of (a) when v = 3 (this automatically proves the result for v = 2 since a double point for (b,(t), b2(t),b3(t))is automatically a double point for (b,(t),b2(t))).See [63,64,66] for (b), (c), (d), respectively. Definition Let K be a compact subset of R'. We define the hitting probability by :
h(y, K ) Theorem 7.20
=
E(y
+ b(s) E K I some s E [0, 00))
h(., K ) is a harmonic function on [W"\K.
Proof Giveny#K,pickRsothat{xIIx-yI I R } n K = @ . L e t 6 < R and define a stopping time z by t ( b ) = inf{sll b(s)( = 6} t<
y
cc almost everywhere by the law of the iterated logarithm. Moreover, - b(z),
+ b(s) # K for s Iz(b) so that if%(,) = b(s + z) h(y, K ) = E(y =
E(y
=
[h(Y
+ b(s) E K I z(b) I s < CO) + b(z(b)) + b ( s ) € K \ O < s < a )
+ x; K ) d S , ( x )
where dS, is the normalized invariant measure on the sphere of radius 6. The last equality comes from the fact thata is a Brownian motion independent of b(z) (Theorem 7.9), that Ib(t)l = 6, and that b(t) has a rotationally invariant distribution. Since h is a bounded measurable function, it is harmonic.
I Remarks 1. The mean value property and h E L:,,, imply h is C" since h * f'is C" i f f € C r . I f j f = 1 andfis rotationally invariant with suppf' c B,, then ( h * f ) ( y ) = f ( y ) by the mean value property. 2. One can also base the proof on the Dirichlet ideas used to discuss hitting probabilities earlier.
7.
83
Regularity and Recurrence Properties-2
3. Results of this genre go back to Kakutani [145, 1461. They are the starting point of "probabilistic potential theory" developed by Doob, Hunt, and others and summarized in Meyer's book [186]. They are central to some rather deep results of Burkholder, et. al. [28] on H P spaces; see the readable lecture notes of Petersen [203]. Via these notions, Brownian motion can be used to prove some rather deep results in complex analysis [42]. The following result has already been proven as part of Theorem 7.1 1, but since we shall recover parts of that theorem from Theorem 7.20, we give an independent proof. Lemma 7.21
Let K be a ball. Then h(y, K ) -+ 1 as y
-+
K.
Proof Let r be the radius of the ball and R the distance from y to the ) , hitting probacenter of the ball. By the scaling relations b(s) a b ( ~ a - ~the bility can only be a functionf(r/R) of r/R. We want to show that f(x) 1 as x 1. Let K" be the ball of radius a about the point ( -a, 0, . . . , 0). Let yo = (1,0,0,. . .). Then -+
h(Y0, K") = f(a/(1
+ a))
so
lim f(x) XT
=
1
lim h(y,, K") O-rW
K")
(since K , c K , for a < j)
=
h(y,
=
E(b,(s) < - 1, some s)
=
1
by the law of the iterated logarithm. Aside This lemma and Theorem 7.20 immediately imply the v = 2 case of Theorem 7.11, i.e., that h(y, K ) = 1 for v = 2 and K a sphere. For h is harmonic and rotationally invariant so h(y) = a + b lnly 1. Thus 0 I h < 1 implies h(y) = u and the lemma implies that u = 1. By a little more argument, one can recover all of Theorem 7.1 1 and also the fact (used in the Example following Theorem 7.11) that a fixed point not equal to zero is hit with probability zero if v 2 2. The argument is as follows: fix r < R and define, for r < IyI < R :
y(y)
=
E(ly
+ b(t)l
is first equal to r before it is equal to R )
i.e., g(y) is the probability that a Wiener path starting at y hits the sphere I x I = r before it hits the sphere I x I = R. By the law of the iterated logarithm, it eventually hits 1x1 = R. As in the proof of Theorem 7.20, g ( y ) is harmonic
a4
II. The Basic Processes
in the region r < 1 y J < R. Moreover, we claim that g(y) -+ 1 (respectively, 0) as IyI -+ r (respectively, R). To prove this let
(
q ( E ) = E max Ibi(t)l 5 c113 (i = 1,. . . 7 v ) ; min b,(t)
-
E
OSf5E
OSfSC
)
~
~
Using (7.6) and (7.6’), one easily sees that q ( E ) + 1 as E 10 so that g(r + &’I3, 0,. . . ,0) 1 as E 0 by a geometric argument. We can therefore conclude that -+
-+
g(y) =
{
( R - Y)/(R - r>7 [lnR - Iny]/(lnR - Inr), [y-‘V-2’ - R-(V-z)]/[r - ( V - z )
v = l - R-(V-
211,
v=2 v 2 3
If we take R -+ co with r fixed (and use the law of the iterated logarithm to note that if a path does not hit the sphere of radius r, it will always hit the path of radius R “first” and use the continuity of paths to note that if it hits the sphere of radius r, it will hit before the sphere of radius R for R large), we recover Theorem 7.11. On the other hand, if we first take r + 0 and then R 00, we see that a path starting at y # 0 will hit zero at some time with probability zero if v 2 2 and with probability one, if v = 1. -+
We now return to the proof of Theorem 7.19(a). Lemma 7.22
Proof
Let y.
If K is a union of closed balls, then h(y, K ) -+ 1 as y -+
yo E K . Since yo E K , it is in some ball K O c K . Then
J I I $
by Lemma 7.21.
K.
-+
h(y,, K ) 2 lim h(y,, KO)= 1
I
The important idea of [145] is to use the notion of Newtonian capacity. We develop the ideas in v = 3 dimensions. Definition Let K c R3 be compact and let A + ,,( K ) be the probability measures on K . Then C ( K ) ,the Newtonian capacity of K , is defined by
C ( K ) = [inf{Jlx
-
Y I - ’ d ~ L ( X ) d ~ L ( Y ) I ~ E (~K +, ) ]1] -
Note that C ( K ) may be zero, if the integral is always infinite, e.g., if K
=
(0).
~
7.
85
Regularity and Recurrence Properties-2
Theorem 7.23
For v = 3 h(y, K ) 2 C(K)[max{ I x - Y I I x E KII-'
(7.21)
Remarks 1. Suitably generalized this is true for any v 2 3. 2. In particular, this says that if C ( K ) # 0, then h(y, K ) # 0 for all y. There is a converse to this; namely, that if C ( K ) = 0, then h(y, K ) = 0. The latter follows from the fact [124, 1641 that if C ( K ) = 0, then any positive function harmonic on RV\K withf + 0 at infinity is either identically zero or unbounded. Since h is bounded, it must be zero. Proof Let {y,} be a countable dense set in K . Let K , be the union of the &-ballsabout the points y,. We first claim that K c K , and
(7.22)
h(y, K ) = lim h(y, K , ) E l 0
For clearly, h(y, K ) I h(y, K,). If b(s,) + y E Kl,,, then since Ib(s)l + co as s + cy3 (Theorem 7.12), the s, are bounded so by passing to a subsequence, s, + s. It follows that b(s)E K ; i.e.,
n {b(s) + y
E
K , I some s}
=
{b(s)
+ y E K I some
S}
E
from which (7.19) follows. (Note: (7.22) fails in two dimensions if K = {O}.) Clearly, C(K,) 2 C(K). Thus, (7.21) for K , implies it for K ; i.e., we can suppose that K is a countable union of &-balls.Henceforth suppose that K has this form. Next, we need a fact from potential theory [124,164]: There is a probability measure dp on K so that the function 6Jx) = Ix - y l - ' dp has the value C ( K ) - ' on K . (Intuitively, one gets p by minimizing the energy
Jddx)
4 4 Y ) Ix
- Y I-
'
subject to p 2 0, j dp = 1.) It follows that h(y, K ) = C(K)6,(y) since both sides are harmonic on R3\K going to zero at infinity and one at 8 K and thus (7.21) comes from $,(x) 2 [max( Ix - y I y E K ) ] -
I
Let v = 3. Fix a, c > 0. Then K(b) Theorem 7.24 has strictly positive capacity with probability one.
=
{b(t)la
t
< c}
Proof It suffices to find a probability measure dpb on K(b) so that jdp,,(x)d~(y)Ix- yl-' < co.Define &,(A)
=
IC - al-'I{s)b(s)EA; a < s
Ic } \
86
II. T h e Basic Processes
Then
so, by Fubini's theorem, it suffices to show that
But b(s) - b(t) 1s - t 11/2b(l)and E( I b(l)l- ') < Thus (7.23) holds since
Ja
GO
trivially (since v > 1).
Ja
Proof of Theorem ZlY(a) We first prove that a E E(b(t) = b(s)IsomeO I t I 1 ; 2 I s I
For let K(b)
=
m) > 0
{b(s)lO I t I l}. Then a = E(h(b(2), K(b)))
since the function b(s + 2) - b(2) is a Brownian motion independent of {b(t)lO I t I 2). For almost every b, h(b(2), K(b)) > 0. Thus a > 0. Next we note that a T = E ( b ( r ) = b ( s ) I s o m e O ~ t <1 ; 2 1 s 1 for some T < co since aT Now let
T)>O
a as T + co.
--f
1,
if b(t) = b(s); some nT It I nT
0,
otherwise
+ 1, nT+ 2 I s I (n
=
+ 1)T
Then the qn(b)are independent random variables since b(t + nT) - b(nT) is independent of the earlier bs. So, by the second Borel-Cantelli lemma (or alternately by the strong law of large numbers) Cdb) almost everywhere in b.
I
Remark By scaling, for v E(b(t)
= 02
=
=
3,
b(s)lO I t I a ; 2a I s I Ta)
7.
87
Regularity and Recurrence Properties-2
is independent of a. Thus picking a, see that (b(t) = b(s)IT, I t I T,
=
T-'2-" and T, = 1 - 2-(,-l), we
+ a,; T, + 2a, I S
I T,,,)
are independent, identically distributed events. This shows that the path 1. Then, by scaling and transhas infinitely many self-crossings for 0 I s, t I lation covariance there are infinitely many double points in any interval a I s, t I b. In particular, any point on the path is a limit of double points! By scaling again there are double points with s - t 2 a for any a. Brownian paths are very complicated indeed !
Bound State Problems
8. The Birman-Schwinger Kernel and Lieb’s Formula
From the earliest days of quantum theory, the semiclassical (WKB) limit has been an important notion. Many rigorous developments of this limiting procedure have relied on the method of Dirichlet-Neumann bracketing, see, e g , [252]. Kac in his original Feynman-Kac paper [1381 suggested that Wiener integrals would be an ideal tool for this problem and his suggestion was implemented by Ray [212] in a suitable problem. More recently, the idea has arisen that semiclassical results are not only the answers as h -,0, but that they, or some multiple of them, might bound the relevant quantum quantity for all values of h. This idea occurred first in the bounds of Golden [11 11, Thompson [275] and Symanzik [270] for partition functions (extended by Lieb [170] to certain spin systems), then in certain beautiful bounds [40, 171, 2241 on the number of bound states and most deeply in the realization of Lieb-Thirring [175] that these bounds were critical for the stability of matter. These works relied on a variety of methods; we want to concentrate on that of Lieb [171] who exploited Wiener integral methods. Indeed, the central tool will be a formula he derived. Apparently, unaware of the technical details of his work, Kac [142] derived some special cases of his formula and applied it to the h 0 limit. In this section, we will present this formula; in the next, we will show (following Symanzik [270] and Lieb [171]) that classical bounds come from this formula and/or Feynman-Kac together with Jensen’s inequality and in Section 10 (following Kac [138, 142]), that h -+ 0 limits come from these formulas and Tauberian theorems. As a preliminary to Lieb’s formula, we must introduce the BirmanSchwinger kernel. To avoid disagreeing with all other conventions (except --+
88
89
8. The Birman-Schwinger Kernel
[142]), we let N,(V) denote the number of bound states (eigenvalues) of -A V of energy less than -a. We will only take a 2 0 and will mainly consider the case V = - W I 0. Since N,(V) I N,(- V - ) for any V , this assumption will not affect bounds; it will affect limit theorems but the machinery could be extended to accommodate the general case. As usual, H , = -+A, so that N,(V) is the number of bound states of H, + of energy less than -+a. Now, for any V ( I 0) which is H,-form compact, one can easily [257] show that (y > 0)
+
(H,
+
=
-Y4
if and only if ($ = lL‘11’24) K,$
(8.2)
=
where K , is the Birman [15]-Schwinger [233] “kernel”: K,
I
= Vll/2(Ho
+ y)-ll
(8.3)
Vl1/2
Theorem 8.1 (The Birman-Schwinger principle) compact.
Let V be H,-form
(a) Let a > 0. Then N,(V)
(b) If a
=
# {eigenvalues of K ( , j z ) ,> 2)
=
0 and v 2 3 (so that K O = lim, N,(V)
=
, K , exists), then
# {eigenvalues of K O > 2)
Proof (a) Let e,(A) denote the eigenvalues of H , + AV suitably ordered with ei(A) = 0 if N0(2AV) I i - 1. Then, the ei(.)are continuous and strictly monotone decreasing in the region {A I ei(A) < O}. Thus, for a > 0 N,(V)
= = = =
#{ilei(A) < -+a; A = +} #{A(ei(A)= -+a; some 1 < #{AIK(,12)a has eigenvalue A-’ with A < +} # {eigenvalues of K ( l , 2 ) a> 2)
t}
where we use the equivalence of (8.1) and (8.2) in the third inequality. (b) Clearly No( V ) = lim, N,( V ) . But since K , is monotone increasing as a 10,
,,
# {eigenvalues of K O > 2)
=
lim( # {eigenvalues of K , > 2)).
Now consider the operator (y > 0): A(A) = W”2(H,
+ AW + y>- ’W”2
I
90
111.
where W (H,
+
Bound State Problems
= - I/ 2 0. On the one hand, ?)-I
=
(H,
+ AW +
y>-1
+ A(H, + AW + y ) - ' W ( H , + 7 ) - '
so that A(A) = K ,
-
AA(A)K,
or A(A) = K,/(l
+ AKJ
On the other hand, if W is in C z , then, by Theorem 6.6, A(A) has an integral kernel :
1
m
A(A; X, y) = W"2(~)W"2(y)
dt e-Y'e-'(HO+A^W) (x, Y)
0
= W'/2(x)W'/2(y)
We would like a more general formula: F(K,)(x, y )
=
6 s
W'12(x)W1'2(y) dt e-,'
(1:
d p o , x , y ~ t gW(w(s))ds
1
(8.4)
The transformation g N F should be linear and, by the above, should take + Ax)-'. We thus try
g(y) = e-lYinto F(x) = x(l
F(x) = x or in terms of
6
e-Yg(xy)dy
(8.5)
+
Theorem 8.2 (Lieb's formula [171]) Let W E Lq([wv) Lp([wv) with q = v/2 (v 2 3), q > 1 (v = 2), q = 1 (v = l), and q < p < 00, with W 2 0. Let y 2 0 or y = 0, v 2 3. Let f be a nonnegative lower semicontinuous function on [O,m) withf(0) = 0 and letf, g, F be related by (8.5)-(8.7). Then (both sides may be simultaneously infinite):
8.
91
The Birman-Schwinger Kernel
Proof
We first note that since F(x) =
Jorn
e - ' " - ' g ( z ) dz
fpositive implies F is monotone increasing. Thus either formula for y > 0 implies the formula for y = 0 by application of the monotone convergence theorem on both sides ( K , increases as y does, so its eigenvalues increase, and so Tr(F(K,)) is a sum of increasing functions). Similarly, since any lower semicontinuous function is a monotone limit of continuous functions, we can suppose thatfis continuous with support compact in (0, co) to get (8.9). As a final preliminary we note that (8.8) and (8.9) are equivalent: for with respect to the measure dv = dx dpo.x. x : , , the variables o(sl), . . . , o(s,) are a) where s " + " a means identically distributed to o(sl "+" a), . . . , o(s, addition mod r. Thus
"+"
is independent of u so t - yo p(u) = p(0) which says (8.8) equals (8.9). Thus we need only prove (8.8) for y > 0, and continuous .f' supported in some [a, b], 0 < a < b < a.Suppose in addition that W is in C:. We have verified (8.4) for g(y) = e-". Since (by Stone-Weierstrass) sums of such functions are dense in the continuous functions on [0, cx,) vanishing at infinity, (8.4) holds for the g in question. As in the proof of Theorem 6.6, the right-hand side of (8.4) is continuous in x and y, so by a general result, Tr(F(K,)) can be evaluated by setting x = y and integrating (F(K,) is a positive operator with continous kernel ; see [216, 2593). Next suppose that W E La' with compact support. Then we can find W, E C: and a bounded set S so that W, I axs, W, W pointwise almost everywhere and in L"" (v 2 3; otherwise take L2).Then K,(W,) -+ K,(W) in norm, so that F(K,(W,)) -+ F(K;,(W))in norm. It follows (see [259]) that the traces converge if they are always finite (including for F(K,(axs)). Thus we will obtain (8.8) if we can show that the right sides converge. But for each t , and almost every x and o,the inlegrands converge so it suffices to obtain a uniform L' bound on the integrand. Write g(x) = xmh(x) and let da = dx d p o , x , x :as t above. Then (x = zs): -+
so we only need that
92
111.
Bound State Problems
But, by Holder's inequality:
s
x(x) - . . x(o(s,)) I da x"(x) so the above inequality is equivalent to lOmdt e - y c t - v ' z t m < oo
which is obvious if we take m large enough. This establishes (8.8) in case W is in L" with compact support. Finally, let W obey the hypotheses of the theorem. Choose W, in L" with compact support so that W, t W . Since the eigenvalues of K,(W,) are monotone and F is monotone, the left sides of (8.8) converge. The integrands on the right converge pointwise, so it suffices to prove domination by an I,'function. Let h 2 g so that h has support in [c, co) (c > 0), h bounded and h monotone increasing. Then we only need i
= Jomdt e-7'
J~p0,.,.+,h(~~W(o(s) ds) < oo
Jdx
But (8.8) with g replaced by h and F by
I0"
e-Yh(xy) d y
H(x) = x
holds by approximating with Wn's(here the integrals converge by using the monotone convergence theorem). Thus i
=
Tr(H(K,))
Notice that since h(x) = xm4(x)with 4 E L" H(x)
I x m + l l l q l l m ~ (+ m 1)
so i < 00 follows ifTr(Ky) < rn for some m. But the hypothesis on W implies that Tr(K;) < co so long as m > p [216]. I Remark By a further approximation argument, one can easily extend (8.8), (8.9) to arbitrary measurablef's in L:oc with the property that - If(x>l < co lim ~
X l O
and
IxIa
for some
ct
>p
93
9. Phase Space Bounds
or
9.
Phase Space Bounds
The main point here is that of Symanzik [270] and Lieb [171] that to get classical phase space bounds from path integrals, one need only apply Jensen's inequality: Proposition 9.1 (Jensen's inequality) Let f be convex on R (convex (1 - 0)b) I Of(a) (1 - O ) f ( b ) ;0 < 8 < 1, a, b E R). Then, meansf(8a for any probability measure, v, on R:
+
+
so long as j 1 x I dv(x) < co.If v is supported on [ol, co)for some c( > - co and f ( m ) = limy+mf ( y ) , then this last condition may be dropped.
Proof
We first note that by convexity, for any y, we can find a with f(x) -f'(Y) 2 4 x - Y )
for all x ; for example, convexity implies that ( f ( x ) - f ( y ) ) ( x - y)-' is O)/dx exists and convexity monotone decreasing as x 1y, so that df(y implies that one can take this value for a. Take y = x dv(x) and integrate the above inequality with respect to dv(x). Then since J ( x - y ) dv(x) = 0 :
+
s
Jmdv(x) 2 f ( Y )
s
The case where x dv(x) = co is handled by a simple limiting argument.
1
By letting dv be the probability distribution for X , we see that
f(E(X))5 ~ ( f ( X ) )
(9.2)
for any random variable X which is either bounded from below or has E(lXl) <
0.
The basic philosophy of how to get phase space bounds is illustrated by the following theorem.
94
111.
Bound State Problems
Theorem 9.2 Let V be in L:,,(R") and bounded from below. Let H ( h ) -A2/2mA + V . Then
=
Proof' By absorbing the cl into V and/or /I and doing the explicit p integration on the right-hand side, (9.3) is equivalent to
s
Tr(exp( - t ( H o + V ) ) )I(271t)- ' I 2 exp( - t V(x)) d v x
(9.4)
Suppose temporarily the following
which is formally "obvious" from the Feynman-Kac formula. Since e - x is convex and ds/t on [0, t ] is a probability measure, (9.2) implies that
so, using Fubini's theorem : r.h.s. of (9.5) 5 J d p o , , o:
=
s: f
i d ' . exp( - t V(x
+ o(s)))
r.h.s. of (9.4)
where we used the translation invariance of dvx in the first equality and the fact that the resulting integrand is independent of s and o in the last step. Thus all we need is to prove (9.5). We prove the more general: Tr(fe-r(H"+"lf)= Jd'x f2(x> / d p o , x , x , ,exp( -s;V(o(s)) ds)
(9.6)
For f~ CF and I/ E C;, this follows from Theorem 6.6. For V EL" with compact support andfE CF, we then obtain (9.6) by a limiting argument of the type used in Theorem 8.2 and then using the monotone convergence theorems for integrals and forms for arbitrary VE L:,,, V 2 a > - 00, and f~ C;. Using monotone convergence again, we can obtain the result as f-1. I
95
9. Phase Space Bounds
Remurks 1. (9.3) is a celebrated inequality of Golden [ l l l ] , Thompson [275], and Symanzik [270] obtained in [ l l l , 2751 from the inequality Tr(eA'2eBeA'2). The above proof is that of Symanzik [270]. Tr(eA+B)I 2. By simple limiting arguments the result extends to any V with V- - A form bounded with relative bound zero.
The most beautiful phase space bound is on N ( V ) = N o ( V ) . In a semiclassical picture, (2n)-"T,.
Nsc(V ) =
where
T~
J
I {(P? X ) l P 2
= volume of unit ball in R'.
Theorem 9.3
Let v 2 3 and let V E L"'2(R').Then N(V) I u,, sl V _ ( X ) ~ ' d'x /~
(9.7)
for some universal constant u,,. Procllf' Since N ( V ) 5 N ( - V-), we can suppose V = - W Birman-Schwinger principle for any f , F related by (8.7)
0. By the
(since F is monotone)
N ( V ) I F ( 2 ) - ' Tr(F(K,,=,))
by Lieb's formula (8.9). If moreover f is convex, then
f(rot W ( o ( s ) )
ds/J) I
Yo (dslt) f ( t W ( o ( s ) ) )As . in the proof of Theorem 9.2, we can now interchange the ds and &(a) integration with the x integration, eliminate o(s), and then trivially do the s and o integrations. This result is N ( V ) 5 F(2)-
'
~0m(2n)-'/2t-
'I2
dr
s
s
d " x f ( t W ( x ) )= 6" d"x W(X)'/~
where 2"
=
(2n)- "'2
'
-
'/2f(s) dslF(2)
(for change variables from t to s = rW(x)). Notice that for v 2 3, 6, < co for we can takef = 0 near s = 0 andf(s) Cs near infinity. But for v = 1,2,
96
111.
the integral diverges at infinity since lim,,,f(s)/s zero identically. I
Bound State Problems
> 0 so long asf 2 0,fnot
(9.7) was proven independently by Cwickel [40], Lieb [171], and Rosenbljum [224]. Lieb [171], whose proof is given above gets the best value for the constant a, among the three. Namely, if we define
Then we have shown that a, I inf{(2n)-'/'
J:"'-''/y(s)
ds lfis convex and I = J:e-Yl(zy)
*} Y
In particular, by minimizing over f's of the form:
Lieb finds [1761 : a3 I0.116 = a,(L)
(9.9)
There are two natural lower bounds on the precise value ofa,. We will prove later (Section 10) that N(AV)/Nsc(AV)+ 1 as A+ 00. This immediately implies that (9.10) the "classical value." Another lower bound on a, was found by Glaser et al. [lo31 : If N ( V ) < 1 then - A + V 2 0, so taking expectation values in the vector 4 = I V l ( v - 2 ) 1 4 we find that ( p = 2v/(v - 2))
< 1 3 lIg/Pdvx 5 JIV+l2 d'x
~ ( - 1 4 1 4 ~ 2 ) )
But, 1141pd'x
=
1I V1'12d
"SO~
Jl+lpdvx < a ; ' =
J141pdVxI JIV#12 d"x
or (using the freedom to scale $), for any 4,
9.
97
Phase Space Bounds
Thus a, yields a constant in Sobolev’s inequality; i.e., if c,
= SUP{lldllPI IlV4llZ = 1; P - l
=
3
-
l/v)
is the best constant in Sobolev’s inequality; then a,
2 c:: = a,.(s)
(9.1 1)
-
The bounds (9.10) and (9.11) correspond to the extreme cases N ( V ) 0 or N ( V ) 00. Lieb and Thirring [176] made the following natural conjecture:
-
The (v 2 8), (v I7) results follow from looking at the precise values of c, and z, all of which are explicitly known. The value a3(L) in (9.9) is fairly close to what should be the exact value since a,(L)
=
1.49a,(s)
(or put differently 0.077 I a3 I 0.116). Recently, Glaser et al. [lo21 have shown by explicit examples that the Lieb-Thirring conjecture is false for v 2 8; i.e., a, is strictly larger than a,(c) [which is the larger of a,(c) and a,(s) for v 2 81 ; in their examples it is never larger by more than 36%. On the basis of their work, they make the weaker conjecture a, = a,($
(v 5 7)
lim a, = I ~
v- w
av(c>
Glaser et al. also prove that for v
=
4 and V spherically symmetric
(strongly supporting the Lieb-Thirring conjecture for v _< 7). Additional information on best constants in the moment inequalities can be found in [O].
*** For further applications of Brownian motion to bound state problems, see [260a].
***
98
111.
Bound State Problems
We want to explain, following Lieb-Thirring [175], how (9.7) leads to a proof of the stability of matter. Since the modification that electron spin makes in the Pauli principle is easily accommodated, we will ignore its effect. W e emphasize that the remainder of this section is an aside not used in the later sections. Fix N and M and consider the operator on L2(R3N).Let N
H,(R,, ..., RY) =
-
N
ZAi
i= 1
-
c
+
c
M
1 Iri - Rj1-’
i=l j=1
Iri - rj1-l
+
1
IRi - RjlP1
l
lSi<jSN
where R,, ..., RM are parameters in R3, and a point in R3N is written (r,, . . . , fN), ri E R3. H N leaves invariant the subspace L: of functions I&,,. . . , r N ) antisymmetric under interchange of the ri)s, so we define
E(N, M )
=
inf{inf spec(HN(Rj) 1 L:) I all Rj}
i.e., the ground state energy of N electrons and M infinitely heavy protons. Stability of matter says that
+M)
E ( N , M ) 2 -c(N
This was first proven by Dyson-Lenard [68] whose proof was difficult and whose value for c extremely large 10” in units where c = 1 is presumably correct). The proof we are about to give due to Lieb-Thirring [175] is not only a considerable simplification, but the constant is only about 30 times too large. It depends on some facts in the Thomas-Fermi theory. Define a functional on p’s in L513with p 2 0 by (5-
€!&pi
R , , . . . , R,)
=M
s
+
p513(x)d 3 x -
;
s
W ( X ) ~ ( Xd3x )
JdX)P(Y)lX - Y I- d3X d3Y
+
IRi - Rjl-’ l
cEl
with W(x) = Ix - RiI-’. The terms in € “approximate” (Y, H N Y )in a suitable semiclassical approximation. We will prove below that €!&)(P; R,,
. . . , RM)
2 -d(a)M
(9.12)
where d(a) is an a-dependent constant; this stability of matter in the ThomasFermi approximation comes from an elementary effect: there is no binding of “molecules” in the Thomas-Fermi theory.
9.
99
Phase Space Bounds
Let Y be a normalized antisymmetric function on R3Nand let (9.13)
We are heading towards a proof of the fact that, for a suitable ci and 4 : (Y, HN(R1,. . . , RM)'l') 2 &&)(py; R,, . . . , RM)
-
4N
(9.14)
whence stability of matter will follow from (9.12). (9.14) follows from
(-
(Y, (Y,
c Iri
i= I
(9.15)
Ai)Y!) 2 a0 [Py3(x) d 3 x
- rjl-
p4I3(x)d 3 x - d($ci,)N
'Y
i<j
(9.17) [for one can take c( = +a0 and 4 = d(&,) in (9.14)]. Equation (9.17) is obvious. The inequality (9.16) comes from (9.12) by the following clever device [175]: Take p = pyI and take N = M and notice that p y ; R,,
. . . ,RN)lY(Ri,. . . , R, 1' dR1
' . . dRN
r
1
+j
/p~(x)pp(y)Ix - y1-l d 3 x d 3 y
+ 1.h.s of (9.16)
(9.18)
But, by (9.12), the left-hand side of (9.18) is greater than or equal to -d(cc)N. This leaves the proof of (9.15) and it is here that the semiclassical bound (9.7) enters: Theorem 9.4 ([175]) Let ej( V ) denote the negative eigenvalues of - A + V . Let v 2 3. Then, for any k > 0 and I/ E L ( 1 / 2 ) v + k :
1 Iej(V)lk I b,(k) with b,,(k)= ka, functions).
sA y k - ' ( l
- y)""
I
d"xl V-(X)~~+('/~)"
(9.19)
d y (which is expressible in terms of
r-
100
111.
Bound State Problems
Proof’ By a limiting argument, it suffices to prove (9.19) for V E C c with VI 0. It is not hard to see that
so that (9.7) implies that Nu( V ) I a,
I
I( V
+ a)-
lvj2
d”x
(9.20)
In terms of Stieltjes integrals:
1Iej(V)Ik = =
k
s
cik
I
dN,
c t k - l N , dci
(integrating by parts)
by (9.20). We obtain (9.19) by scaling (i.e., change variables from
tl
to y
=
4- V ( x ) I > . I Remark There are also bounds of the genre of (9.19) for v
=
2, k > 0 and
v = 1, k > *(see [176]):
Theorem 9 5 ([175]) Let Y(rl, . . . , r,) E L2(R’”) be normalized and antisymmetric with respect to its ri E R”variables. Let py be given by (9.13). Then (H,= Er= - AJ (Y,H o Y ) 2 1, J p bv + 2)/V d”x
where I ,
=
b,(1)-2”(2/2
(9.2 1)
+ v)(~+”)/~(v/~).
xr=
Proof’ Let V(x) = py(x)2/”. Let H , = ( - Ai - AV(x,)). Then, one can write down the eigenfunctions of H , explicitly in terms of eigenfunctions of - A - AV(x) and so determine that H , rL; 2 -ELl l e j ( - i V ) l , so by (9.19)
(Y, H N Y ) 2 -b,(l)A1+(l‘z)v
9.
101
Phase Space Bounds
or (Y,H o Y ) 2 [A - by(1)A1f(1/2)V] Jp$’2”‘(*)
d”x
(9.21) follows from a maximization with respect to A. [ We summarize what we have accomplished [modulo the proof of (9.12)]: Theorem 9.6 stability of matter ([68, 1751) There is a universal constant c so that for any normalized antisymmetric ‘P in L 2 ( R 3 Nand ) any R,,. . . ,
R,
E
R3
2 -c(N
+ M)
(9.22)
For further discussion, including how to go from (9.22) to a proof of noncollapse as N , M + 00, see Lieb [172]. It is very easy to see that E(N, N) I -dN
for a suitable d > 0 and thus - E ( N , N ) grows like N (indeed, limN+mE ( N , N ) / N exists and is strictly negative; see [172]). If the antisymmetry restriction is removed, then the behavior is very different. Namely, let E , ( N , M) be defined analogously to E ( N , M ) , but with L; replaced by L2. Then, various methods, including those used above, show that E , ( N , N ) 2 -c‘N5’3
and Lieb [172a] has proven that E , ( N , N ) I -d”5’3
Therefore, if electrons were not fermions, bulk matter would collapse and we would not be here! We note that if the protons are given a finite mass, then the analog of E , ( N , N ) is believed to behave like N7I5 rather than N5I3,but a lower bound of this form has not been proven.
*** All that remains is a proof of (9.12). We sketch the argument; for details, see Lieb-Simon [173] whose proof of (9.12) uses heavily ideas of Teller
102
111.
Bound State Problems
[274]. By a scaling argument, the ci in the definition of &$) can be absorbed, so we consider only the case CI = 3 ; i.e., &,(p; R,,. . . , R ,)
'I
=-
5
pSi3(x)d'x
+ 51 Jp(x)p(y)lx W(X) = 1 I X - Ri1-l
-
I
W(X)~(X) d3x
-
y1-l d 3 x d3y +
C IRi - Rj1-I i<j
(9.23)
We will consider trial p's in
It isusefultoconsiderthenorms II.IlS1,and llpll+ on F. Lemmu9.7
=
[jp(x)Ix
-
~l-'p(y)]~'~
Fix R,, ..., R,. Then, there are linear functions I, and
I 2 on F with lIl(P)l 5 aMIIPlI5/3;
1/2(P>l 5 bMllpll.
(a and b independent of M and Ri) so that
Proof It is easy to see that one need only consider the case W(x) (i.e., M = 1, R, = 0). Let
=
Ix I
i.e., the potential due to a charge of one uniformly spread in a sphere of radius one. Then W2(x) = [ X I - ' for 1x1 2 1 and W2(x) 5 1xl-l for 1x1 < 1. Thus W, = W - W2 is in so I,@) = 1W,(x)p(x) d'x is a bounded functional on L5I3.By the positive'definiteness of I x - y I-' and the Schwarz inequality:
9.
103
Phase Space Bounds
Theorem 9.8
Fix R,, . . . , R,. Then
E(R,, . . . , R,)
inf &,(p;
=
R,, . . . , R,)
(9.24)
P S Y
is finite (and bounded from below independently of Ri but depending on M ) . Moreover, there is a unique p in cFfor which the infimum in (9.24) is realized. This p, denoted by p(x; R,,. . . , R,,,,), obeys: 4(x; R,, . . . , R,)
= p =
Proof
W(x)
-
sp(y)jx - yI-'d3y 2 0
(9.25)
4312
By the lemma
&,(PI
2 $llPll;j:
- aMllPll5/3
+ 911Pll:
-
bMllP/l+
is clearly bounded from below. Moreover, 8, + +a as either llpllSi3or llpli + + co so if we choose a sequence pn with b,(p,) -+ E(Rl, . . . , R,), then supllp,ll 5,3 < rn and sup/lp,Il+ < ixI. Thus, by the Banach-Alaoglu theorem, the infimum in (9.24) is taken on a set S which is compact in the topology of weak convergence, i.e., the topology in which pa -+ p if and only if I(p,) -+ l ( p ) for all I's continuous in either 11 + or 11 5 1 3 . Suppose that pa + p in this topology. By the lemma, p,(x)W(x) d3x + p(x)W(x) d3x. Moreover, by the Hahn-Banach theorem, limllpaII + 2 llpll+ and limllp,llsi3 2 llp115,3.It follows that lim b,(p,) 2 b,(p); i.e., 8, is lower semicontinuous. Since every lower semicontinuous function on a compact set takes its infimum, Its uniqueness follows from the strict convexity there is a minimizing p in 9-. of 6;i.e.,
6,(h
+ (1 - O ) P 2 )
< O&,(Pl)
+ (1
- O)&,(p2);
0 < 0 < 1; p1 #
P2
(which is easy to check). Since p(x; Ri) minimizes b,, it must obey the Euler-Lagrange equations d&,/dp(x) = 0 at points where p(x) # 0 (d&,/dp(x) 2 0 at points where p(x) = 0). Thus p
= [max($, O ) p 2
where 4 is given by (9.25). Thus, all that remains is the proof that Notice that the lemma shows that
4 2 0.
for any po. This implies that 4 - W is bounded. It is also continuous since near any xo # Ri it is the sum of a harmonic function and the convolution of
104
111.
Bound State Problems
1xI-l with a i?E Lmhaving compact support. Moreover, if we write p = p1 + p o where p1 has compact support and p o has small 11. II + and 11. norm, then we see that 4 - W 0 as I x I co (for pl(y) I x - y I - d 3 y + 0 as x + 00 since p1 EL'). Thus 4 + 0 at infinity. Clearly 4 + + 00 as r + Ri. Now, let -+
S
=
-+
s
{x I4(x) < 03
which is open by continuity and disjoint from the R i . Since A 4 = 4np (distributional sense) and p = [max(+, O)I3/*, 4 is harmonic on S, and therefore it takes its minimum value at aS u { 00). But clearly 4 = 0 on as and 4 -,0 at infinity, so the minimum value on S is zero. This is clearly impossible unless S = Qr. 1 Remarks 1. The beautiful argument that 4 2 0 which is reused below is due to Teller [274]. 2. By more work [173], one can show that 4 > 0, and that 6,p are C" away from the Ri and that p(x) dx = M .
Now we generalize the problem slightly by adding parameters zi 2 0, and let
+ 1 /p(x)p(y)~x -
Y I - 1 d3X
d3y
+ c z i Z j ~-~ Ri ~ I -1
i< j
xzl
with W,(x) = zilx - Ri1-'. As above b M ( p ; Ri, zi) has a minimum E(Ri; zi) and a minimizing p(x; Ri, zi) and corresponding $(x; Ri, zi). Lemma 9.9 region zi > 0:
E(Ri, z ~ ) ~ , ,-+ E(Ri, z ~ ) ~ , , - as z, aE -(Rj;zj)
azi
=
+ 0.
Moreover, in the
lim[q5(x;Rj,zj)-ziIx - Ri1-'] x+Ri
Proof The first assertion follows easily from the bounds in Lemma 9.7. Formally the second assertion comes from writing
and noting that the ap/azi terms are multiplied by aS/ap = 0 since p is a minimum. The terms from a€/dzi give precisely Ix - Rilp(x) d3x + zjJRj- RiI-' which is the limit in question. The rigorous points of this formal proof may be found in [173]. I
cjsi
-s
105
10. The Classical Limit
For fixed x, R , , . . . , R,, 4(x; Ri, zi) is
Lemma 9.20 (Teller’s lemma) monotone increasing in zi.
Proof Let Zi 2 zi 2 0. Let S
= { X ~ + ( X ; Ri,
Zi) < 4 ( ~Ri, ; zi)}
S is open and avoids those Ri with Zi > z i . On S, the function II/ = 4(.,.Ti)4(., zi)obeys:
A$
=
471[~(. Zi) - P ( . zi)l .)Zi)3/2 - 4( .)Zi)3’2] < 0 9
7
= 471[4(
so $ is superharmonic and thus it takes its minimum on dS u {a}. Since rC/ = 0 on dS and II/ -, 0 at infinity, this minimum is zero so S = 0. I Theorem 9.11 (Teller’s theorem)
E(R1,. . . , RN+M; 2 1 , . . . , z N + M ) 2 E(R1,. . . ,RN; ~ 1 ,. . , zN) + E(RN+ 1, . . . RN+M ; zN+ 1, . . . ZN+M) 2
3
(9.26)
Proof Let AE = 1.h.s. - r.h.s. of (9.26). By the continuity as zi -, 0, we need only prove dAE/8zi 2 0 in the region zj > 0. But, by Lemma 9.9, the derivative is a difference of 4’s which is nonnegative by Teller’s lemma. I Remark Teller’s theorem says that molecules do not bind in the ThomasFermi theory. For a physical “explanation” of this, see [274, 1731. Corollary 9.22 [Equation (9.12)]
E(R1,. , . ,R,; zi
=
=
1) 2 - d M
Proof By Teller’s theorem, E(R,, . . . , R,; z i = 1) 2 - d M since E(R, z = 1) is independent of R and finite.
I
E(Ri, zi = 1)
10. The Classical Limit
In the last section, we obtained classical bounds by managing to replace (with respect to dpo,o,o,,)by x. In the limit as t -, 0, one expects w(s) to go to zero so that classical limits should be connected to t -, 0 limits. This idea of Kac [138,142] will dominate this section. For another approach x
+ o(s)
106
Ill.
Bound State Problems
to the classical limit, see [252]. For a discussion of the classical limit for the pointwise solution of the heat equation, see the end of Section 18. The simplest result to prove is the folllowing. Theorem 20.2 Let
Let I/ be a continuous function on Rv with exp( - B V )
E L’.
ZQ@)= Tr[exp( Z,(h)
=
- B [ - A;
+ V])] +)]).(/I
Jdvp __ (2nR)’ d“x exp( - fi[&
Then (10.1)
Proof Let W = PV. By doing the p integration explicitly, (10.1) is seen to be equivalent to: t””Tr[exp( - t [ H ,
+ t-
W])]
+
I
( 2 ~ ) - ” / ~exp( - W(x)) d”x
as t LO, where H , = -+A as usual. By (9.5), we need to show that
obeys f(t)t’”
I
+ (27~)-”/~exp( - W(x))
d”x
= fo
(10.2)
By Theorem 9.2,f(t)tV” I fo for all t so limf(t)t”/2 Ifo. Given 6, let At be the hypercube of side 6 centered at x. Let W,(x)= max(W(y)lyEAf) and let Qv(6, t ) = E(b(s)EA;, all 0 I s I tlb(t) = 0). Then, taking the contribution of those paths which stay in At, f ( t ) 2 (2nt)-””Qv(6,t ) /exp( - W , ( x ) ) dx
so
lim t’”f(t)
2
[lim
I
Qv(6,r ) ] ( 2 ~ ) - ” / ~exp( - W,(x))dx
107
10. The Classical Limit
Now Q'(6, t ) = Q1(6, t)' by the independence of the components and as in the proof of Lemma 7.10,
Q '(6,
t ) = P,:
0; t )
where PD has Dirichlet conditions at x = k 6. By the method of images (see Proposition 7.17), we have that for each fixed 6, Q1(6, t ) 1 as t 10, so lim t ' / 2 f ( t ) 2 ( 2 7 ~ - ' ~exp( ~ - W,) d'x. Now let 6 + 0. Since exp( - W,) 5 e&( - W ) and W,(x) --* W ( x ) pointwise (by continuity), exp( - W,) d'x --* exp( - W ) d'x. I
s
-+
s
Remarks 1. Results of this genre, proven by different means, go back at least as far as Berezin [12]; see also Combes et al. [37]. The method of Lieb 11701 can also be extended to prove this theorem [260b]. 2. One can replace the continuity assumption by one that V is bounded from below and locally L' if one uses the fact that for any L'-function W (which will be V restricted to a finite region)
as 6 + 0 for almost everywhere x.
*** The remaining classical limit results will involve information on the growth of the number of eigenvalues of H , + V which are less than E as E --* co,or of H , + AV which are less than zero as A + co.The latter case is the number of eigenvalues of K;:, less than A -+ m. In both cases, we will want to relate the divergence of dim E(o,a)(,4)as a -+ cc for some operator to the divergence of Tr(e-'A) as t 1 0 since it is the latter that we will compute using (9.5) or using Lieb's formula. More generally, we want to relate the divergence of dp as a + co to that of eCrx d p ( x ) as t 10. One direction of this relation is easy:
s:
Theorem 20.2 (Abelian theorem) Let d p be a (positive) Bore1 measure O ,+ C as a --* co. on [0, co)and suppose that for some y 2 0, C 2 0: U - ~ ~ [ a) Then
lim t~ J e - 1 ~d p ( x ) = 110
where r(u) =
:s
'
e-xxa- dx.
cr(y+ 1)
108
111.
Bound State Problems
Proof Let G(a) = (a + I)-? j: dp = (a + l)-?F(a). By hypothesis, G(a) C as a + co and go = sup,I G(a)l < co. Then, by a Stieltjes integration by parts, +
t y Se-l" dp(x) = t y + ' Je-'"F(x)
- tY+'
s
e-'"(x
dx
+ l)YG(x)dx
+
+
F o r t d 1, e - Y ( y + t)?G(y/t) d e-Y(y l)ygois in L ' ; as t + 0, (y t)?G(y/t) + CyY for y fixed, so the result follows from the dominated convergence theorem. 1 The converse direction of this last theorem is much deeper. It depends critically on the positivity of d p (for example, if dp(x) = (- 1)"6(x - n), then j; d p does not have a limit but j e-'" d p = 1/(1 + e-') --t as t + 0). Fortunately, a beautiful argument of Karamata [1481 exists which makes the proof fairly easy:
c:=,, +
Theorem 20.3 (Tauberian theorem) Let p be a (positive) Bore1 measure on [0, co) and suppose that e-'" dp < co for all t > 0 and that for some y 2 0, D 2 0 :
lim t? s e - ' ~cip(x) = D 110
Then (10.3)
Proof(version of Karamata [148] due to M. Aizenman, unpublished) If y = 0, the result follows from the monotone convergence theorem, so suppose henceforth that y > 0. Let dpf be the measure given by pt(A) = typ(t-'A) and let dv = x y - ' dx (so that vf = v). (10.3) says that lim p,[O, 1) = cv([O, 1))
(10.4)
1-0
where c
=
D/T(y). Suppose that we can show that lim f+O
s
f(x) dpl(x) = c
s
f(x)
dv(x)
(10.4')
109
10. The Classical Limit
for allfE C?[O, m). Then using the fact that ~ ( ( 1 )=) 0, one easily sees that (10.4) holds (see, e.g., Proposition 17.2). By hypothesis lim t+O
s
e-” dpr(x) = c
s
e - x dv(x)
so the measures e-” dpr are uniformly bounded. Thus (10.4‘) follows from
lim f-rO
s
e-”g(x) dpL,(x)= c
s
g(x)e-” dv(x)
(10.4”)
for a dense (in 11.11 set of g in C,[O, m), the continuous functions going to zero at infinity. But ( 1 0 . 4 ) holds for g(x) = e - n x by hypothesis. Since polynomials in e-” are dense in C,[O, m) by the Stone-Weierstrass theorem, we have proven ( 1 0 . 4 ) for the required set. Remark The above results are called “ Abelian ” and “Tauberian” a,6(x - n ) and we because of the earliest prototype results. If dp(x) = let I = e-‘, then lim, j e-‘” dp(x) = lirn, a,P in which case Theorem 10.2 with y = 0 just asserts that if a,, is absolutely summable then its “Abelian sum” is its ordinary sum. This is a famous result of Abel. Tauber was the first to consider converses in this case. Results of the genre of Theorem 10.3 were first obtained by Hardy-Littlewood but only with considerable effort. There is one more result related to the above which we will need below:
c:=o
Proposition 10.4 Let dp be a positive measure on [0, 00). Suppose that for some y 2 0, C 2 0 ~ = lim a - Ioadip(x)
c
a+ m
Then ra
Proof
Let F(a) = j:dp(x) and G(a)
=
j:
x dp(x). Then
110
111.
Bound State Problems
Letting H ( a ) = (a + l)-YF(a),we see that (as in Abel's theorem) a - y - ' I i F ( y ) d y = IolH(ya)(y
as a
+
+ ~ - ' ) ~ d yC(y + 1)-' +
co (by the dominated convergence theorem).
*** We can now return to the "classical" limit problems. Results of the following genre go back to Titchmarsh [276]; this kind of proof goes back to Ray [212]. Theorem 20.5 Let V be a continuous function on R", going to infinity at infinity, so that for some y > 0 and some C > 0
lim(271)-'~-~I{(p, x) I +p2
+ ~ ( x I) E } I = c
E-03
Suppose, moreover, that lim(271)-"~-~I {(p, x) I +p2
E+ m
+ &(x) I E } I = Cd
(10.5)
exists and limb Cd = C where V,(x) = max{ V(y) I y E A:, the hypercube of side 6 about x}. Let H = -+A + V and let n(E) be the number of eigenvalues of H less than E. Then lim K Y n ( E )= C E-m
Remark For reasonable Vs (e.g., polynomials), the quantity in (10.5) is &independent and equals C.
Proof By the Tauberian theorem, Theorem 10.3, we need only show that lim tYTr(e-'H) = Cr(y
+ 1)
(10.6)
110
Now, as in the proof of Theorem 10.1, Tr(e-'H) 5 (271)-'
J
exp[ - t o p 2
Tr(e-fH) 2 Qv(S, t)(271)-"
I
+ V(x))] d'x
exp[ - t ( i p 2
d'p
+ V,(x))]
d'x d'p
so that (10.6) follows by using the hypotheses and the Abelian theorem, Theorem 10.2. 1
111
10. The Classical Limit
Rernurks 1. Interestingly enough, Theorems 10.1 and 10.5 are intimately related. Both involve the behavior of Tr(e-'(Ho+a"))as t 10. In one case, A is fixed, and in the other, it is taken proportional to t - ' . 2. Using Proposition 10.4, one can immediately control the divergence of the sum of thejth powers of the eigenvalues of H which are smaller than E.
We can also recover the famous result of Weyl [285] : Theorem 20.6 Let R be an arbitrary open set in R" with laRl = 0. Let H t , D= (the Laplacian with Dirichlet boundary conditions in an).Let nn(E) denote the number of eigenvalues of Hf,D which are less than E. If the Lebesgue measure, I Rl, is finite, then
-+
) 2-v'271-v~vlRl lim E - " ' 2 n n ( ~ = E- m
where z, is the volume of the unit ball in R". Remark See Section 21 for the definition of for general R. We assume here the relevant Feynman-Kac formulas proven there and in Section 22.
Let P,(x, y; t ) be the integral kernel for on R x R so that Proof
c7-'Ho,D
which is continuous
By a Tauberian theorem we need only show that as t 10, tvi2
lPn(x, x; t ) d"x
-+
(27~-~'~1RI
because
Now let Po be the P for R P,(x,
=
R'. Then, since (see Section 21)
E{x + b(s)ER; 0 5 s 5 t I b(t) I P,(x, x; 1 )
X; t ) =
=
O}PO(x, X; t )
(10.7)
and C ~ ' ~ P x; ~ ( tX )+ , ( 2 7 ~ - " ' ~we , need only prove that for almost every x E R: t"'ZPn(~, x; t ) + ( 2 7 ~ - ' ' ~and apply the dominated convergence theorem. But clearly, by (10.7), if A: c R:
QY(B7t)Po(x, X; t > I Pn(x, x; t ) I PO(X,X; t ) so the result is proven since Qv
-+
1 for any 6 > 0.
I
112
111.
Bound State Problems
Remarks 1. Actually, Wiener integrals, per se, are irrelevant to this proof; all that is relevant is the Tauberian theorem and the inequality for (A: c 0): P,;(x, x ; t ) 4 P,(x, x ; t )
s P,(x,
x; t)
which can be proven using potential theory. In this form the proof is just that of Kac [140]. ~ t 10. By 2. In the above, we established that P n ( x , x ; t ) ( 2 7 ~ t ) - " /as related means, Kac [140] obtained the first few terms in an infinite asymptotic series Pn(x, x; t ) (27~t)-"'~ an(x)tn. By very different means, Kannai [147] has obtained asymptotic series
-
-
pn(x, y; t )
-
m
( 2 7 ~ t ) - " /exp(~ I x - y12/2t)
1 b,(x, y ) t n n=O
(with bo = 1) for very general x and y (e.g., if R is convex, all x and y). 3. There is a version of Weyl's theorem for unbounded regions due to Majda-Ralston [179a]: it involves the asymptotics of the S-matrix for acoustical scattering. The proof that Jensen-Kato [136a] gave of their result relates it directly to the ideas above; while they do not use path integrals for their estimates, one can. Theorem 20.7
Let V 5 0 be in L"/2(R")with v 2 3. Then lim A-"'2N(AV)
=
(27~)-"z,
A+ m
1
V ( X ) ~ "d"x '~
Proof We use Lieb's formula, (8.Q with g(y) = 1 (respectively, g(y) = 0) if y > p (respectively, y I p) so that F ( x ) = x j$ eCYg(xy)d y = xe-B'x. Thus letting p j denote the eigenvalues of K,=o, we see that
(10.8)
x E(
V(x
+ b(op))( do > 1
1
b(pz)
=
)
0
113
10. The Classical Limit
As B -+ 0, E(S', . . . I b(Bz) = 0) is easily seen to go to zero (respectively, one) if V(x)r < 1 (respectively, V ( x ) z > 1) if V is continuous with I {y I V(y) = a } I = 0 for all a. Thus for VE C: (in this case, the reader can provide the necessary justification of the interchange o f f d'x dz and lims, o), we see that
Applying a Tauberian theorem to the measure dcc(x) =
we find that as A
-+
c
P j 6 ( X - p;
'1
cc
so applying Proposition 10.4:
But, by the Birman-Schwinger principle (Theorem 8.1)
N(AV) = # {p;
< +A}
which proves the result for nice V's. A limiting argument using Theorem 9.2 ([249]) handles the general case. 1 Remarks 0)
o! >
1. An identical proof shows that ( v
=
1, 2 is allowed for
114
111.
Bound State Problems
c
From this formula and Iej(V)Ik = k c r k - N .(V) dcr, one easily obtains limit theorems for 1Iej(AI/)Ikas A -+ 00. 2. In the next section we will need the y # 0 analog of (10.8). Namely, if p j ( y ) are the eigenvalues of K , and v is arbitrary (y > 0), then
x E(Sd1 V(x
+ b(s))l
1 )
ds > 1 b(t) = 0
(10.9)
3. Results of the genre of this theorem using rather different methods go back to Birman-Borzov [16], Martin [lSl], and Tamura [273]. The above proof is due to Kac [142]. 4. The limit here should be classical since N(AV) is the number of bound I/ with h = A-l" so A + co is the same as h + 0. states of -h2A
+
11. Recurrence and W e a k Coupling
There is a striking difference between quantum mechanics in v 2 3 and E L"", then for v 2 3, M(AV) = 0 for A sufficiently small (this follows from Theorem 9.3, if we note that N < 1 implies N = 0). On the other hand, if v = 1, 2 and V I 0 (V 0), then N(AV) > 0 for all A > 0. There are a variety of ways of seeing this: v I 2 dimensions. If V
+
+
(a) If V is a spherical square well, one can solve ( - A AV)u = Eu explicitly in terms of Bessel functions. (b) (P. Lax, unpublished) One can construct rather simple trial functions u with (u, ( - A AV)u) < 0 in case V is a spherical square well. (c) (Simon [251]) An elementary trial function argument shows that llKJ -+ co as y -, 0. Since K , is compact, positive, and self-adjoint, its norm is an eigenvalue. Thus for any fixed A, K , has at least one eigenvalue larger than 2A-I for all small y. (d) (Simon [248]; see also [17, 1611) One can develop a theory which gives explicit "series" for the actual weak coupling eigenvalue and which shows at the same time that such an eigenvalue exists.
+
Given all these relatively simple proofs, the considerations below have something of the character of using a sledgehammer to crack a peanut but the swing is still somewhat illuminating.
115
11. Weak Coupling
First, we mention some poetry of Kac [143] which has not yet been made into a rigorous proof. Let I/ be in C;, negative and strictly negative on some open set S, say, V < - u on S(u > 0). Then, if H = -+A + AV, andf'(x) = 1 : ( e p f H f > ( O=) I e x p ( - 2 /:V(b(s))
d s ) Db
2 Iexp(uAl{s 5 t ( b ( s ) E S } I ) D b
Recurrence (Theorem 7.13) tells us that I {s I t I b(s) E S}I + co as t + almost everywhere in b, for v = 1 or 2 and thus in those dimensions (eCtHf)(0) + 00 as t + co.Thus, as a map from L" to L", lle-fHlldiverges as t + co no matter how small A is. The occurrence of a negative bound state is equivalent to lle-'Hll diverging as a mapfrom L2 to L2. Unfortunately, being unbounded as a map from L" does not imply that the semigroup is unbounded on L 2 ; see [260a] for further discussion. A proof that llKyll + co as y 10 can be based on (10.9) as follows below.
Theorem 22.2 Let v = 1 or 2. Let V = - 1 on some cube C, 0 off C. Let p j ( y ) be the jth eigenvalue of K ( y ) , counting from the largest downwards. AV has at least one bound state Then pl(y) -+ co as y 1 0. In particular, H , for any A > 0.
+
(zjm,,
Step 1 p j ( y ) e - 1 ' p ~ (is 7 )bounded as y 10) Let H , (respectively, H 2 ) be -*A on L 2 ( C ) [respectively, L2(R"\C)] with Neumann boundary conditions on aC. Then (see, e.g., [217])
(H,
+ y)-'
+ + y)-' = ( H I + y)-' 0(H2 + ?)-' ( H , + y)- 00. Thus, if en is the nth eigenvalue of H , ,
5 (Hi
H2
It follows that K , I listed in increasing order, p j ( y ) I ( e j + y)CjliVwe conclude that
'. Since only el = 0 and e j
-
a,
lim
C p , (1.1 J
<
~ - ~ / P J ( Y )
(11.1)
00
710 j = 2
The strategy now is to use (10.9) to show that
c''=, . . .
+
co so that p,(y) -+ a.
Step 2 (independence of finite times from infinite time) Let A be an event depending only on {b(s) 10 I sI t o } .Then
E ( A I b(t)
=
0) + E ( A )
as t
+ 00
(11.2)
116
111.
Bound State Problems
For E(A) =
and for t > t o : E ( A 1 b(t)
I
E ( A I b(to) = ~ ) ( 2 n t ~ ) exp( - " ~-+x2t01) ~ dvx
0) - 1E ( A I b(to) = ~ ) ( 2 n t ~ ) exp[-+x2(t;' -"~~ 1(2nt0)-"'' exp[ -+x2(to1 + ( t
=
+ ( t - to)-')] dvx -
to)-')] d"x
on account of the fact that b(t) - b(to) is independent of {b(s) 10 I s I (1 1.2) now follows from the monotone convergence theorem. Step 3 (recurrence for paths conditioned on b(t)
=
to).
0) We claim that in
v = 1 or 2 dimensions, for each x :
E(J:l V(x
as t + 00. Clearly for t > E(I:l V(x
to
+ b(s))l
2 E(l 'J:
Fix t o , take t lim E(
-, 00
V(x
I ) 1 ) 1 )
+ b(s))l ds > 1
b(t) = 0
-, 1
ds > 1 b(t) = 0
V ( x + b(s)) I ds > 1 b(t)
and use (11.2):
+ b(s))l ds > 1
(11.3)
E(I':/
=
V(x
0
+ b(s))J ds > 1
t-+ a,
Now take t o + co and use Theorem 7.13 to get (11.3). Step 4 (completion of proof) Looking at (10.9) and using (1 1.3), one immediately sees that for v = 1, 2
as y 10. Thus pl(y) -+
00.
I
Remarks 1. The proof actually shows much more. Namely
This is actually the precise behavior (see [248]). 2. The reader might think that v = 1,2 enters critically in the above proof in Step 3. This is actually wrong, for while (11.3) will not hold if v 2 3, the
117
11. Weak Coupling
lim-inf will have a nonzero value, and that was all we really needed in Step 4. The crucial place that v = 1,2 enters is in the t K V i 2at infinity. Thus, in some sense, the above proof does not so much derive weak coupling bound states from recurrence but rather shows they both come from a common cause (compare Proposition 7.17).
*** Although we have given a path integral proof that for v 2 3 and V "nice" one has that N@V) = 0 for 1 small, it is not a very direct proof via recurrence. Suppose that W(x) = - V(x) is the characteristic function of { x I 1x1 I l}. We have already remarked in our discussion of recurrence that (see the Remark following Proposition 7.18)
and one sees by a similar argument that supE( r"W(b(s) x
+ x) ds
\Jo
) < co
(11.4)
On the other hand, suppose we know that a
= sup E x
Then
A ( [J: exp
allf
W(b(s)
+ x) ds])
< 00
(1 1.5)
IIm
Thus e-t(Ho+'")is bounded by a as a map from L" to L". By duality and interpolation it is bounded by a, independently oft, as a map from L2 to L2. We conclude that H , + AV 2 0 if (11.5) holds, so N(AV) = 0. At first sight it seems unlikely that (1 1.4) implies (1 1.5) since E(exp(1X)) < co is much stronger than E ( X ) < co. That makes the following result of Portenko [204a] especially striking: Theorem 22.2 ([204a])
Let W 2 0 be measurable. If
118
111.
Bound State Problems
then supE(exp[l:W(b(s) X
+ x) d s ] ) I(1 - y)-'
< co
Proof Expanding the exponential, it suffices to show that
I
E(W(b(s1)
0 <s, <
"' < S " <
m
+ x)... W(b(s,) + X))dsl...ds,
I y"
(11.6)
since we can use the monotone convergence theorem to interchange E and the expansion. Fix 0 < s1 < ... < s,_ Let b(t) = b(s,_, + t) - b(sn-l). Then
s.. l m
E(W(b(s1)
=
+ X) . . . W(b(s,) + x)) ds,
+
~oxE(W@(sl) x)... W(b(s,-,)
I YE(W(b(s1) + X) . . . W(b(s,-
1
+ x)W(b(t) + b(s,-,) + x))dt
+ x)))
since bis independent of b(s,), . . . ,b(s,- l). (1 1.6) now follows by induction. Remarks 1. If viewed properly, one sees all that was really used was that b has the Markov property. 2. This result was rediscovered by Berthier-Gaveau [13] from whom we learned it. The proof is somehow a probabilistic version of some of the ideas of Kato [lSl] (see also [247]). 3. The above results can be rephrased in L p language as saying that if (-A)-'W is a strict contraction on L1, then N ( V ) = 0. This can be seen without recourse to path integrals as follows: By duality W(-A)-' is a strict contraction on L", so the Stein interpolation theorem applied to F ( z ) = Wz(-A)- W" implies that W1I2(- A)- W1" is a strict contraction on L 2 .By the Birman-Schwinger principle N ( V ) = 0. We note however that Theorem 11.2 is stronger than N ( V ) = 0, for N ( V ) = 0 says that e - t H is a contraction L 2 ;Theorem 11.2 says it is a contraction on L". 4. This result will play a central role in Section 25.
Inequalities
12. Correlation Inequalities
Most of the inequalities discussed in this section and the next do not absolutely require path integrals for their statement or proof. Indeed, we V ( o ( s ) )ds by (t/n)V(o(jt/n)) and will prove them by replacing looking at the joint distribution of the {o(jt/n)}j”= Thus we will undo the Trotter product formula proof of the Feynman-Kac formula; clearly, we could just use the Trotter formula and state all inequalities in terms of the semigroups. Path integrals are notationally and, more importantly, conceptually clarifying for the results. In this section, we will consider a variety of inequalities called “correlation inequalities” after their original occurrence in the statistical mechanics of lattice gases. There the earliest results were obtained by Griffiths [I1151; a bibliography on the subject including the large number of inequalities and their often impressive application to lattice gases would run to hundreds of papers! The applicability of the most general of these inequalities to Euclidean quantum field theory was discovered by Guerra et al. [120]; in essence, the results of this section are the specialization of their idea and its development to zero space dimensional quantum fields, i.e., P(4),-processes ! Consider the following generalization of the P($),-process of Section 5:
yo
cj”=
Fix 5 g E Lz(R) positive functions and V which we suppose continuous and nonnegative. Then Z - ‘f’(o(O))g(o(t))exp
(12.1)
119
120
IV.
Inequalities
is a probability measure on the paths { o ( s ) 10 I s 5 t}. If I/ = +x2 + P and g =.f= R,, then d p is just the restriction of dv, to C(o,l,(see Theorem 6.7). Now fix n and let us approximate d p ( o ) by
2, = (f,( e - ' H o / n e - ' " l n ) n g )
and consider the joint distribution of {o(jt/n)}~,, with respect to dp,,. It has the form (12.3) where
dv,(x) = f ( x ) e - ( " 2 ) J x 2d x J = nit, d v j ( x ) = e-JX2e-(l/n)V(x) d x ( j =1, ..., n - 1) dv,(x)
=
g(X)e-(1/2)Jx2e-(t/n)v(x)
dX
The reader who has seen the theory of lattice gases [129, 133, 2261 will recognize (12.3) as the probability distribution of an Ising-type ferromagnet except that the dv(xi) = +[&xi - 1) + &xi + l)] of the Ising model is replaced by a more general type of distribution. The following result of Ginibre [1011 generalizes the clzissic inequalities of Griffiths [1151 as extended by Kelly-Sherman [1561; hence, called GKS (Griffiths-Kelly-Sherman) inequalities. Below, in dealing with measures like (12.3), we will expand exponentials without concern about convergence. All the theorems below should have additional hypotheses (usually eax2d v j ( x ) < 00 for suitable a will do) which justify the convergence of these series. In applications, one can prove the additional hypotheses or can use a limiting argument.
,
Let 9 be the functions on R which are nonnegaTheorem 12.1 ([loll) tive and monotone on [0, co) and either even or odd (denoted respectively, F e 90 ,). Let 9" be the functions on R" of the form f l ( x l ) .. .fn(xn)with J;- E P I .Let d p be a probability measure of the form of (12.3) where J 2 0, each dvj has the form exp(fj(x)) dAj(x) with ji E 9, and dAj even. Then
,,
((.>
= j.44
for allf, g E F,,+ 1.
(GKSI)
( f )2 0
(12.4)
(GKS 11)
(.f's) 2 (.f>(s>
(12.5)
121
12. Correlation Inequalities
Proof ( G K S I) We first note that for anyf,, . . . ,fn measure dA on [w
E
Fl and any even (12.6)
For the integral is zero if there are an odd number of factors from F; since then nf, is odd and dA is even. If there are an even number of factors from 4;,then f j is even and nonnegative for x 2 0 and so for all x, so (12.6) 1, then holds. If g = gl(xl) . . . gn+l(xn+,) E 9,+
nj.=,
Expanding each exponential and noticing x E F1,the integral is a sum of factors of the form of (12.6) and thus positive. ( G K S II) Let dp(x) be the measure on R"" and consider .*theduplicate system," i.e., dp(x) dp(y) on (R"' 1)2. Then
Let Then
x ~xP(C .fl(-xi)
+.flbi))
dQ(x) dQ(y)
Expanding the exponentials, we see that it suffices to prove that
J f~(Fix) + ciFi(y)) ~ Q ( Xd) ~ ( y ) 0 2
(12.7)
i= 1
for each choice of
f 1 and each Fi E F n +(since xixjE 9,+ l). Now
E~ =
~b f db'
= %U
+ u')(b f b') + +(a - u')(b T b')
By repeated use of this equality we can reduce (12.7) to the special case where each Fi(x)has the formf(xj) w i t h f E F , . But then (12.7) breaks up into a product of one-dimensional integrals; i.e., we only need to prove that for d A even on aB andf,, . . . ,fm E 4 ,: r
m
(1 2.8)
IV.
122
Inequalities
Now, since each ji is even or odd
for o = & 1, and ~ ( i = ) 1 or 2. Using dL(x) left-hand side of (12.8) as
1
=
u.3=
*
[I +(-l)Z=(i)] 1
=
dL( -x), we can write the
J n [ f i ( x ) + ci4"!fi(Y)I d X x ) a y ) x,ytn
where o3 = 0102.Thus it suffices to prove (12.8) where the integral is over ei) times the integral over the region x, y 2 0. But this integral is (1 + the region x 2 y 2 0 (consider the x tf y interchange) and thus we need only prove (12.8) in case the integral is over the region x 2 y 2 0. But in that case A(x) -fi(y) 2 0 sincefis monotone, andf,(x) +,f;.(y) 2 0 sincefis positive on LO, a).I
ny=,
By taking limits, we have some inequalities for path integrals: Suppose that V = V, + V2 where V, is even Theorem12.1' ([120]) and - V2 E .F;S(i.e., V2 odd, negative, and monotone decreasing on [0, a)). Let5 g be positive even functions. Let A = u(o(t,),. . . ,~ ( t , ) with ) a E F,, and similarly for B. Let ( .) = f . d p ( o ) where dp(w) is given by (12.1). Then, if ( A 2 ) < co,( B 2 ) < co, (A) 2 0
(12.4')
(AB) - (A)(B) 2 0
(12.5')
+
In particular, this holds for any P(q5),-process with P ( x ) = P , ( x ) P,(x) with P , even and Po odd, negative and monotone decreasing for x 2 0.
1. To get the P(4)l result for the result fromf, g even, we take Q(S - +r) and t + co using Theorem 6.9 to be sure we recover the P($),-process. 2. To handle unbounded a's, we replace a by truncated functions and take limits. Remarks
.f, g = Q, and then take w(s) -+
Looking at the proof of (GKS l), one sees that the monotonicity of the functions was never used. Thus, we have the following.
12.
123
Correlation Inequalities
Theorem 22.1" ([173a]) Let V be a function on R" so that for x 2 , .. . , x, fixed and x I 2 0 we have that V(x) I V ( - X I ,
x2,.
. .)
(12.9)
and let (. ) = j . dp(m) where dp(w) is given by (12.1) extended to R" withf, y even and positive. Then (A) 20
(12.4")
if A = a ( o ( t l ) )with a(x,, . . . ,x,,) a function which is odd in x 1 for x2,., . , x,, fixed and positive for x 1 2 0. Proof After discretizing, fix the x2, . . . , x, coordinates in each variable. Expand in the x 1 couplings and get positivity by the proof of (GKS I). If one then integrates in the x 2 , . . . , x, variables, positivity results. I Rernurk If V is unbounded below, one proves (12.4") for an approximating sequence of K's bounded below and takes limits.
As an application we have the following result of [173a]:
Theorem 12.2 ([173a]) Let e ( R ) be the ground state energy of -$A z l l r l - l - z 2 ) r - Re1-I where e is a unit vector and zl, z 2 2 0. Then e ( R ) increases as R does in the region R > 0. Proof Without loss take e eigenvector :
de dR
-=
1,0,0). Then, if R, is the ground state
= (-
(RR, [z2(xl
+ R)lr
= z 2 ( Q R ,x I I r I
where
a, is the ground state for l? P=
-zllr
=
-
Rel-31RR)
3QR)
-$A
+ Re1-I
+ P with -
~~1rl-l
and we have used translation covariance. Since 1rl-I is monotone, P obeys x 1 I r I 3 Q R ) 2 0. I (12.9), so by (12.4") and a limiting argument
(aR,
Now suppose that W E F 1and we let ( replaced by V ( x ) + AW(x). Then
be of the form (12.1) with V
124
IV.
Inequalities
for the ( A W ) , term comes from the derivative of the Ae ... term and ( A ) , ( W ) , from the derivative of 2-l. We have thus proven (taking suitable limits for P(4)l and unbounded W ' s ) :
CoroIlaryI2.3 Let 3.' + P , Q be fimctions which are bounded from Q, ( P , even, etc.) with - P o , Q,, Q,, below with P = P , P o , Q = Q, - P o - Q o ~ F l Suppose . that the R(+),-process exists for R = P AQ, A E [O, 11, and use (. ), to denote its expectations. Then, for any A = a(q(t,));
+
+
+
a E .Fn,
if 1, 2
5 (A),,
('4)a1
A0
Our first application of this result is taken from [120] $
Theorem 22.4
E,(P)
=
Let P , Q be even, Q E 9,. Let E , ( P ) = inf spec(Lo inf spec(L, P 1 Qf). Then
+
E2(P
+ AQ)
- Ei(P
+ P)
+ AQ)
is monotone increasing in A.
+
Proof By adding an E X , to P , we can suppose that all L = Lo AQ have purely discrete spectrum in which case E l and E , are the first two eigenvalues. Let Q , , Q2 be the corresponding eigenvectors. We claim that (xQ,, Q,) # 0, for Q, is nodeless and Q, has exactly one node at x = 0. Thus, making an eigenvector expansion of xSZ, and noticing (Q,, x Q , ) = 0 - ( E ~- E , )
=
Iim t ~ In[(xQ,, ' e-"L-E~)~Q1)] 1-w
=
lim t - ' In(q(t)q(O)), 1-a0
is monotone decreasing in A.
Example
I
If Ei(ao, a , , . . . , a,,,) is the ith eigenvalue of - ( d 2 / d x 2 ) + > 0, E,(a) - E,(a) is monotone in the a's.
Cjm,, a j x 2 j ,then in the region a,
Our second application is from [ 6 ] : Let V be an arbitrary even function on R with V- EL', Theorem 12.5 V+ E I,:,=. Let W be an even function monotone increasing on (0, co). Suppose that - ( d 2 / d x 2 ) + V + AW has a normalized ground state (that is, an
12.
125
Correlation Inequalities
eigenvector corresponding to lowest eigenvalue), for each a
for each A E [0, 11.Then
SI,
1$1(x)l2 d x 2 s"$o(x)12 dx
Proof Let F(A) = 1 - s"u l$i(x)12 d x . We want to show that F(A) is monotone decreasing in A. But
F ( 2 ) = (G(q(O))), where G(x) = 0 (respectively, 1) for 1x1 < u (respectively, >a) and is the P(4),-process for -d2/dx2 + I/ AW. Since G E Fl,this follows from Corollary 12.3. I
+
Martin [182] has discovered an "elementary" proof of Theorem 12.5 that avoids path integrals; see also [289,290]. There is one other family of inequalities that holds for all even d X s , namely the FKG (for Fortuin, Kasteleyn, and Ginibre [SS]) inequalities. We state the following without proof (see, e.g., [258]). Theorem 12.6 ing
Let F be a nonnegative function on rWY F(x
where (x
A
A
Y)F(X v Y) 2 F(X)F(Y)
=
[0, a)"obey( 12.10)
Y ) ~= min(x,, yi); (x v Y ) ~= max ( x i ,yi). Let
Letf, g be functions on Ry which are monotone increasing in each xj (with xi for i # j fixed). Then
(.fig> 2
(.f>(s>
Let us note some cases where (12.10) holds: Lemma 12.7 ([S])
(a) If F(x) only if
=
exp( - W(x)) with W a C2-function, then (12.10) holds if and
(12.11)
126
IV.
(b) Let d S , be the usual measure on unit sphere in R’ with v
J
for x
E
1 or 2, and let
n dS,(ai)
C J i j x i x j a i cj
F(x) = exp
=
Inequalities
) ni = I
(i+j
Rm+.Then F obeys (12.10) if J j i = J i j 2 0.
Proof (a) (12.10) is equivalent to w(X A
y)
+ w(X
V
y) -
w(X)- w ( y ) 5 0
(12.12)
and this differencecan be written as a sum of integrals of a z W / d x i a x j . Thus (12.11) implies (12.10). By taking y = x + Eei - Eej, (12.12) yields (12.11). (b) Let ( .), = j . exp(. . .)n d S / l exp(. . .)n dS where the centered dot is a function of the IS’S. Let W = -In F . Then
aw = -
~
azw a x i axj
=
axi
-( J i j G i *
(7
Isj), -
J i j x j o i*
.>
x
c Ji1JjkX,Xk((Ti *
0 , ;Gj
-
Ok).l.,x
k. I
where ( A ; B ) T = ( A B ) - ( A ) ( B ) . Thus (12.11) holds if (ai-aj), 2 0, (oi - c l ;C T ~ - ( T ~ ) ~2, , 0. For v = 1, this is just the usual GKS inequality; for v = 2, it is an inequality of Ginibre [loll. I As a typical application of this consider the following.
Let V be a function on R3which is only a function Proposition 12.8 ([S]) of p = (xz + y2)112and IzI, with a 2 V / a p az I 0 in the region p, z 2 0. Suppose that I/ is monotone in p, 1 z 1, that G is another function of p, I zI which is monotone, and let W be a function of p alone. Suppose that - A W ;iV has a ground state for each I E [0, 11,and let R, be the corresponding ground state eigenvector. Then (RA, GR,) is monotone decreasing as I increases.
+
+
Proof Pass to the path integral [P(q5),-process] for - A it ( .),. As in Corollary 12.3, it suffices to prove that
+ W + I V ; call
(G(4(O))V(4(s)))2 (G(4(0)))( V(ds)))
and this follows if we can prove the FKG condition (12.10) for the measure obtained by first discretizing and then “integrating” over the angles in the
12.
127
Correlation Inequalities
(x, y)-planes and signs of z’s. The F that results is a product of three kinds of factors :
exP( - mW(/li)>,
c
Jexp(-li
~ X P-(
@
(ri
- rj)’)
v(Pi I zi 1)) >
d4i d(sgn zi)
It is enough to check (12.10) for each factor: The W-factors trivially obey (12.10), the I/-factors are all right by hypothesis and Lemma 12.7(a), and the final factors are all right by Lemma 12,7(b). An interesting application of this last result is to the Zeeman effect in hydrogen, i.e., hydrogen in a constant magnetic field. In the approximation of infinite nuclear mass, with suitable units and a magnetic field B, in the z-direction, the Hamiltonian is given by (1.2) which can be written: H
=
-A
+ (B2/4)p2 - BL, - r - ’
where L , = (l/i)(d/d4) is the z-component of angular momentum. H commutes with L, and one can ask what is the value of L, in the ground state of H . Consider the Hamiltonians H(m, A)
=
-A
+ m2p-2 + (B2/4)p2- mB
-
Ar-’
r
Now H (functions with Lz = m) is isomorphic under the natural association to H(m, A = I ) 1 (functions with Lz = 0), so that E(m) = inf(H L,
=
m) = inf(H(m, A = inf(H(rn, A
= =
1) t L, 1))
=
0)
where the last equality comes from the fact that H(m, A) has a strictly positive ground state eigenvector and hence one with L, = 0. Now think of m as a continuous parameter. Then
dE(m, A ) dm
=
(2mp-2)m,A- B
where ( . ) m , A denotes expectations in the ground state of H(m, A). V ( p , l z l ) = - ( p 2 + z 2 ) - ‘ j 2 is easily seen to be monotone in p, z and to obey d 2 V / d p d z < 0. Since - p - 2 is monotone in p , Proposition 12.8 implies that is monotone increasing in A. Thus for m 2 0,
1 = 0)_ dE(m, A = 1) - >- dE(rn, . _ = am dm
o
~
128
IV.
Inequalities
where the last equality comes from the fact that E(m, 1 = 0) = 2 8 for all m 2 0 [the ground state in the p-variables is a multiple of p" exp( -&I*)]. This shows that E(m) 2 E(0)
(12.13)
for m 2 0. Since E( - m ) = E(m) + 2mB for m > 0, (12.13) holds for all m. By an additional argument [6], the inequality can be shown to be strict. We thus have proven the following. Theorem12.9 ( [ 6 ] ) The ground state of the Hamiltonian (1.2) of a hydrogen atom in a magnetic field has m = 0 for any B. is replaced by any function Remark The above proof will work if V ( r ) with V negative, monotone increasing and concave on [0, 00). By an additional trick, one can avoid the hypothesis of concavity [6]. If V is not monotone, then the ground state need not have m = 0 (see [4] and references therein).
*** The above results (at least in one dimension) give one some control on differential operators - d 2 / d x 2 + V for general even Vs. One could get control over such general Vs because there was no restriction other than evenness on the "single spin distribution," dA. One can get much greater control by placing stronger restrictions on V . The earliest result of this genre is the following obtained by Simon and Griffiths [261]: Theorem 12.10 Let (.) be the P(+)l expectation for P ( x ) = ax4 + bx2 (a > 0, b in R). Then any multilinear inequality true for an arbitrary spin-3 ferromagnet is valid for ( .). Remarks 1. By an arbitrary spin? ferromagnet, we mean the measure on { - 1, l}" given by (f(o))
=
C ua= a=l,
* ..., m
.f(a,)exp(C J,, a,rr&/Normalization
1
with J,, 2 0. By a multilinear inequality we mean one of the form
1 d(P)(o,,, . . .,
Ga,)P
20
PEb
where 9'is the family of all partitions, P, of { 1, . . . , n} into disjoint subsets k
/
\
129
12. Correlation Inequalities
2. Examples of such inequalities are those of Newman [196]
with the sum over all pairings of { 1, . . . , 2n}, the GHS inequality discussed below and certain inequalities of Lebowitz [165]. 3. For the proof, see [261]. See [76, 621 for further discussion including multicomponent inequalities. The idea in [261] is the following: If ci are independent spins, then the distribution of o i I f i is
Z’’’
(2n)- ‘ I 2 exp( -x2/2)
in the limit of N -, 00. If one couples the spins together with a Hamiltonian (2N)-1(1r=1 CT~)~, then this leading behavior is precisely canceled and it ai/N3’4approaches c exp( -ax4) for a, c suitable. In this turns out that way one can approximate a “single” spin with e - x 4 dx distribution with a sum of ferromagnetically coupled spin-$ spins. Thus ferromagnetically coupled e P x 4dx spins can be approximated by a larger array of spin-i spins. Since multilinear inequalities extend to sums of spin-f spins, the theorem is proven.
cr=
*** The most interesting applications of Theorem 12.10 involve the GHS (for Griffiths, Hurst, and Sherman [l 171) inequalities which have been proven for many Vs by Ellis, Monroe and Newman [74] whose treatment we follow below. Similar results have been obtained by Sylvester [269]. As a preliminary, we discuss the Ursell functions or cumulants. Given n random variables, X I , . . . , X,, we define
130
IV.
Inequalities
There are explicit formulas for u, but of more value is an axiomatic characterization of Percus [202] : (PI) u, is multilinear in X , , . . . , X , (in the sense described after Theorem 12.10). (P2) un(X1, . . . , X,) = ( X I . . . X,) + sums of products of two or more factors (P3) If the X’s break up into two mutually independent sets, then u, = 0.
Proposition 22.12 (Percus’ lemma [202]) Fix n. Then u, obeys (P1)(P3). Moreover, any expression obeying (Pl)-(P3) is identical to u,.
Proof ( P l ) and (P2) are easy to check. (P3) follows if we note that if the X’s break up into independent sets then In(exp.. .) is a sum of two functions each of which is independent of some hi. Conversely, given a function ii,(X,, . . . , X,) obeying (Pl)-(P3), we can write an(xi) =
C d(P)(X>p
PE B
by (Pl). Here d(P) is a number, B is the family of all partitions of { 1, . . . , n}
and (x>P=
? ( n Xj) i=l
jtn,
By (P2), d ( P ) is determined for # ( P ) = 1. We claim that (P3) then determines d ( P ) inductively in # ( P ) . For fix a partition Po and let P a P, indicate that P, is a refinement of P. Let P , = {xi}.Then
c d(P)
I’ -P.,
=
0
(12.15)
cp4p,,
for suppose that {Xj}jt,, are independent. Then d ( P ) is coefficient of (X),<,. But (12.15) determines d(P,,) in terms of d(P)’s with # ( P ) < #(Po>. I As a typical application of Percus’ lemma, we note the following formulas p), take n of Cartier [33]: Given random variables X,, . . . , X , on ( X , F, independent copies of X (i.e., the n-fold Cartesian product of X with the
12.
131
Correlation Inequalities
product measure), let X ! j ) be the copy of X i on thejth factor. Let w = eZniin be a primitive nth root of unity and let Ri = wjX?). Then (Cartier's formula)
xr=l
u,(X, ) . . . )X , )
=
tz-1E(21, . . . ) X n )
(12.16)
To prove (12.16), note that ( P l ) and (P2) are easy; (P3) follows if we note that under the measure preserving map X?) X ! j - I ) (with X'O) = X'")), Ti H Thus -+
-
-
E ( X j , , . . . , X j , ) = t d E ( X j , , . . . , Rj,)
is zero if I < n.
Theorem 22.12 ([74,269])
Let dp be a measure on
[w"
of the form
+ c hizi + c Jijxixj d"x
1
n
dp(x)
=
( ,
Z - ' exp -
v(xi)
i= I
i= 1
i, j
(12.17)
where Z is a normalizing factor (assumed finite). Suppose that J j i = J i j 2 0; hi 2 0 and that each v(x) is an even function of x, C' on ( - co, ~ m with ) Vl convex in the region (0, m). Then with respect to dp (GHS inequality) u,(x,, If all hi
=
X',
(12.18)
Xj) I 0
0, then (Lebowitz inequality) ( 12.19)
u,(x,, . . . , xq) I 0
Remarks 1. If we take v.(x) = V(x) with V(x) take a m,
=
a[(x' - I)']
and
-+
e-"'") dx/Normalization
-+
+[6(x
+ 1) + 6(x
-
l)]
and one recovers the original GHS [I171 inequality for the usual spin-* Ising model. 2. Since u3 = 0 if all hi are zero and by (12.18), uj I 0 if h, > 0, hi = 0 ( i # 4), we have duJdh, 5 0 at all hi = 0. That is, (12.18) implies (12.19). (12.19) was originally obtained by Lebowitz [165] as one of a large number of new inequalities; it is a remark of Newman (unpublished) that it follows from (12.18). 3. At first sight, it may be surprising that for spin-+ models u1 2 0, u2 2 0 but u3 I 0. There is a good physical reason for this: Typically, the magnetization of a magnet is as drawn in Figure 1, so that in the region h > 0, it is positive, monotone, and concuue. Since m = (a1), this says ( a l ) 2 0, d2(al)/dhi dh, I 0. t3(o,)/dhi 2 0 but
xi
IV.
132
Inequalities
m
Figure 1.
A typical magnetization curve
Proof([74]) GKS I was proven with one copy of d p , GKS 11 with two ) ) components copies; now we will use four copies, i.e., d p ( x ( l ) ). . . d p ( ~ ( ~with XI'). Let B be the orthogonal matrix
(12.20)
and let wlk)= C bk,$. We first claim that u , ( x , , x 2 , x 3 ) = - ~E(w\"w~~'w',~))
(12.21)
This follows, either by explicit expansion of the right-hand side of (12.21)into 64 terms or by using Percus' lemma: the right-hand side of (12.21) is multilinear, the leading coefficient is ( x I x z x 3 ) , and (P3) is obeyed since E(wY)) = 0 for j = 2,3,4. Thus (12.18) and (by Remark 2) (12.19)follow if we show that (12.22) where Fi(w('),. . . ,w ( ~ is ) )a function positive in the region with all w(")2 0 and odd or even in each w ( ~ ) . Now B is an orthogonal matrix, so d"x(j) = d"w(') and
n:=
4
Thus,
4
n:=
133
12. Correlation Inequalities
If we now expand the first exponential and note that products of the allowed F's are allowed and that wjk)is allowed, we see that (12.22) is implied by onehalf of the lemma below. I
c;=
Lemma 22.23 Let x ( ~=) I Bji w ( j ) where B is the matrix (12.20). (Note: B - ' = BT.) Let W consist of all functions of w ( ' ) , . . . , w ( ~which ) are ) under some, even under odd or even under change of sign of each w ( ~(odd others is allowed) positive if all w ( ~2) 0. Let /
4
\
4
where V is an even C'-function. Then (12.23)
for all F
EW
if and only if V' is convex on (0, a).
Proof Suppose that F(w) is multiplied by (-1)"1 if the sign of w ( ~is) changed. dp is left invariant under w ( ~+ ) - w ( ~( )i = 1,2,3,4), since I/ is even, ) and also under reversal of two w's (e.g., w ( ~-+ ) - w ( ~ ) , w ( ~+) - w ( ~corresponds to interchanging x 1 with x 2 and x 3 with x4). Thus the integral in (12.23) is zero unless all mi are even or all are odd. If all are even, then F 2 0 so (12.23) is trivial. We are therefore restricted to consideration of all mi odd. The invariance of dp under reversal of two signs then yields the integral over the sixteen "quadrants" as two sets of identical contributions; i.e., [ F ( w ) dp(w)
J
=
8
[
J w ,t
F(w)[dp(w) - dp( - w ( l ) ,w('), w ( ~ )w'"))] , o
For this integral to be positive for all F 2 0 and w(j) 2 0, it is necessary and sufficient that the measure in [. . .] be positive. Thus, (12.23) is equivalent to the syrxmetric condition V(w(')
+
+ w ( ~+) w ( ~ )+) V ( w ( ' )+ w(') - w ( ~ )
-
+ w ( ~+) w ( ~ ) w ( ~ ) )
- V(w(')
-
-
+
w ( ~ ) ) 2 others
3 others 2 0
(12.24)
for all w ( ~2) 0. Suppose that (12.24) holds. Fix a. Then (12.24) holds for w ( l )= a + y ( I y I < +a), and so it holds for w ( ' ) = a and V replaced by V,, the convolution of V with a positive C"- function supported in ( - $ a , ;a). V, is C" so we can choose w(l) = a, w ( 2 )= w ( ~=) w(4) - c and expand in Taylor series; (12.24) says : l$(a
+ 3E) + 3 q a
- E) -
I/h(u - 3E)
-
3&(a
+ E) 2 0
134
IV.
Inequalities
so taking E -,0, V y ( a ) 2 0. Taking 6 + 0, we see that the third distributional derivative of V is a measure on (0, co), so V’ is convex there. Conversely suppose V is C1 with V’ convex on (0, 00). By symmetry, we . need only check (12.24) for w ( l ) 2 w ( ~ )Then:
1.h.s. of (12.24) = w ( ~ ) G(w(’)+ r ~ ( w(’), ~ ) ,w ( ~ )dr)
s_’
1
with
+ + + +
G(x, y, Z ) = V’(X y Z ) V’(X - y - Z ) - V’(x y - z ) - V‘(x z - y)
+
so we need only prove G is positive in the region x 2 y 2 z 2 0 (G is symmetric). Now since V’ is continuous on all of R and convex (respectively, concave) on x > 0 ( x < 0), V” exists for all but countably many x’s and V’ is the integral of V”. Thus
s
1
G(x, y, z ) = z
[IV ” ( x
+ rz + y ) - V ” ( x + rz - y)]
dr
1
Since x 2 z, we only need that x, y 2 0 implies V ( X
+ y) 2 V”(x
-
y)
and this follows from the fact that Y ” is even and monotone on (0, co).
I
As usual, we immediately obtain inequalities for the P(4),-process:
Theorem 12.24 Let P have the form Q - hx with h 2 0 and Q a C1 even Then function with Q’ convex on (0, a). (a) (q(t)q(s)) - ( q ( t ) ) ( q ( s ) ) is monotone decreasing as h increases. (b) I f h = 0, (&l
) q 2 ( t 2 ) ) - ( q 2 ( t1) X q 2 ( t 2 ) ) 5 X q ( t 1 ) q ( t 2 ) ) 2
Remark (a) comes by noting that u3 = i3u2/ah, and (b) by using Lebowitz’ inequality with o1 = 0 2 ,o3 = 04. Corollary 22.15 ([261]) Under the above hypotheses on Q, let Ei(h) be the ith eigenvalue ofi(d2/dxz) + Q ( x ) - hx. Then E,(h) - E,(h) is monotone increasing in h.
135
12. Correlation Inequalities
Proof If Ri is the ith eigenvector, then (R,, $2,) # 0 since R, has one node say at a, so (R,, 4Rl) = (O,, ( q - a)RZ,)# 0. Thus, if $ = 4 0 1 (R,, 4R,)R, and fi = H - E l : -(E, - E , )
=
lim t - ' In[($, K~H$)] 1-00
lim
=
f-
W(q(f)q(O))
-
(dt))(dO))I
1+m
Now use Theorem 12,14(a).
I Under the above hypotheses
Corollary 22.26([107]; see also [264,196]) on Q and notation,
E3(h) - E,(h) 2 E,(h) at h
=
-
El@)
0.
Proof Since Ri( -4) since R, has nodes at 4
= =
( - l)i+'Ri(q), (R,, q2Rl) = 0. On the other hand,
&&,
(R3, q2Ri) =
(R3,( y 2 - a)Ri> # 0
so, using Theorem 12.14(b): -(E3
- El)
=
lim t - ' ln[(42(f>42(o)) - (42(f>>(42(o)>I t-00
I lim t - ' In[2(y(t)q(o))2] = -2(E,
-
El)
1-m
so E3 -
E , 2 2(E, -
El)
I
There is a final aspect of Theorem 12. I2/Lemma 12.13 of interest: Let V be an even C'-function on R with V ' ( x ) Theorem22.27 ([75]) convex on [0, 00). Suppose that H = -d2/dx2 + V ( x ) has an eigenvalue at the bottom of its spectrum with eigenvector ,-Icx). Then f is C' and f" is convex on [O, 00).
Proof Let ( .) be four independent copies of the path integral associated to H . By the generalized GHS inequality, (12.22), we have that (F(w)) 2 0 for any F E W , the family in Lemma 12.13. Thus (12.23) holds with the *' V" in d p replaced by 2f Since V is (locally) C', a general argument implies that
136
IV.
Inequalities
f i s (locally) C' (actually C 2 can be shown), so by the converse direction of Lemma 12.13,f' is convex on [O, 00). Remark V' continuous up to zero is not needed. All that is needed is lim, L o V'(0) I 0, since one can then obtain V' as a limit of W ' s with W'(0) = 0 and W convex.
Example We want to describe an example with V even and C", with V c m ) ( x2) 0 for all x 2 0 all integers m 2 0 but for which the ground state I)= e - ' ~ i t h f ' ~ negative ) for large x. V will just be x4. If E is the energy of the lowest eigenvalue, then theformal WKB form for the asymptotics of I)is
and thus for V
= x4
f
1 3
--x3
+ In x + O(x-')
and f'(4)
N
- f j ~+ -O ~( X - ~ )
The point is that using ODE methods, one can prove these asymptotic formulas.
13.
Other inequalities: Log Concavity, Symmetric Rearrangement, Conditioning, Hypercontractivity
A. Log Concavity
Definition A function F : R" -+ [0, log convex) if and only if F(lx
+ (1
-
A)y) 2 F(x)'F(y)'
00) is
-'
called log concave (respectively, (resp., I F(x)'F(y)' - ')
for all x, y E R", 0 I A I 1.
Examples 1. If C is a convex set, its characteristic function is log concave. xixjaij)is log 2. If {aij};,j = is a positive definite matrix then exp( concave.
-1
13.
137
O t h e r Inequalities
The following is an elementary consequence of Holder's inequality. Theorem 13.1 y in R", then
IfF:Rmf".+ [O, m) withF(., y) logconvex foreach fixed
is log convex in x. By taking limits (or using Holder directly on the path space), one sees that V(q(s))ds) = (Qo, exp( - t(Lo + V))Qo) is log the map V w 1dq exp( convex so that using (1.9), we recover the result (which also follows from the Rayleigh-Ritz principle) that V H inf spec(& + V ) is concave. Much subtler is the following result proven by Prekopa [205] and then independently by Rinott [220] and Brascamp-Lieb [191.
-yo
Theorem 13.2
If F : R m f n+ [0, G(x)
=
'x))
is log concave, then
JF(x, Y> d"Y
is log concave. Proof([19]) By induction, we need only consider the case n = 1. Moreover, since log concavity is an expression about G over lines, we need only consider the case rn = 1. Fix x o , x' E R and A E (0, 1). If G ( x o ) or G(x') is zero, the inequality G(Axo + (1
-
A)x')
2 G(x,)'G(x')'-~
is trivial. Moreover, by replacing F by F ( x , y)exp( - & y 2 )we can suppose G is everywhere finite. Then replacing F by ea+hxF for a, b suitable we can suppose that SUP
F(xo,y )
= SUP
F(x', y )
z0
(13.1)
Y
Y
Fix 0 < z < zo and let C(z) = { ( x , y ) I F(x, y ) 2 z } . Log concavity says that C(z) is convex and nonempty. Thus for x = Axo (1 - A)x'
+
C(x,z )
=
CY I (x, Y ) E C(z))
is nonempty (by 13.1) and an interval [ a ( x , z), b ( x , z)] with a and - b convex in x . In particular, g(x, z ) = meas(C(x, z ) ) = b(x, z) - a(x, z ) is concave; I.e., g(Ax0
+ (1
-
A)x', z ) 2 Ilg(x0, z)
+ (1 - A>g(x',z )
138
IV.
Inequalities
But
with equality for x
= xo or x'.
G(Ax0
+ (1
Thus
- A)x') 2 AG(x0)
+ (1 - A)G(x')
2 G(x,)~G(x')~- I
Corollury23.3
I
The convolution of two log concave functions is log
concave. Proof on R2".
If F and G are log concave on R", then F(x - y)G(y) is log concave
I
Most of the results below can be obtained by systematically using Corollary 13.3 and the Trotter formula. As usual, we give instead a path integral result. For convenience, we state things for the Wiener process. Similar results hold for any Gaussian process and for Wiener measure.
Theorem 23.4 Let F(b, A) be a function on v-dimensional Wiener paths depending on an additional parameter A E R". Suppose that F(b, 1) = limm+mF,(b, I.) in L'(Db) for each fixed A for F , obeying F,(b, 3L) = G,(b($"i),
. . . ,b(sF)), A)
with G , log concave on Rn+mv.Then H(A)
=
W ( b , 2))
is log concave. Proof E(F,(b, 3L)) = G,(xl, . . . , x,, I.)Q(x) dmVx where Q(x) is a Gaussian. Since G, and Q are log concave, so is G , Q and thus so is the integral by Theorem 13.2. I The following applications are from [19,20]. If I/ is convex on R' and H e - f c x )thenfis , convex.
Corollary 23.5
R(x)
=
=
-+A
+ V has ground state
13.
139
O t h e r Inequalities
Proof Let Q, be the ground state of the harmonic oscillator, Lo. By a limiting argument using (1.10) it suffices to show that (eFtHQ,)(x) is log concave for each t. But
(e-‘HQo)(x)= ,(ex,(
-
+ b(s)) ds)Q,(x + b(t))) cj”=V(x + uj))Q,(x + urn)is log con-
JiV(x
and G(u,, . . . , urn,x) = exp(-(t/m) cave on R(”’+’)”. Thus the result follows from Theorem 13.4.
I
Remark Let S be an open convex set in [w” and let V , = n dist(x, S). Then V , is convex. Taking n + ‘x3, we see that the ground state, $, of the Dirichlet operator -A; is log concave; in particular, the level sets {XI $(x) = a } are the boundaries of convex sets (x I $(x) 2 a } . This is a result of BrascampLieb [20].
Corollary 23.6 Let V(x, A) be a convex function on inf spec(Ho V ( x , A)). Then E(A) is convex in A.
+
Let E(A) =
[w””.
Proof The proof follows from Theorem 13.4 extended to dpo and the formula (1.9):
-E(A)
=
i
lim t-’ In ~Qo(co(0))Qo(o(t))exp( - J:V(w(s), A) ds dpo(w) t+m
I
This result is useful to obtain lower bounds on ground state energies which complement Rayleigh-Ritz results.
Example V ( x , A) = Ax2 + x‘ + $A2 is jointly convex in the region A 2 0 (compute the matrix of second partial derivatives). In particular, if u(A) = inf spec( -f d2/dx2 + Ax2 + x4), then for 0 I A: -3 I u”(A)I 0
+
+
If b(A) = inf spec( -3 d 2 / d x 2 x2 Ax4), then [241] b(A) = A”3a(A-2’3) by a scaling argument, so we also obtain information about b. Brunn-Minkowski inequalities for Gaussian measure have been extensively studied by probabilists; see [2a, 118a, 18a, 18b, 18~1.Our treatment follows that in [191 and begins with the classical Brunn-Minkowski theorem and its relation to Theorem 13.2.
Theorem 23.7 (Brunn-Minkowski inequality) Let C,, C , be nonempty compact convex sets in R” and let C , = AC, + (1 - A)Co.Then ( 1 . I indicates Lebesgue measure) lClll’” 2 A ~ C ’ ~ ’ ’+ n ( I - A)lColl’n
140
IV.
Inequalities
E } ) we Proqf By a limiting argument (replace C iby {x 1 dist(x, Ci)I 0. Take may suppose that I C iI ;-
C,
= ICII-I'nC1;
and suppose we know that
for suitable
[CAI2
c, = ICO
1. Then
,i. But then
proving the result. We are thus reduced to the case ICoI =
Then D is convex by construction, so its characteristic function, x, is log concave. But then I C , I = ~ ( xA), dx is log concave, so I C A1 2 I C , I - A I C Id = 1. I
Remarks 1. The Brunn-Minkowski inequality is known to be true for nonconvex C ialthough a different proof is needed (see, e.g., [20]). 2. The isoperimetric inequality is one consequence of the BrunnMinkowski inequality: For let C be an arbitrary convex set with smooth boundary and let B be the unit ball. Then the surface area, s(C), of C is
where we have used the Brunn-Minkowski inequality. Since s(B) = v J B I , we see that if ICI = IBl, then s(C) 2 s(B), which is the isoperimetric inequality.
13.
141
O t h e r Inequalities
Theorem 23.8 (abstract Gaussian Brunn-Minkowski [19]) Let o n " = co) with the measure ((271)- exp( Let C,, C, be measurable convex sets in and let C, AC, + (1 I
( - m,
R"
[w"
=
-$xi) dx,) = d p .
'Iz
=
- A)Cl.
Then p(C,) is log concave in A. Proof
Let,f(A, x) = 1 (respectively, 0) if x E C,. Thenf(A, x) is log concave
so
j,(n, x) = (2n)-
1'2
f
j(a,y)e- 1 ' 2 ~ : dyl
y, = x , ( i > I )
is log concave as in the proof of Theorem 13.2. Integrating over n-other variables and taking n + co (Co, C , measurable implies convergence) we obtain the result. I Remark Since Wiener measure is "isomorphic" to the last measure on by a linear map, this theorem extends to, say, Wiener measure, realized on continuous functions. [w"
Log concave functions can also be used to prove Theorem 12.2: Alternure proqfqf Theorem 12.2 [ 173al We will prove a stronger result: Namely, fix R , , . . . , R, and let r(A) be the ground state of rn
-+A
z,Ir
-
-
ARi1-'
i= I
Then e(A) increases as A increases. By the usual kinds of limiting arguments, it suffices to prove that F(A) = (0,( , - W ~ i n , - t P / n ) r ~ ~ ) is decreasing for SZ a Gaussian and P a cutoff V arranged to be bounded below. Since e + z z l x l - ' i nis monotone decreasing, we can write the cutoff V's as an integral of characteristic functions of balls centered at zero [see (13.2) below], so it suffices to prove that
is monotone in A if each x j , i ( j =- 1, . . . , m ; i = 1, . . . , n ) is the characteristic function of some ball centered at zero. Since Zi(r - ARi) is log concave in r and A and eKff'O/' has a log concave integral kernel, p(A) is log concave. Since it is also clearly symmetric ( F ( A ) = p( - A ) ) it is easily seen to be monotone decreasing.
***
142
IV.
Inequalities
B. Symmetric Rearrangement
A function f on R" is said to be symmetric decreasing if and only iff is a function, g, of I x 1, and g is positive and monotone decreasing as I x I increases. There is a particularly useful way of writing symmetric decreasing f 's. Namely, r
(13.2) J
where xi is the characteristic function of the ball about zero of radius r(1) max{ 1x1 I f (x) 2 A}. The proof of (13.2) is immediate, it just says that
f(x)
=
=
Jof(x'dl
It realizesfas a "sum" of characteristic functions of balls about zero. If x, and x2 are two such functions, then x1 * xz is seen to be symmetric decreasing by a simple geometric argument. Thus (13.2) implies the following lemma. Lemma 23.9 Theorem 13.20 state of H = -$A
Iff and g are symmetric decreasing so arefg and f * g. If - V(x) is symmetric decreasing and $(x) is the ground
+ V , then $(x) is symmetric decreasing.
Proof By (l.lO), it sufficesto show e-'HQo is symmetric decreasing where Q, is the ground state of Lo. But, by the Trotter formula, it suffices that (e-'Hoine-'"in)nQo be symmetric decreasing, and this follows from Lemma 13.9 if we note that e-tHoi" is convolution with a symmetric decreasing function. I Remark Theorems 12.17, 13.5, and 13.10 say if V is a C3 even function on R and F fis the ground state of d 2 / d x 2 + V ( x ) ,then for k = 1,2,3, we have that Vk'2 0 on (0, m) implies thatf(k' 2 0. This does not hold for k 2 4; see the example at the end of Section 12. Given any nonnegative measurable ,f on EX" with the property that
I {x I f(x) > p } I < co for each p > 0, there is an essentially unique function f * that is symmetric decreasing and obeys
13.
143
Other Inequalities
I
It is called thesymmetric decreasing rearrangement off[ifp ++I {x f ( x ) > p } I is continuous,f* is uniquely determined ; otherwise it is only almost everywhere determined and can be fixed by demanding f * ( x ) = lim, ? f*(Ax), say]. The earliest results that sums increase, under symmetric rearrangement, go back to Hardy and Littlewood. The ideas were developed by among others, Hardy, Littlewood, and Polya and by Sobolev. The strongest version is the following. Let,/;, . . . ,fk be positive measurable functions on Theorem 23.22 ([22]) R" and let ,fT, . . . ,,f'tbe their symmetric decreasing rearrangements. Let a i j x j ( x jE P). a,, . . . , akE R" and define li: R"" + R' by l i ( x , , . . . ,x n ) = Then
s fi
s fl
Cy=
k
fi(li(x)) P.Y I
f;(li(x))
dnvX
i= 1
i= 1
For a proof, see [22]. The idea is to use (13.2) and its analog for nonsymmetricfto note that thef's and,f* can be taken to be characteristic functions in which case the inequality is a geometric statement. The analog of this statement with * replaced by symmetrization about a single plane ("Steiner symmetrization") is proven using Brunn-Minkowski and then * is realized as a limit of Steiner symmetrizations about different planes. Typical of the applications of this inequality to Wiener integrals (really Trotter formulas) is the following result (it was proven in [178] assuming Theorem 13.1 1 as a conjecture and served as motivation for [22]). Let V = - W be negative and let V* = - W*. Let Theorem 23.22 E( V ) = inf spec( -+A V ) and let P,( V ) = exp( - t( -+A V ) ) .Then
+
+
E( V * ) I E( V )
(L Pl(QJ)
5
( f * 7
P,(V*)g*)
( 1 3.3)
(1 3.4)
Proof (13.3) follows from (1 3.4) and (1.9). (13.4) is easy to prove in a Trotter approximation using Theorem 13.11 and the fact that exp(+tA) is convolution with a symmetric decreasing function. I Remurk One might conjecture that N ( V ) I N ( V * ) on the basis of this. However, this is wrong (Lieb, private communication): If V is a sum of very shallow square wells in one or two dimensions very far apart, then N ( V )can be large while N ( V * ) = 1. Even in three dimensions, two wells far apart, each of which just binds a state, will yield a V with N ( V ) = 2 and N ( V * ) = 1 (M. Klaus, private communication).
***
144
IV.
Inequalities
C. Conditioning We discuss briefly some inequalities due to Guerra et al. [120, 1211, which are of some technical use in quantum field theory (e.g., [94,96,110, 1211). Let { X i } i G Ibe a family of objects which are Gaussian random variables with respect to two different probability measures p and v. We say that p is obtained from v by conditioningand write p < v if and only if
z j X i j l Zdp I
s
1 1z j X i j l 2dv
for all zlr.. . ,z, E C ;i.e., if and only if a” - a” is a positive definite matrix where arj = 1X i X j dv. Examples 1. I f X = cr(t), the Brownian bridge and b(t), the Brownian motion, then Dcr < Db. For cr(t) 0tb(1) b(t) implies that JlxZib(ti)lz
Db
=
Jlx
zia(ti)l2 DU
+1 1
ti~il’
2. Let L,(o) = - i d 2 / d x 2 + & d x 2 - i w . The corresponding path integral has covariance (2w)- exp( - w I t - s I). Since Qo)-l e - 4 - Y l
=
( 2 7 q 1 / ( k z + sz)-
leik(x-y)
dk
we see that as w increases, Dq, decreases in conditioning sense. The description above is an active one; i.e., we fix the Xi’s (thought of as coordinates) and vary measures. It is useful to change the point of view to a “passive” one where we fix the measure as dv and think of changing from X i to Without loss, we can extend the indexing set for { X i , d v } to a Hilbert space and suppose that 1X i X j dv = 1g!~(J;:)+(fj) dv = (,h, f,)for suitable J;:,fj E 2.Now define an operator A from 2 to Yf by
xi.
Then 0 IA II since p < v. Let B = A’’’. Let 41(.f) = & B f ) , &(.f) = 4((1 - A)”zf).Thenzi = ~,(,l;.)withthemeasuredvisa“model”for(X,, dp). Consider the process 410 42ri.e., the product of dv 0 dv with (41
04 2 ) ( f ) ( X , Y ) = 4 l ( . f ) ( X > + 4z(f)(Y)
145
13. Other inequalities
0 4, is a model for (Xi, dv). Now Jensen's in-
It is not hard to see that equality says that
Thus given any function F of the X i , there is another function G of the X i given by
so that Jexp(G) d p
s
(1 3.5)
lexp(F) dv
Can one make the map F H G more concrete? The answer is yes, and the point is that j . . . dv, is the same as taking a conditional expectation with respect to &. Thus, by (3.9'), F
=
:exp
1
Cajxj
( j
:
v
=
)
:exp ~ C ~a j X j :,, (i
where we use :-:" and :-:,,, because in the original active picture :-: depends on covariance; e.g., :exp(X,):,
=
( ;
exp X I - - Jx: d A )
for A = p, v. A little more useful in applications than :exp $(f'):are the Wickordered polynomials :P(4):defined by the formal generating functional m
C tfl:@':/n!= :exp(t4): = exp(t4 i t 2 ( 4 2 ) ) -
fl=O
for Gaussian
6's. Thus, e.g.,
:44:= 44- 6(4')4' + 3(4')' Since the map F
H
G is linear, it takes :4fl:v into :@"?.
We thus have the following theorem. Theorem 23.23 (conditioning inequality [120]) Gaussian processes and if P is a polynomial, then
If p
< v,
for two
146
IV.
Example
Inequalities
If : x ~is: the ~ polynomial :x4:0 = x4 - 3 o - ' x 2
+20-2
then the above and Example 2 above imply that the ground state energy of :~, L,(w) + L X ~ where L,(w)
+
=
-
'
lim t - In Sexp( - J:~q(a)4: ds) Dy,
1-
a,
increases as o increases. Thus the ground state energy of 1 d2 2 dx2 increases as w does. This can be turned into an inequality on the derivative of the ground state energy of -3 d2/dx2 ax2 x4 under change of a.
+
+
***
D. Hypercontractivity Finally we state an estimate which has been very useful in quantum field theory: it has not yet had any striking application to (one-dimensional) path integrals. Theorem 23.14 (hypercontractive estimates) Let E ( . 1 q(0)) be a conditional expectation with respect to the oscillator process. Suppose that p and r are such that
( r - 1) I e2'(p - 1) Then
I
IIE(f(q(0) q(0))IIr
Ilf(4(t))llp
where 11 . l i p is the Lp-norm with respect to dq. Remarks
1. This inequality for r - 1 < e2'(p - 1) with a constant
a(p, r, t ) was first proven by Nelson [192]; Glimm [lo41 showed that for p , r fixed by choosing t large, one could take a = 1 ; this was significant for
applications to systems with an infinity of variables such as field theory. The full result is due to Nelson [194].
13. Other Inequalities
147
2. For proofs see Nelson [194], Gross [118], Brascamp-Lieb [21], and Neveu [ 195b). 3. Generalizations to other P($),-processes are due to Rosen [222], Eckmann [70], and Carmona [31]. 4. Certain kinds of improvements to Holder's inequality ("checkerboard estimates") in special cases come from this estimate and the Markov property; see [120].
Magnetic Fields and Stochastic Integrals
14.
lt6’s Integral
l
Our goal here is to define f (s) db and more generally J g(b(s))db. This integral cannot be viewed as a Stieltjes integral, for b is never of bounded variation (equivalently the curve b(s) is not rectifiable):
Theorem 14.2 o(
For one-dimensional Brownian motion, define for
> 0 and n a positive integer f(b; n, a)
=
5I
k=l
(a) (b) (c) (d)
148
b(k)
-
b ( 7 )
1
If a < 2, thenf(b; n, a) + co as n + co with probability one. If a = 2, thenf(6; n, a) + 1 as n + cc with probability one. If a > 2, thenf(b; n, a ) -+ 0 as n + cc with probability one. The convergence in (b), (c) is also in any LP-space with p < co.
14.
149
lt6’s Integral
By the indepencence of the lb(k/2”) - b((k
-
1)/2”)1 as k varies
where we have used E ( [ X - E ( X ) I 2 ) I E ( X 2 ) . Thus
If CI < 2, then for any fixed k , ~ ~ , 2 “ ( ~ > k for n sufficiently large. For such n :
E ( f ( b ;n, a) < k ) I E ( f -
2”(1-(l/2)a)ca <
-Lc2
a 2”(1-(l/’)a)
1
I
by (14.1). Thus, by the first Borel-Cantelli lemma,f’(b; n, a) 2 k for n large. Since k is arbitrary, (a) is proven. (b) and (c) have similar proofs. For (b), one replaces E ( f < k ) by E( I f - 1 I > E ) and for (c) by E( I f I > c). L p convergence in case (b) for p = 2m (m an integer) is by direct calculation of E( I f - 1 12”) (or one can use a suitable corollary of hypercontractive estimates). In case (c), we just note by the triangle inequality for ~ ~ ~ ~ ~ , , :
E( I f ( b ; n, a) 1”)
goes to zero.
5 2””( I b( 1/2”) Ip“) < - pP(l-(1/2)a)
I
Remurks 1. By the extra devices used in the proof of the strong law of large numbers (Theorem 7.14), one can replace 2” by n and take n to infinity. 2. The same result (except the 1 in (b) is replaced by v) holds for vdimensional Brownian motion; indeed, all but (b) follow from Ib, I I Ibl I Ibil, and (b) comes from Ib12 = C 3. If “probability one” is interpreted to mean “except for a set of measure zero,” the same result holds for Wiener measure. By local equivalence, similar results hold for the oscillator process and the Brownian bridge.
150
V.
Stochastic Integrals
For suitable f’s, functions of’s alone, Payley et al. [1991 succeeded in defining 1; f ( s ) db(s) = G ( f ) by the following device: Iff is C’ and f(1) = f ( 0 ) = 0, then de$ne G(f) = -
lo1
f‘(s)b(s)ds
a formal integration by parts. With this definition rl
E(G(f)’)
=
rl
ds J dt f ’ ( s ) f ’ ( t ) min(t, s)
0
0
lo 1
=
I.f(s>12 ds
since (a2/& a t ) min(s, t ) = 6(s - r). Thus G ( f ) extends to a linear map from L2(0, 1) to L2(C(0, l), Db). We plan to extend this definition vastly using ideas of It6 [134]. What we learn from the above is that despite the fact that b is not smooth, one can define f db for moref’s than one might expect. Moreover, L2 calculations and extensions in L2 are useful. Our presentation below will actually settle for considerably weaker results than obtained by Jt6, for we will settle for an L2 definition of 1f ( b ( s ) )db similar in flavor to that above; i.e., we will define the integral for nicef’s and extend by L2continuity. It6 has a “pointwise” definition which is more complicated but yields more information about the result. For example, F(b, t ) = f0 f(b(s)) db can be shown (for anyffor which it can be defined) to be continuous in t for almost every b. All we get is continuity in L2-sense. For our purposes, this L2 definition will suffice. See McKean [183] for the “pointwise treatment. The basic idea of lt6’s definition is to take ”
f ( b ( s ) )db
=
lim
n-m
m= I
f(br+))k(:)
-b
r G ) ] (14.2)
Before discussing the limit, we want to note some important and perhaps surprising aspects of this formula. First, the differentials stick out into the future; i.e.,f(b(s)) is evaluated at the left endpoint. This is convenient because ,f(b((m - 1)/2”)) is independent of b(m/2”) - b((m - 1)/2”)whilef(b(m/2”)) is not. For this reason, we shall be able to define this integral for nonsmooth f ’ s ; if we had taken the differentials pointing into the past, it turns out that f‘ would have entered naturally. Secondly, the ordinary formal rules of calculus do not hold for the stochastic integral; e.g., 1: b(s) db = &b2(1) - 1) not i(b2(1))! [Note that h(0) = 0.1 The computation of this integral is
151
14. lt6's Integral
illuminating since it also illustrates the difference between differentials pointing into the past and the future. Define
We have that for any n
I+(n)
+ I - ( n ) = b(1)2
since
[h(s)+
b(m+)][b(s)
-
b(mG)]
=
h2(g)
-
b ' r 4 )
and the sum telescopes. On the other hand, by Theorem 14.l(b),
I + ( n ) - IL(n) + I pointwise as n -+ 00. It follows that lim I k ( n ) = +(h(l)'
2 1)
n+m
showing the difference between I + and I - and their unintuitive answer. Of course &It + I - ) gives the intuitive answer here but if we took 1hk dh, this prescription would not yield ( k + 1)- bkt I . Theorem 14.2 Let f be a C' function on R withfand f' bounded. Then, the limit on the right-hand side of (14.2) exists in L2-sense and ( l l . l 1 2 = L2-norm)
(14.3) Proof'
Define
Now for i # j , E(q,, iqn.j ) = 0, since for .j > i there is a factor of h(j/2") h((j - 1)/2")independent of the rest ofq,, q,,, j , and E(b(kl2")) = 0. Moreover, sincej(b((m - 1)/2")) is independent of h(m/2") - h((m - 1)/2"), we have
152
V.
Stochastic Integrals
Thus, (14.4) Since Emf(b((m - 1)/2”))21/2“converges pointwise to j: f ( b ( ~ )ds) (f~ and b are continuous) and the sum is dominated by I l f l l i , (14.3) will follow if we show that J n ( f ) converges. A calculation similar to the above shows
1
2”-1
C
m=O
where C L2. I
=
Ilf’llk
so I f ( x ) - f(y)12 I C l x
-
yI2. Thus J , ( f ) is Cauchy in
We now extend the definition of j f(b(s)) ds from C’-functions to an arbitrary f’with
finite; e.g., f’ bounded near zero and L z will certainly suffice. This integral is in L2 for any suchf. We can actually go one step further: Let xa be the characteristic function of ( - a , a ) and suppose that
as will be true iffis Lm near zero and locally L2 elsewhere. Then for each a, we can define 1: fx,(b(s)) ds. We claim that for almost every b ( 14.5)
exists pointwise and is finite. For b(s) is continuous on [0, 1) and so bounded. Thus for any fixed b, (14.5) is independent of a for a large. We also want to extend the scope of definition. First we can clearly use
14.
153
it6’s integral
v-dimensional Brownian motion and define j h f(b(s)). db, and we can easily define jt in place of jh. More importantly for the applications we have in mind, we can define j h f(o(s)). do for the &,-Wiener paths. The construction is identical to the above but now
since o(s) has the distribution of Lebesgue measure. Extending via the analog ~ fo f(o(s)) do can be defined for of (14.5), we see that for any . f ’ L:,,(R”), popalmost every path. This is the integral we will need in Section 15. This random variable is called the Ita integral. If one looks at the construction above, it is clear that we can allowfto be a function on R” x R and define 1; .f(b(s), s) db so long as E(jA I f(b(s), s)l’ ds) < 00. It is a simple exercise that sh f ( s ) db agrees with the Payley et al. [199] definition. In fact, one can deal with general nonanticipatory functionals, i.e., functions ,f(b,s) with the property that f(.,s) is only a function of {b(t)1 t I s}, i.e., is BS-measurable;for example, if z is a stopping time one can define & f ( b ( s ) ,s) db. The only subtle step in the construction for general nonanticipatory functionals is the density of ”smooth functions” for which one can obtain the integral as a limit; in fact, it is more useful to deal initially with f’s withf(-, s) constant for k/2” I s < (k + 1)/2” and take limits; see, e g , [183]. For calculation and understanding of stochastic integrals a lemma of It6 [135] is particularly useful. We state it for the full Wiener integral since it is somewhat easier to give optimal conditions o n f ; a similar result holds for the Brownian path if, for example (to state an overly strong condition), fis C2 globally. This is a special case of a more general result of It6 which will be featured prominently in Section 16.
Let f ( x , t ) be a function on R” x [0, t] Theorem 24.3 (It8’s lemma) and use Vj; etc., for x derivatives and ,{for t derivatives. Suppose that j i s in ,!& and that its distributional derivatives Vf, Af, and,flie in I&. Then
(Af)(o(.s),s) ds
for almost every o.
+
.f(o(s), s) ds
(14.6)
154
V.
Stochastic Integrals
c for Proof By a limiting argument we can suppose that f ~ C and notational simplicity that t = 1. Write
+ Bn + Cn
.f(a(lX1) - .f(a(O), 0) = A n
with
=
o(m/2") - o ( ( m - 1)/2"). By construction An
in L2 as n
+ 00.
+
Jol(~f)(a(s)~ s>* dm
Using Y) -
I c l x - y13 and Theorem 14.1, we see that up to a term going to zero in L2-norm, B, equals
As in the proof of Theorem 14.I ,
B, equals
plus a term going to zero in L2. Thus B, using
I f ( x , t ) - f ( x , s) - fcx, one easily sees that C,
-+
Sof(a(s),
-+
s>(t -
s) ds.
3 Jb (Af)(o(s),
$1
5
s) ds. Finally,
c I t - s12
I
(14.6) is often written in infinitesimal form as the formal expression df
=
V f do
+ (4Af + f ) d t
(14.7)
155
14. It6's Integral
The surprise is that second derivutives o f f uppear in thejrst order differential. (14.7) can be expressed even more succinctly by dto; d t O j = dij dl
(14.6) can be used for calculational purposes: Example (v = 1) f ' s with &f" stochastic integrals. For example, if
+ ,f = 0 will have particularly
simple
f(x, t ) = exp(ax - $a2t)
then df = Vf do.Recognizing the Wick-ordered exponential f ( o ( s ) , s) = :exp(ao(s)): and taking w to b, we see that j::exp(ab(s)): tlb
=
a- :exp(ab(t)): - ix-
(14.8)
Thus, it is the Wick-ordered exponential, rather than the ordinary exponential which stochastically integrates like an exponential ! It is a curious historical coincidence that the work of It6 leading to (14.8) is approximately simultaneous to that of Wick on "normal ordering." If (14.8) is expanded in a, one finds that j::b"(s): dh
If we take n
=
=
(n
+ I)-l:b"+'(t):
I we find that j:b(s)db
=
):bz(t):= i ( b 2 ( t )- t )
a result already obtained. It6's lemma provides directly the connection of Brownian motion and = H , and also a new proof of the Feynman-Kac formula (somewhat related to the second proof we gave). A key observation is that because the differentials point into the future, we have that E ( f ( b ( s ) )db) = E(f'(b(.s))E(db) = 0 since E(db) = 0; i.e., E(j! f'(b(s))db) = 0, more strongly
-4A
(14.9)
if g is measurable with respect to {b(u)lu I u}. Choose f ( x , s ) = ( e - c t - s ) Hgo)(x), so .f = H , f. Thus f + 9Af It6's lemma implies that
=
0 so that
156
V.
Stochastic Integrals
Using (14.9), we see that
E ( f ( b ( 0 3 9 ) = E(f(b(O), 0)) Since f(b(t), t ) = y(b(t)) and f(b(O),0) again that
=
(e-'H"g)(0),we have proven once
E(g(b(t))) = (e-'H"g)(0)
The proof of the Feynman-Kac formula is similar but more complicated. V and Here, we take H = H ,
+
f(x, s) = ( e - " - s ) H g ) ( x ) Then, at least for sufficiently nice V's (say, V E Corn), f(b(sX
= f(b(tX
t ) - fV(b(u))f(b(u), u ) du - l(Vf)(b(u), u ) . db S
Notice that by (14.9) the last term has zero expectation. Iterating this equation once:
If one recognizes the object in [. . .] as exp( that
-Po
V(b(s)) ds) one has proven
In Section 16, we will use the general It6 lemma for an even slicker proof of the Feynman-Kac formula.
***
157
14. It6's Integral
We next make a simple calculation: Theorem 24.4 (v
1)
=
(14.10)
CI
=
SI f(s)12ds + 2
Proof CL =
The object in the exponential in (14.10) is clearly linear in b, so
E([J f ( s ) d b + !g(s)b(s)ds12). We recognize the first and third terms
~ .compute the as just the expectation of ( j f ( s ) dh)' and of ( jg(s)b(s)d ~ )To middle term pass to the Riemann sums defining 1f ( s ) db and note that t < s
E([b(s + As)
-
b(s)]h(r))=
-
s,
AS,
s
+ As
t>s+As
I
We note that iffdepends on some random variables independent of b, and E represents expectation with respect to b, then (14.10) still holds, e.g., for v = 2:
which is J exp( - ( H o - *f2))(x, 0) dx.
*** Finally, we want to discuss the definition of stochastic integrals with respect to the Brownian bridge, ct, and the oscillator process, q. The former is especially important since it will allow us to define the integral for almost every Wiener path with fixed endpoints. At first sight, this seems difficult, for the increments of CI depend heavily on the present: since c1 must reach a(1) = 0 at t = 1, it must know where it is at a given time to decide where to go next; more prosaically, one checks that for s I r E([a(t
+ At)
-
~(t)]ct(~)) =
-S
AZ
and E((q(t
+ At) - q(t))q(s))= ~e-"-s'[e-At 11 -
The key to deciding how to overcome this problem is (5.6)
158
V.
Stochastic Integrals
and u(t) = ( 1 - t)b
:t)
(14.12)
__
(I
s.
These formulas would allow us to define j . dcr and d q in terms of db but, more simply, they suggest to us simple "increments" which are independent of the past; namely, ~ ( tA[) ,
=
+
At
+
~ ( t At) - ~ ( t ) --a(t) 1-t
(since (14.12) says the increments of a(t)/( 1 - t ) are independent of the past) and
A(t, At)
=
q(t
+ At) - e-A'q(t)
One trivially checks that for s < t : E(a(s)y(t, At)) = 0 = E(q(s)A(t,At)) and that E(ly(t, At)12)
=
E(lA(t, At)I2)
= $(1
Both expectations go to At as At nice enoughf
( 1 - t)-'(l -
-
t
-
At)At
eC2*')
+ 0 and
are bounded by At for all At. For
SA
exists and has L2-norm, E ( , f ( ~ ( s ) )ds, ~ )for we need only mimic the construction of db. But clearly, by the continuity of paths:
exists and equals fA ,f(a(s))cr(s) (ds/(l - s)); there is no divergence at s = 1, since ~ ( s )-,0 faster than (1 - S)''' -'by a law of iterated logarithm. As a result, for nice enough f's,
exists and, since
15.
159
Magnetic Fields
we have
In this way, it is easy to see that one can define j h . f ( c ( ( s ) ) da for almost every a, iffis in L" near zero, and in Lp,, for some p > 2. Similarly, one can define J: f ( 4 ( s ) )4. One can summarize the relevance of the above considerations for the full Wiener integral by: Theorem 24.5 Let g(to) be a function of the Wiener paths up to time t with the property that llgllm < 00 and
is continuous for almost every o.Let F be a bounded continuous function on R" and letfbe a C'-function on R". Then
is continuous in x, y for each fixed 1. Proqf
Rewrite the integral using (4.12) as
The integrand is uniformly bounded and thus continuity for almost every suffice. This is easy by hypothesis and construction of dcc. I
ci will
15. Schrodinger Operators with Magnetic Fields
Consider the quantum mechanical energy operator for a particle in a magnetic field B(x) with vector potential a(x) (B = V x a) H(a, V ) = +(- iV - a)2 + 1/ where we will later discuss hypotheses o n a, V and the precise definition of H. We want to prove the following which we call the Feynman-Kac-It6 formula: (15.1)
160
V. Stochastic Integrals
where
F(o, t)
= -i
s:
1so .
a(o(s))
*
do
-
-
r
(div a)(o(s)) ds -
s:
V ( o ( s ) )ds (15.2)
In this section, we will give an analytic proof of (15.1) and in the next section a “stochastic integral” proof. We first learned of (15.1) from Nelson [195]; a probabilist would view it as a Cameron-Martin formula for “imaginary drifts” (see the discussion in Section 16) and so of considerable antiquity. There is even discussion of a’s which are noncommuting matrices such as would arise in nonrelativistic couplings to Yang-Mills fields; see [6a, 40a, 168al. Before turning to the proof we want to remark on several aspects of the formula itself. First, even though H has an u2 term, no u 2 appears in F. The i(div a ) term is a reflection of the fact that the kernel of e-IH is Hermitian symmetric: Changing o(t)and o(0) by taking s into t - s does not merely change the sign of a do because the differential do has a preferred end. For this reason, the integral term in F should more naturally be the symmetric integral
-
S
= nlim -m
‘y
m = l 2
+ a(or+))]
[a(o(;))
*
[ a ( ; )
-
o(?)]
It is not hard to see that in terms of the It8 integral
S
=
s:
a(o(s)) * do
+
-
2 l
S0
(div a) ds
That is, the div u term restores the ”time-reversal symmetry” destroyed by the It8 convention of having differentials point into the future. One can also see the effect of this div u term if one considers gauge transformations. The change of a to I = a + grad 1 should not change any physics since B = V x a = V x I. In fact
eiAH(a,V ) e - i A= H ( I , V ) Thus, if a is replaced by 5 we must have (1 5.3)
To verify (15.3), we use It6’s lemma (Theorem (14.3): [:(VA)(w(s)).
do = 1(o(t))- A(o(0))-
15.
161
Magnetic Fields
Since div(Vll)
=
All, we have that
sdyll. do +
1,
1 ’
div(Vll) ds
=
ll(o(t)) - ll(o(0))
verifying (1 5.3). Note the div a term was essential for gauge invariance. We now turn to the analytic proof of (15.1): Lemma 25.2 K,(x, y)
=
real-valued. For s > 0, let
Let a E C;(R’’). (27~s)-”’~ exp( - I x
y I2/2.s) cxp(fi[a(x)
~
+ a(y)]
*
(x
-
y)) (1 5.4)
Then
( 15.5)
( 1 5.6)
Proof As usual, let P,(x, y) = (2ns)-”” exp( - Ix - yI2/2.s) which is the integral kernel for e-sHn.Then I K,(x, y)J I P,(x, y) implies that
I(Q,f>(x)I I ( e - s H ” l f l ) ( ~ )
(15.7)
so Q, is as bounded operator with llQsll I llepsHoU= 1. Since (d/ds)P, = +Ay P, and we can integrate by parts (since a E C; and,f E D ( H O ) )
d ds - (Q,,f)(x>
Jp,(.,
=
Y)Ay[e(l/2)i(a(x)+a(y)).(x-y)
f(Y>l d”Y
Using A(ei’ff)= ~j’~[Af+ 2iVh- Vf’ + (iAh vy.i[+(a(x)
+ a(y>).(X
- Y)I =
A,,[. . .] and the fact that P,(x, y)
+
=
6(x
-
~ ( L I , (+ x ) ai(y))
-
(Vh)’)f]
1
+ 1 (viuj>(y)(x ~
2,
-(div a)(y) + t(Aa)-(x - y)
-
y) as s 10, we obtain (15.5).
1
-
y)j
162
V.
Theorem 25.2
Stochastic Integrals
With Ho(a), Q, as above, we have that s-lim(Qi,,)" = e - t H n ( e )
(15.8)
n-rm
One need only mimic the proof [189] of Theorem 1.1. Let For anyfED(Ho), we have that limi-ro(d/dt)((St- Q,)f) = 0 by (15.5) and thus forfE D(H,)
s
Proof -
e-tH
lim Iln(Sf/n -
QtdfII
=0
n- m
with convergence uniform on compact sets in the D(Ho)-norm. Since, as in (1.9, II(Q:/n - St)fII 5 n
SUP
II(Qt/n - Si/n)SsfII
oss5t
the proof is complete.
Theorem 15.3
Proof
I
For a E Corn, I/ = 0, (15.1-2) are valid.
Clearly
where
As n
-, 00, we have the L2 convergence
-
F,.(w, t ) -+ - i Jb'a(o(s)) do
'
i
1 lo
- -
(div a)(o(s)) ds
by the usual arguments. By passing to subsequences, we can be sure of convergence pointwise almost everywhere. Since
we see that ( f , e-rHn(a)g) = lim (f;QYing) n- a,
(by Theorem 15.2)
15.
163
Magnetic Fields
Now let a E Li,([w").Then - iV, - a j is a symmetric and thus closable operator on C;. Let G, be its closure. Let H,(a) be the unique positive D(Gj) with form self-adjoint operator with quadratic form domain
n;=
Y
(f*Ho(a)g) = C ( G j f , Gjg) j= 1
The following result of Simon [260] which we state without proof is motivated in part by work of Kato [l55]. Let a E L~oc([w"). Then C r is a form core for Ho(a). Theorem 25.4 Moreover, if a,, + a in L i C (i.e., Il(a,, - a),f1I2 -+ 0 for all f~ Cr), then (H,(a,,) I)-' + (Ho(a) l)-I strongly.
+
+
This result allows us to prove: with div a = 0 (distributional sense), and Theorem 25.5 Let a E L~oc([wv) let V obey the hypotheses of Theorem 6.2. Then (15.1-2) are valid for H(a, V ) = Ho(a) V.
+
Proof First take V = 0 and let a,, E Cr be chosen so that div a,, = 0 and a,, + a in Lit. By passing to a subsequence, we can assume that a,, + a pointwise as well. Then 1a,,(oJ(t))*do--t a(o(t)) .do almost everywhere in (0,and so, by the dominated convergence theorem, the right-hand side of (15.1) converges. By the strong resolvent convergence of Theorem 15.4 and the continuity of the functional calculus [214], the left-hand side converges. This establishes (15.1-2) for a in & and V = 0. Bounded Vs then follow by the Trotter formula and the first proof of Theorem 6.1. By following the proof of Theorem 6.2, one can accommodate general V's so long as one knows that V-H,-form bounded with relative bound u < 1 implies it is H,(a)-form bounded with relative bound a' < 1. We will prove this below. I One important consequencc of ( 1 5.1-2) is the following. Theorem 25.6
Under the hypotheses of Theorem 15.5,
I exp( - tH(a, V))4 I
I exp( - w
o
+ V ) )14I
point w ise.
Proof' (1 5.9) is equivalent to ((f,e-'"'"*")g)I
I
(If[,
e-t(Ho+V)
Isl)
( 15.9)
164
V.
Stochastic Integrals
which follows immediately from (15.1-2) if we note that leFl = eReFand Re F is independent of a. Remarks 1. (15.9) was originally conjectured by Simon [255] as a special case of a general conjecture on when le-rAdl Ie-rBIqhI.Nelson (reported in [255]) remarked that this conjecture was an immediate consequence of the Feynman-Kac-It6 formula. The general conjecture was then proven independently by Hess et a/. [127] and Simon [256] providing a purely functional analytic proof of (1 5.9) relating it to an inequality of Kato [1531. 2. As is so often the case, one can avoid functional integrals and only use the "Trotter formula." For (15.9) follows easily form (15.7) and (15.8). 3. There is a circularity in the above in that the proof of Theorem 15.4 [260] begins with a direct proof of (15.9). In any event, (15.9) for smooth a's can be proven as indicated (and this is essentially all that is proven in [255, 127, 2561). In some sense, the proof of (15.9) in (I2601 is the most elementary !
Simon's original motivation for (15.9) was concerned with problem (3) of Section 1:
00
+
If H , I/ has the property that Tr(e-t(Hot"') < Theorem 15.7([255]) for all t, then so does H,(a) V and T ~- W( d a~) + U ) T ~- W( o +~ U) (15.10)
+
Proof If 1,461 I BI 41 pointwise and Tr(B*B) < 00, it is easy to see that Tr(A*A) ITr(B*B). Thus (15.10) follows from (1 5.9). I
The real power of (15.9) in studying H,(a) + V was realized by Avron et a/. [4] and Combes et a/. [37]. By combining (15.10) and (9.3) we recover the following theorem. Theorem 25.8 ([37]) Theorem 9.2. Let
Let a E Lkc and let V obey the hypotheses of
(15.11) Then
15.
165
Magnetic Fields
The proof in [37] does not use path integrals but rather the GoldenThompson inequality. By following the arguments in Section 10, we can recover the result' of Combes et ul. [37], originally obtained by different methods (see also [260bl). Theorem 15.9 Let V beacontinuousfunctionon R"withexp(-/jV)~L'. Let a be an R"-valued C'-function on R" with div a = 0 (for simplicity). Let H ( h ) be given by (15.1 1) and let
ZQ(h)= Tr(exp( - BH(h)))
Then Z,/Zc
-+
1 as h + 0.
Proof As in the proof of Theorem 10.1 by letting W = PV and choosing a suitable multiple of a, we need that t V i 2Tr(exp[-tH(t-'l2a,
r-'W)])
-
i r:J
-+
-
( 2 7 ~ - " J/ ~e - W d v . ~
1
'/2a(o(s)) * do dpo,x, y,
for almost all x,y. By 'Theorem 14.5, the right-hand side of this last expression is continuous in Y and y so as usual:
Rewriting everything in terms of the Brownian bridge, a, we see that we must show that
166
V.
as t 10. Formally, this is true since a(1) - a(0) = 0.
fA a(x)
*
Stochastic Integrals
da = a(x) * A j da = 0 by
f da =
The actual proof is not much harder than this formal proof. Let
Let
and F(a, t ) = FO(a,t)exp
a(x
+ ,,ha(s))
*
da
Then
By Theorem 10.1, the first term converges to j e - w c x dx ) as t + 0 and as in ) + j e - W ' x ) d x as t + 0. Since E(llallm I R ) 1, that proof, j e - w - c x ; R J 1dx as R + GO we can, given E , find T and R so that for t < T : -+
On the other hand, as t
10,
E,(F(a, t ) ; llallm I R ) + e-Wcx)E(Jla(lm I R)
for each fixed x since IF(a, t)l I exp(- W+(x; R f i ) ) which is dominated for t < 1 by a function in L'(Da) and for each fixed a, 1
F(a, t ) -+ enp( - Jo W(x) ds - i Jola(x) . da) = exp( - W(x))
For each fixed K , we conclude by the dominated convergence theorem that
J; xl
e-W(x)E(llall, < ~ ) d x
15.
167
Magnetic Fields
Finally, as in the first part, I
n
I
which can be made small if K is large, R is large, and t is small.
I
A number of applications of (1 5.9) to spectral and scattering theory for H(a, V )have been made by Avron el ul. [4].Typical is the following which we used in the proof of Theorem 15.5.
Theorem 25.20 ([4]) Let I V1 be a multiplication operator which is H,-form bounded with relative bound a. Then for a E L:,,, it is H,(a)-form bounded with relative bound at most a.
Proof By (15.9) and ( A constant c), we have that I(Ho(a)
+ A)-1i2
=
~ ~ e - ” e - ~ ‘dtt -(for ~ / suitable ~
+ 2)- ‘”4 I s ( H , + 2)-
1/2
I4 I
so that
II I ~11’2(Ho(a)+ 1)-1’24115 II I W i 2 ( H o + A)-1’2141II which implies that
11 I V11/2(Ho(a)+ 2)-1i211 I 11 I VI1I2(HO+ A)-1/211 Since B is A-form bounded (B, A 2 0) with relative bound B’”(A is bounded and limA+m~~B1’2(A + A)-1’2112 is complete. I
+
tl
if and only if the proof
= a,
For fixed V and a, let K,(a) = I V11’2(H,(a) + y ) - l 1 VI’l2. By following the proof of Lieb’s formula, one finds that for any F related to a positivef by (8.71,
I F(K,(a))4 I 5 F(K,(O)) I4 I Since F(K,(a)) is a positive operator, this implies that Tr(F(K,(a)N on account of the following.
Tr(F(K,(O)))
( 15.12)
168
V.
Stochastic Integrals
Lemma15.11 Let A and B be bounded positive operators on an L2space with IA$I I Bl$l pointwise. If B is trace class, then so is A and Tr(A) I Tr(B). Remark
This is false if A is not positive.
Proof Let (S,}Y=, be a family of disjoint sets of finite measure and let P, be the projection onto all functions of the form a i x s , .Then, P, A P , = A , and B,7 are trace class and since Tr(As) = p(Si)pl(xs,,Axsi), the Tr(B). One can find a net of hypothesis implies that Tr(A,) ITr(B,) I S’s, S,, so that A , + A strongly. That A is trace class then follows by a general result [259]. 1
c;=l
We can now follow Lieb’s proof of the Cwickel-Lieb-Rosenbljum bound and then the proof of stability of matter to obtain the following theorem. Theorem 25.12 ([4, 371) Matter is stable in an arbitrary magnetic field; i.e., (9.22) continues to hold if - A i is replaced by -(idi - ai(x))’. The constant c is independent of a and can be chosen to be the constant of the proof of Theorem 9.6. Remark N ( V , a)
One might think that (15.12) implies that
= dim(spectra1 projection for (- co,0)
for H(a, V ) ) IN ( V )
but this is not true; see [4].
*** Finally, we do a calculation of the explicit kernel of e-tHo(a) when a l ( x ) = - 3 B x 2 , az ( x ) = @x,, a3(x) = 0 [constant field (0, 0, B ) ] . The kernel is [84,4l
x exp{
+ (x,
-
-
&
(x3 - y,)’
-
coth
G) -
Bt [(xz - y J 2
1 - X’Yd} Y1)’l - -iB(X,Y2 2
Since the a’/dz’ term commutes with the remainder, without loss, we restrict to two dimensions. For notational simplicity we consider the calculation with t = l, x = (0, 0), y = (0, y). Actually using scaling covariance,
15.
169
Magnetic Fields
translation invariance (not the usual realization of translations [ S ] ) , and rotation invariance, one can go from this special case to the general case. Thus, we need to compute
z
(2.nt)-
l,-Y*/2f
j e - i x D a , Da,
where a, and a2 are two independent Brownian bridges and ' x = 21 B[ jo cc,[dd* + y d s l
1
-
-
jo(a,
+ sy)da,]
Since X is quadratic in a, this integral is, in principle, explicitly calculable. First we note that
X
=
-B
5:
(u,(s)
+ .~y)dal
The integration by parts is justified since
With f' = a, + sy and g = a i the left-hand side is zero, and the first two terms converge to It6 integrals and the last term to zero since a1 and a, are independent. Next, we note that E(d~,(.y)d . l ( t ) )
=
6(~ t ) ds
-
ds dt
which follows from b(s) = a(s) + sb(1) (orthogonal direct sum) and db = dcc + (ds)b(l). Since X is linear in a l ,
so
with
Y
=
/oi(a,(s)
+ sy)2ds -
(so1
(a,($
+ sy)ds )2
170
V. Stochastic Integrals
The integral for Z can now be computed as we did in Section 6 in our second proof of Mehler's formula.
16. Introduction t o Stochastic Calculus
In this section, we will present a "formal" treatment of some aspects of calculations with stochastic integrals. We will be formal in that we will use the general stochastic integral for nonanticipatory functions alluded to in Section 14, and that we will not give a proof or even precise hypotheses for the main result: the general lt6 lemma. The reader can consult [86, 1831 for more details. A Stochastic integral is a random function c(b, t ) obeying
c(b, t ) = c(b, 0 ) +
Sd
f(b, s)' db +
sd
g(b, S ) ds
(16.1)
where f and g are nonanticipatory functionals with suitable Lp properties and c(b, 0) is independent of b. One writes (16.1) in the shorthand:
dc
=
f * d b+ g d s
(16.2)
The following result is critical. Theorem 16.1 (It8's lemma) If ct, . . . ,c, are stochastic integrals and u is a C2-function on R" x [0, co) with some mild restrictions on growth at infinity, then x = U(C,, c 2 , . . . ,c,, t )
is a stochastic integral, and
au m au dx=C-dCi+-ds+at
i = t aYi
-
-
where dci dcj = fi fj ds(dci = fi db
1 m aZu ___ dci dcj 2 i, j = ayi ayj
1
(16.3)
+ gi ds); i.e., db, dh, = d,, ds.
The pattern of proof is identical to that of the special case Theorem 14.3. Indeed, if the c:s are approximated in the right way, it is the identical proof (see [1833).
+ gi ds, then d(ctc2) = c t dc2 + c2 dc, + f t fi ds
Example 1 (product rule)
If dci = fi * db
*
(1 6.4)
171
16. Stochastic Calculus
Let us compute the differential
Example 2 (Feynman-Kac formula)
of
where V , f~ C;. Since z = V(b(s)) ds is a stochastic integral with d z = V ds, we have by Itij's lemma and ( d ~ =) 0~ that d(e-') = -e-"V ds. Again by It8's lemma, cff = Vf. db + t ( A f ) ds. Thus, using (16.4), ds)[Vf- db - ( H f ) ds]
dc = exp( - J:V(b(s)) where H
=
-3Af
+ vf. Thus, if we define
and we use (14.9) that E(db)
=
(Q,f)(x)
=
which yields Q,
=
0, we see that f(x)
.
=
/:ds (QAHfMx)
e-'" and so a new proof of Feynman-Kac formula.
Example 3 (Feynman-Kac-It6 ferential of c
-
exp[ - i J:a(b(s)).
db
formula)
Let us compute the dif-
1
I
1J"
- -
with a,fE C:. Then, by It6's lemma d(e-iz) = - i e - " d Z - &-"(dz)2 - -ie-'"[a-db
= e-'"fb(t))
(div a)(b(s)) ds f(b(t))
+ &V*a)ds]
- 3e-iza2ds
so using (16.4) dc = e - " [ ( .. .) db
+ ($Af - ia
*
Vf
-
&V * a)f
-
f a ' f ) ds]
recognizing the occurrence of - Ho(a)f [see (15.6)], and writing (Q, f ) ( x ) = E(exp( - i :a/
- db + /:
we find that
yielding the Feynman-Kac-It6 formula.
div a d s ) f(x
+ b(s)))
172
V.
Example 4 (drift)
Stochastic Integrals
Let a E Cp and let
x(t)
=
b(t)
+ J:a(b(s))ds
which for obvious reasons is often called the drift generated by a. Then dx = db + ads so that
d(f(x(t)))
=
Vf
*
db + (a * Vf
+ *Af)
ds
Following our usual procedure ~ ( f ( x ( " ) = (e-'Xf)(O>
where X is the differential operator
X f = -$A,f
-
a-Vf
(16.5)
which is thus called the generator of the drift.
Example 5 (Cameron-Martin formula [30]) and c = ezf'(b(t)).Then dc
=
ez[(Vf
+ af)
*
db
Let dz
+ (a - Vf + $Af)
=
a * db - $u2 ds
ds]
so in terms of the generator X in (16.5):
E(ezf(b(t)>) = (e -'"f)(O) This is usually summarized in the Cameron-Martin formula
Dx (Radon-Nikodym derivative) Db
-
=
exp
We can now explain how a classical probabilist would view the FeynmanKac-It6 formula. If one writes H,(a) = --+A + ia * V +u2 *i(div a), then one sees that H,(a) is the generator of a (complex) drift - ia and a potential $u2 $i(div a). Then the combined Cameron-Martin formula for a * V and Feynman-Kac for $u2 + i div a yields the Feynman-Kac-It6 formula. There is a connection between the Cameron-Martin formula and the Feynman-Kac formula which shows a connection between P(4),-process and drift processes-a connection which will reoccur in Section 19. Suppose that 1 d2 H=----+1/ 2 dx2
+
+
+
where we have added a constant to I/ which guarantees that inf spec(H) = 0. Let HR = 0 and let L be the operator L f = R-'(HRj). L is the generator
16.
173
Stochastic Calculus
of a contraction semigroup on L2(R, Q2 d x ) and L1 Kac formula, e P t Lis given by ( e - ' l f ) ( x ) = SZ(x)-'E
=
Q(x
0. By the Feynman-
+ b ( t ) ) f ( x+ b(t)) (16.6)
But notice that
where u formula: (e-'Lf)(x)
=
Q- 'Q'
=
=
h' with h
E(exp( J:u(x
=
In Q. Thus, by the Cameron-Martin
+ b(s))dh
-
(16.7) Let us check that (16.6) and (16.7) agree. For simplicity take x lemma (Theorem 14.3)
/:h'(b(s)) db
=
h(h(t)) - h(b(0))-
1 '
=
0. By It6's
h"(b(s))ds
0
By explicit calculation
h"
+ u2 = (1' + u2 = R-Z[nW = Q-l,,,
=
- (n,)Z
+
(n1)2]
21/
Thus
f u ( b ( s ) )db
-
1 ' /ouz(b(s))ds
=
In Q(b(t)) - In SZ(b(0)) -
0
establishing the equality of (16.6) and (16.7).
s:
V(b(s))ds
VI Asymptotics
In this chapter, we discuss a variety of connections between Brownian motion and asymptotic behavior, either of other processes or functions of Brownian motion. This is a subject developed by Donsker and his collaborators. In Section 17, we show that normalized random walk approaches Brownian motion in a strong sense; in Section 18, we extend the formula ]im
(Jolpi”’
dx)l’n
=
exp[ - min(f)]
n- m
to various Gaussian processes and describe its relation to the asymptotic behavior of the Rayleigh-Schrodinger coefficients of certain perturbation problems; and in Section 19 we describe some deep results of Donsker and Varadhan and their relation to the Gibbs’ variational principle of statistical mechanics.
17.
Donsker’s Theorem
At the beginning of Section 4, we introduced Brownian motion as an intuitive limit of random walks and we noted that = b(t) lim n - 1i 2X~n,l n-
(17.1)
m
in some intuitive sense. Here we want to examine the sense in which (17.1) holds. The central limit theorem immediately tells us that if we consider fixed t , , . . . , t, then the joint distribution of n-1’2XInt,l converges to that for b(ti).This simple result leaves open some important and natural questions. 174
17.
175
Donsker's Theorem
For example, we found in Section 6 the probability distribution, P , for 1 {s I h(s) > 0; 0 I s < 1) I. One would guess that as N + co, N - ' # { n I NIX, > 0} has a distribution approaching P but since this involves infinitely many X's, it is not clear that just knowing the convergence of the finite distributions is enough. In fact, we will see shortly by explicit example that it is certainly not enough for some reasonable functions. Thus, we need a stronger form of (17.1). The first hints of this strong form go back to Erdos and Kac [77] who proved an invariance principle namely if y i are independent copies of some random variable with E(yi) = 0, E(y?) = 1, then the limiting distribution of a wide variety of functions of (n- ' I 2 I yi) is independent of which yi is chosen. The reason for this is that a strong kind of convergence of (17.1) is involved-this was discovered by Donsker [53]. Kolmogorov-Prohorov [162] and then Prohorov [206, 2071 introduced the key notions of weak convergence and tightness which yield an elegant framework for Donsker's result. For more information, the reader can consult Billingsley [141, whose treatment we follow closely in places, or Parthasarathy [198al. Let y , , y,, . . . be independent, identically distributed random variables with E(yi) = 0, E(yZ) = 1, and let S, = y i . For fixed n, we want to define a process X,(t) with continuous sample paths by letting X,(k/n) = n - l/'Sk for 0 4 k _< n and interpolating linearly in between; i.e. (with [a] = integral part of a),
cy=l
X,(t)
=
n-'/2Sc,,l
+ n - "'(nt
-
[nt])yLnrl+
(0 I t I 1) (17.2)
It is trivial that the process X,(t) has continuous paths; indeed, with probability one, each y j is finite and X,(r) is piecewise linear. Thus X, induces a measure P, on C[O, 11. To state Donsker's theorem, we need the following notion.
DeJnition Let { P , ) , P , be a family of Bore1 probability measures on C[O, 13, the continuous functions on [0, 11. We say P, converges weakly to P , if and only if r
r
for any bounded continuous functionj'from C[O, 13 to R. WARNING "Weak" convergence is considerably stronger than convergence of the finite distributions. For example, if g n is the function O < t I l/n l/n I t I 2/n nt, t 2 2/n
176
VI.
Asymptotics
and if P , is the point mass at g,, then the joint distribution of any o ( t l ) , . . . , w(t,) converges to that associated to P , , the point mass at zero. But, there is no weak convergence since, e g , ,I?,(e-~l"~~rn) = e-1 + 1 = E , ( e - ~ ~ " ~ ~ m ) Let P , be the measure on TheoremZ7.Z (Donsker's theorem [53]) C[O, 13 induced by (17.2) (with y i independent, identically distributed, and E(y,) = 0, E(yf) = 1). Let P , be the measure on CEO, 13 induced by Brownian motion. Then P , converges to P , weakly. The remainder of the section is devoted to the proof and discussion of this theorem. First, we have some discussion of weak convergence. (See [I41 for extensive discussion.)
Proposition 27.2 CEO, 11 with P [ a A ]
Let P , converge to P weakly and A be a Borel set in 0. Then P,(A) -+ P(A).
=
Proof C[O, 11 is a metric space, so every Borel measure is regular (see, e g , Theorem 1.1 of [14]) and moreover Urysohn's lemma holds. It follows that for B open, any Borel probability measure, Q,and W, the family of continuous functions on C[O, I ] :
By taking complements, we see that for any closed set D
Thus, since P ,
+P
I
weakly
lim sup{ J f d P , I "f€ v, . .
.}
lim P,(Ai"') I lim P,(A) where we used f ,f dP = limn 1.f'dP, I lim sup(s f d P , I . . .). Similarly P(A) 2 lim P,(A) =
If ,(Ai"')
=
P(& then the limit exists and equals P ( A ) .
I
17.
177
Donsker's Theorem
Typical of the consequences of Theorem 17.1 and Proposition 17.2 is the following corollary. Covollury 17.3 4. Then
(a)
Let X , be the partial sums of the random walk of Section
lim E ( n - ' # { , j I n 1 X , > 0 ) I (x) n-
=
2n-l Arc sin dL
a)
max I X , I 4
(b)
fi
=
r.h.s. of (7.19)
Proof' Iffis any function in Z , then Proposition 17.2says that if P, + P weakly, then Pn(J I a ) + P(~'I a ) so long as P ( f ' = a ) = 0. Thus (a) follows from Theorem 6.10 and (b) from Proposition 7.16. I We now lurn to the proof of Donsker's theorem. Dejinition A sequence of Borel probability measures { P,} on C[O, I ] is called tight if and only if for any r, > 0, there exists a compact K c C[O, 13 with P,(K) > 1 - E for all n.
Remark I t is an interesting exercise to show that any single measure on C[O, 13 is tight. Theorem 17.4 (Prohorov's theorem) Let { P , } be a sequence of Borel probability measures on C[O, 11 with the following two properties: (a) The finite distributions converge: explicitly for any 0 I t , < < f, I 1, there is a measure d p r , o n R"' so that as n -+ GU /F(dfl),
(b)
t,
< ...
. . . , ~ ( t , ) )~ P , ( ( [ J ) Sp(x1, . . . x,) dPr,(X) +
for all bounded continuous F on R". The { P , } are tight.
Then, there is a measure P , on C[O, finite distributions of P , are the d p , .
13 so that
P,
-+
P, weakly and the
Proof For r = 1, 2, . . . , pick K , with K , c K , c . . ' and P,(K,) 2 1 - r - ' for all n. Since the positive measures of mass at most one on a compact set are weakly compact (see, e.g., [214]), we can find a subsequence P,(,, so that Pnci,1 K , converge for each r to a measure Pm,,. Since the P T , r
178
VI.
are easily seen to be consistent, we can define P , on Given f E W, we have that JK:dPn(i)
+
u K,
Asymptotics
and so on C[O, 13.
IK./dpm
Since Pm(C\K,) I l/r by a simple argument and P,(C\K,) hypothesis, we see that for any r
I l/r by
Since r is arbitrary Pn(i)-+ P , weakly. Clearly, the dpf are the finite distributions of P , . By the above argument, any subsequence has a weakly convergent subsubsequence and the limit must agree with P , since Kolmogorov’s theorem implies that there is at most one measure on C[O, 11 with the dpL,as finite distribution. By a general argument on sequential convergence, P , + P , weakly. I Remark Prohorov’s theorem is usually stated in the form: A family of Bore1 probability measures on a complete separable metric space has a compact weak closure if and only ifthe family is tight. We have isolated the half we need and added the extra argument. The central limit theorem (Theorem 4.1) implies that under the hypotheses of Theorem 17.1, the finite distributions of the P , converge to those of Brownian motion. Thus, Donsker’s theorem is reduced to the proof of tightness. Notice that by proving this we will, at the same time, have a proof that Brownian motion has a continuous version independent of the arguments in Section 5. The compact sets needed in the proof of tightness will be uniformly bounded, uniformly equicontinuous families which are compact by the Arzela-Ascoli theorem (see, e.g., [214]). The key estimate is the following whose proof is closely patterned on that of Doob’s inequality (Theorem 3.5) and Levy’s inequality (Theorem 3.6.5). Lemma 27.5
Under the hypotheses of Donsker’s theorem:
(
lim E maxISil 2 I& isn
1
I c1p3
(17.3)
for each I > 0, and a suitable c < GO. The same result holds if, in the expectation, A is replaced by a sequence, converging to I as n -,GO.
179
17. Donsker's Theorem
Proqf
Clearly, it suffices to consider the case A 2 2$. I I S i [ } .Then
Fix n. Let
A , ={maxj
(1 7.4)
since IS,
-
Silis independent of S,, . . . , S i . But A E(IS, - Sil 2
+A&)
I E(IS,
so
2 2$,
- Sil 2
J2n)
< ( ~ ~ ) - ' E ( I -s , SiI2) ,
3
= +n-l(n - i) I
Thus, since the A i are disjoint and contained in {max,, implies that
(
+E maxlSil 2
A$
isn
1
I E(IS,I2
I S, I 2 A,/%}, (17.4)
+A$> e-rzi2 d x
.+ (271)-1'2
( 17.4)
XI 2 j. 2
as n + 00,by the central limit theorem (Theorem 4.1). Using (3.4), we obtain (17.3) trivially. The extension to the sequence is easy if we note that for any E > O
(
maxlSiI 2
A,&
i t n
for n sufficiently large.
1sil 2 (1 -
1
Notice that if y , has an even distribution, one can replace the argument leading to (17.4') by Levy's inequality. ProoJqf Theorem 17.1 To prove tightness, we claim it suffices to prove that for each positive integer k and each t: > 0, there is a S(k; E ) > 0 so that for all n SUP
~ X , ( U-)
0 < u - t
For assuming (17.5) holds, let
x,(t)l I
k-'
(17.5)
180
VI.
Asymptotics
Then K , is compact by the Arzela-Ascoli theorem and by (17.5) and the fact that X,(O) = 0, P,(K,) 2 1 - E . Now fix n. By the polygonal nature of the paths, the paths are piecewise C' with maximum derivative S U ~ ISi ~- S i~+ lIn+'i2so ~ ~ ~ ~ 1
-
E
(
IX,(u) - X,(t)I I k - '
sup
Osu-fld
< 1 - E(
-
)
ISi - S i + l l_< n - 1 i 2 k - 1 6 - 1
sup Osi$n-
1
1
n- 1
1
I E ( J S i-
Si+1J 2
n-li2k-'6-')
i=O
I n[nk26'E( ISi
-
Si+
I')]
=
n2k2h2
As a result for each fixed n, we can assure (17.5) by taking 6 small. It follows that if (17.5) is proven for all n 2 no(k, E ) , we can conclude it for all n by shrinking 6 ( k , E). Now suppose that for each n 2 no we can find 0 = to < . . < t , = 1 with sup I ti - ti- I I 6(k, E )(the t's may be n-dependent) so that
-
(17.6) Then (17.5) holds for n 2 no, since any u and t with u of some common ti I min(u, t). Now given E , k , fix 6 so small that
-
tl I darewithin26
c([xq+ 1)(24363/2 I E/2k+l where c is the constant in (17.3). For fixed n, let 0 (u - t ) < 6 and let t polygonal nature of the paths IX,,(u)
-
x,(t)I I max
=
(17.7)
m/n for some m. Then, by the
n-li2
ISi+rn - SmI
(1 7.8)
O$i$[n6]+ 1
Now suppose that k , = [6n] 2 1. Let I = [n/[n6]] 1. Let t o = 0, t , = ko/n, . . . , r l - l = ( I - l)ko/n, t , = 1. Then I ti - t i _ 1 I ko/n I 6, so bearing in mind that (17.6) just needs to hold for large n we see that it suffices that
+
18.
181
Laplace’s Method
But, by (17.3), this limit is asymptotically bounded by
c{[n/[nS]]+ 1}{(2k)-’&([nS]
+ 1)-”2}-3
which approaches c ( [ ~ - 11
by (17.7).
+ I ) ( 2 4 3 ~ 2< Pk
I
18. Laplace’s Method in Function Space To understand the results we seek in this section, consider the following method for obtaining the leading behavior in Stirling’s formula for the asymptotics of the gamma function
r(a)= JomxuIf one changes variables to x
r(a)= a‘
=
dx
c1y one finds that
6.-
exp[ - a(y - In y ) ] dy
To find the asymptotics, one locates the minimum of y - In y which is one (occurring at y = 1). This suggests that the integral w e p a and indeed one can prove easily that
This method can be extended further by making a Gaussian approximation ( y - In y ) 1 + i ( y - 1)2 + O((y - 1)3) and controlling the error. One obtains Stirling’s formula T(a 1) = aa+’i2e-a(2n)1’2(1+ O(a-’)). Here we want to develop an analog of this method for functional integrals. Suppose we have a Gaussian measure dpo on some function space with covariance (. , A . ). Formally
-
+
dpo = exp( -i(x, A - l x ) ) ” d x ”
(18.1)
where d x is a formal uncountably infinite product measure and includes an infinite normalization constant. The formal nature of (18.1) is indicated by the quotation marks about dx. In the papers quoted below, the formal nature is indicated by using a special symbol (“flat integral”) for integrals with respect to “dx.”
182
VI.
Asymptotics
Suppose that we want to compute the asymptotics of
Using (18.1) and changing variables x/& normalization !)
a(n) =
s
=
exp(-n[gx, A-’x)
y, we see that (with change in
+ F(x)]) “dx”
(18.3)
which suggests that lim [ - n - ’ In a(n)] n-
=
min[$(x, A-‘x)
m
+ F(x)] = b
(1 8.4)
X
Of course, one must expect to have fairly strong continuity conditions on F since lhe set of’x with (x, K ’ x ) < 00 has &,-measure zero. [This assertion (27-~-~’’ exp(-$a;) da,, the set of a’s with a,’ that with respect to nFZl
1
(4)-
lim n-tm
+
[
sG(x/&) exp( -nF(x/Jn))
1
dpo /a(n) = G(x*)
whenever the minimum in (18.4) occurs at a unique point, x*. If anything, the problem we want to discuss is easier than this, and our arguments are a modification of those of Pincus. Moreover, the methods are such that it should be possible, by combining the work of Pincus and Schilder, to develop rigorously Gaussian approximations and further results on a(n) when F is F(x) has a unique minimum or a set of smooth enough and g x , A-’x) minima with a simple structure. (In the argument below, regions 11 and 111 are unimportant in any further terms in the asymptotic series.) We note that the general machinery of Donsker and Varadhan, discussed in Section 19, is applicable to these problems yielding (1 8.4); see, e.g., [57].
+
18.
183
Laplace's Method
Recently, the kind of problems discussed above have produced considerable interest among particle physicists (who were unaware of the earlier work of Schilder and Pincus). The earliest work was done by Lipatov [I641 whose ideas were developed by the Saclay school in a series beginning with Brezin et a/. [24]. The ideas are well described by looking at the ground state of the quantum anharmonic oscillator; i.e., E(P) indicates the lowest eigenvalue of Lo + /3x4 where as usual Lo = - $ d 2 / d x 2 +'.3 - 3.E ( p ) has a formal perturbation series (see Section 20)
-
Wj)
u)
1
anBn
n=O
This series is divergent but it can be summed [I1 131 by a method known as Bore1 summability. BrCzin et a/. begin by writing [see (1.9)]
E(b) =
-
'
lim T - In Tr(e-T(L"+flX4)) T-a,
Consider first the asymptotics of
or more simply ( 18.5)
The point is that (1 8.5) can be written in terms of a suitable Gaussian integral,
by a Feynman-Kac formula. From this, one sees that
or (18.6) with (18.7)
184
VI.
Asymptotics
Our results below and Stirling’s formula provide a rigorous proof that [n-”bn(T)]’l” + a(T) for an “explicit” u(T). By going several steps further, it is possible [24] to obtain the Bender-Wu formula (limT.+wa(T) = 3). (18.8)
but (18.8) has not yet been proven by these methods since it involves a formal interchange of n and T limits which is discussed further at the end of this section. In the case at hand, (18.8) is certainly correct since Bender and Wu [7] computed the first 75 an’s and originally obtained (18.8) on a purely numerical basis. They also gave several “demonstrations” [S, 9, 101 of (18.8) using different ideas. Recently, a rigorous proof of ( I 8.8) has been found [123a]. The importance of the work of Lipatov and the Saclay group is twofold: first, they realized the relevance of (1 8.8) to doing numerical calculations with the Bore1 sums; secondly, their path integral method isformally applicable to field theories. We now turn to a statement of the main result of this section: Below (. , .) is the L2 inner product; we use 11 rn and 11 .I) for the LP-norms. We will consider Gaussian processes { q ( t ) } o s 1 5 with a covariance p(s, t ) obeying two conditions: (a) For some K <
o0,O
I $: I d s , t>l I K
I p(s,
(18.9)
t ) - p(s’, r ) I 5 K 1s - s’ 12’
(18.10)
(b) p is real and strictly positive definite; i.e., forfE C[O, l],f+ 0,
For the function F we will require [the F of (18.7) does not obey these conditions but see below]. (c) F is a Holder continuous function on C[O, 11, in the following sense. IF(x) -
F(Y)l
5
CRIlX -
YIIP
(18.11)
R where C, may depend on R and for x, y E C[O, I] with IIxllrn, llyllm I 0 < O,, I 1. Note that (18.11) is much stronger than ordinary Holder continuity since it involves the L2-norm and I(. 11 I 11 (d) For all x E C[O, I]: F ( x ) 2 -$c,llxll;
- c2
(18.12)
185
Laplace's Method
18.
where c 1 11 A 11 < 1, and A is the operator on L 2 given by
Sometimes it is useful to replace (d) by (d') For some fixed c 1 and A and for any E > 0 and all x E C[O, 1): F(x) 2
-3C111x11:
-
( 1 8.12')
c ~ ( c )- c~IA-'xII:
Moreover, c 1IIAIl < 1 and A is such that A < 3 and Tr(A'-2A) <
cx)
Theorem 28.2 Suppose that (a)--(d) [or (d')] hold. Let d p o be the Gaussian measure on C[O, I] with covariance p [supported on C[O, 11 by (18.10) and Kolmogorov's lemma (Theorem 5.1)]. Then D ( A ' I 2 ) , the L2-operator domain of A P 1 " lies in C[O, 13, the functional H(x)
g ~A -,' x ) + F(x)
=
( 18.14)
is bounded below on D ( A - ' I 2 ) and takes its minimum on at least one point x* in D(A-'I2). Moreover, (18.4) holds with 6 = H ( x * ) . Before turning to the proof of this theorem we note a slight extension: TheoremZ8.2 Suppose that (a), (b) hold and that F is a function on C[O, 11 taking values in ( - co, 033 so that (e)
F(x) < cc for some x E D(A - ' I 2 ) .
(f)
For each m < co,F,(x) = min(F(x), m) obeys (c) and either (d) or (d'). Then all the conclusions of Theorem 18.1 remain true. Proof
Clearly F obeys (d'), so for E small H(x) 2 +(x, A - l x ) - +C,(X, x) - c2 - E(X, A--ZAx)2
since c111 A /I < 1. By (e), H(x) < c;o for some x E D(A M
=
1 + inf H ( x ) = 1
'I2).
-c2
Thus
+ b < co
x
Let a(n) = j exp( - nF(x/&)) d p o , and u,(H) = j exp( - nFm(x/&)) dpo. Then, form > M , (a,(n))"" 4 e-' by (f) and Theorem 18.1. But clearly la(n) - u,(n)l I e - n M
since le-nF- e-nFmI 5 e - n M .
I
186
VI.
Asymptotics
Example Let us demonstrate the rigorous proof of the leading behavior of bn( T ) for T = 1 (b, given by ( I 8.5); the general T case is similar). Define p(s,
r)
= Tr(xt)-Is-fIbxt)-(I
-l"-'l)"")[Tr(e-LO)]-
1
( 18.15)
I t is easy to check that p obeys (a), (b) with 8 = f by obtaining an explicit formula for p (one can get the explicit formula by making an eigenfunction expansion for Lo or more simply by noting that
(- $ +
l)p(s, t ) = 6(s
-
t)
with the condition that p be periodic at zero and one). Now let Q(x)
ik x4(s) ds. Then since I u4 - b4 I I4[max(u, b)I3(u - b ) : I Q ( ~)
QW
=
I 4 ~ ~ -1 Yii2 1 ~
if R = max(/lxl/,, llyllm). l t follows that F,(x) = min(rn, -In Q ) obeys (c) with O0 = 1. Now, A - ' has eigenvalues 1 + (27cn)' with eigenvectors (27c)-l / 2 e 2 n i n x so Tr(A'-2A) < co for A < By a Hausdorff-Young and then a Holder estimate, if,fis in L2 with (periodic) Fourier coefficientsf, and 2 < p < m [and q = (1 - p-')-']:
a.
so long as a(l/q - f)-' > 1. Since p < 00,we can take c1 < f. But 5 /I A-(1'2)"f11$. Thus, we have the simple Sobolev inequality: Q(x)
=
Ilxll: I cllA-""xll;
so
verifying (d'). We summarize with the following. Theorem 18.3
Let bn(T ) be given by (1 8.5). Let
subject to the boundary condition that x be periodic. Then lim[n!n-2"(- l)"b,(T)]"" n-ai
=
e-a(T)
xn
18.
187
Laplace’s Method
Remurk It is easy to establish that the minimum in (18.16) obeys the nonlinear equation 2X _ d_dt2 +x
4x3 -
jg x4(t) d t
=o
so after scaling, it involves classical motion in a potential x4 - x2 which means that a ( T )can be computed exactly up to quadratures which are given “explicitly” by elliptic functions; see, e.g., [24]. Now, we begin the proof of Theorem 18.1.
Lemma 18.4 (a) A is Hilbert-Schmidt and maps L2 into C[O, 1) and A [ x compact in C[O, 13. (b) A is trace class, A’” maps L 2 into C[O, 11, and A ’ j 2 [ x compact in C[O, 11. (c) One has the estimates: llA-4a, 5 ~ l l x l l 2 :
I I ~ L ’ 2 x l I5mW
l l 2
(18.17)
IIAxllm 5 D11A’i2xl12
P r m f (a) By hypothesis (a) ( 1 8.18a)
(1 8.18b)
Thus A maps L2 into C[O, 11 and the image of the unit ball is a family of uniformly bounded, uniformly equicontinuous functions and so pre-compact by the Arzela-Ascoli theorem [214]. By the weak compactness of the ball, the image is closed. A is Hilbert-Schmidt since { p2 rfs dt < m. (b) A is trace class since its kernel jl is continuous, A is positive and j h p(s, s) ds < m; see [259]. Define orthonormal eigenfuncrions ,fn and eigenvalues A: by
(4f,)(f) = Xf”(0
(18.19)
We claim that for each s, f (1 8.20)
188
VI.
For (18.20) clearly holds for almost all s and identity in C[O, 11, and let
t.
Asymptotics
Let j , be an approximate
.fn.dx) = fn* j E ( X ) pE(s,t ) =
I
p ( d , t ’ ) j e ( s - s’)jE(f - t’) ds’ dt‘
We have that P h ? s> =
c 1; I
f,,
E ( 4 l2
so since thefn and p are continuous:
I lim p,(s, s) = p(s, s) c
As a result,
1 Innfn(s)12 5 K n
for all s. Similarly,
C
I .fn(s)
- fn(s’)
1’
I 2K I S -
S’
n
IZe
It follows that the right-hand side of (18.20) converges uniformly as n and so (18.20) holds. Now, if x(t) = &fn(t), then I(A1’2X)(t)12
I I C n n p n f . ( t ) 1 2Ilxll: ~
00
--f
1I.zn.f,(t)12 = p(t, t>llxll:
Similarly, I(A”2X)(s) - (A”2X)(s‘)(2
I Ilxll:[p(s, s) + p(s’, s’) I2KI.7 - .
-
p(s, s‘) - p(s’, s)]
S ’ ~ ~ ~ ~ ~ X ~ \ ~
Thus A ‘ j 2 maps L2 into C[O, 11and the image of the unit ball is compact as in (a>. are (c) The first two estimates are just the assertions that A and bounded from L2 to C. (18.17) is the second estimate for A 1 ’ 2 ~I . Lemma 28.5 H ( x ) = $(x, A - ‘x) + F(x) is bounded below on D(A and H ( x ) takes its minimum H(x*) at one point, x*, at least.
~
‘I2)
18.
189
Laplace's Method
Proof
By (18.12') [hypothesis (d')], H(x) 2 +(x, A-'x) - +c,l\xll; - c2 - E(X, A-I"x) 2 -c2 +6(X, A-lx)
+
if 1 - c 1 IIAIl - ~ E ~ ~ A2I 6.I ' Thus ~ ~ H is bounded below and on the set where H(x) I inf, H ( y ) + 1, (x,A - ' x ) is bounded from above. Thus N(x) takes its infimum on a compact subset of C[O, 11 (by Lemma 18.4(b)). Since H(x) is easily seen to be lower semicontinuous, we conclude that H takes its minimum value. The key to the proof of Theorem 18.1 is the following:
Lemma 18.6 ous, and
Let y > 0. Then d p O ( x + y A x ) is dpo absolutely continu-
+ yAx) = D ( y ) exp[ -+?;'(Ax, and D ( y ) = det(1 + ? A ) = n;= (1 + ?A;). dpo(x
Proof
x)
-
y(x, x)] dpo(x) (18.21)
Introduce the realization of the Gaussian process dpo: 02
x(t> =
1 an fl=
An
(18.22)
fn(t)
I
where the a, are random variables distributed by m
)( (271)- '1' exp( -+a:)
da,
1
fl=
Then x + (1 + yA)x corresponds to a, + (1 1 2 (2n) exp( - Tm,) da, :
=:()
+ yAi)m,,.
Thus if dp!
=
N
d~!(x
+ yAx) =
n((1 + ~1,") exp[ fl=
1 2 4 2 -
-Ty ~ , a , ,
I
y~iccil} dp!
c
Now as N -+ GO, Aiai + (x,x), 2 A:ai -+ (x, Ax) so the exponential converges in L 1 ( d p o )by the monotone convergence theorem to exp(-+y2(Ax, x) SinceCAi < r n , n r T l (1 Remark 2.5.
+ yAf)+
-
y(x, XI)
D(y) < m.
1
Lemma 18.6 is essentially an explicit form of one-half ofTheorem
190
VI.
Asymptotics
Therefore, we can write a(n), given by (18.2),
+ &A)E[exp( = &x, A x ) + F ( A x +
u(n) = det(1 Q,(x)
(18.23)
-nQ,(x))]
+ n-'I2(x, x)
(1 8.24)
since both sides of (18.23) are JexP(-nw/&)(x
+ &Ax)])
&O(X
+ &Ax)
Now since A 2 0:
so that (18.23) says that to conclude the proof of Theorem 18.1, it suffices to show that [E(~xP(-nQ,(x)>)I""
+
~ X P-(H ( x * ) )
Since Q,(x) is almost H ( A x ) , this is beginning to look controllable. We need three more preparatory lemmas:
Lemma 18.7
For some y and C: (18.25)
~(llxll, 2 u ) 5 Ce-7"'
Remark Marcus and Shepp [lSO] and Fernique [82a] have shown that for any Gaussian process with paths in C[0, 13 and covariance p :
See also [57]. Proof This proof is a simple extension of the idea behind Kolmogorov's lemma (Theorem 5.1). By hypothesis (a):
E( Ix(s) -
x(s')l2)
I 2K 1s -
so, since the x(s) are Gaussian, for C , 2 1, n 2 0
S'pJ
18.
191
Laplace's Method
where y
=
C0(2K)-1122"'/2and D , , D , are suitable constants. Thus,
a)
1 2" exp(
I DI ,I
-
D, C i 2"')
=0
I D , exp( - D, C i )
If lx(k/2") uJ/2j, a,
1
-
=
x((k - 1)/2")l I C02-'10'2for all k and n, then writing t 0 or 1 we see that Ix(t)
-
x(0)I I ~U,Co2-J''2 =
=
c0 B
for some constant B. Thus, E ( l x ( t ) - x(0)I 2 C, some t ) I D, ~ X ~ ( - D , B - ~ C " )
If we note that E(llxll, 2 a) I E(lx(0)I 2 4 2 )
we obtain (18.25).
Lemma 18.8
+ E(lx(t)
-
x(O)( 2 4 2 , some t )
I Suppose Tr(A
2A)
<
1x1. For E
sufficiently sinall:
E(exp[c(x, A - 2 A x ) ] ) < co
Proqf
In terms of the realization (18.22) (x, A
-
2%)
1n,z
=
-4AcY,2
"= 1
If one looks at Theorem 3.11 and its claimed extension to nonpositive matrices, one sees that
if (and only if) 8, < for all n and Lemma 18.9
For any y E(x
E L2
1 IJA"2X
1 /j,
< co. I
and /j > 0, there is a c with
- yIl2
I
p ; llxll,
I c) > 0
(1 8.26)
192
VI.
Asymptotics
Proof Since llxllm < co, almost everywhere in x, we need only prove (18.26) with the condition llxllm I cdropped. In terms of the realization (18.22) m
IIA1i2x - YII: =
C
YJ2
-
n= 1
N
Choosing first N with 2
x;+
ly, 1' I p2/4,we see that it suffices to show that
'(1A:)-
If I En I I ($/4)Bn,
m
m
then
1
E(la,l I y&l,
( C I , , ~ , "I ) ~ p2/8,
so we only need
all n) > 0
(18.27)
for any y > 0. But
so (18.27) follows from 1 E( I CI, I 2 y/An) < E(IcI,I 2 a) I Cexp(-*a2)
00. This
s 2CC2
-
< co
and
Proof of Theorem 18.1 Lower bound Given find /3 so that )I A li2x - A - li2x* 11 5 fl implies H(Ax)
is easy from
E,
1
we claim that we can (18.28)
H(x*) I E
For H(Ax) - H(x*) 5 )I(A"2X, A"2x) - (A-'/Zx*, A - ' / 2 x * )I IF(Ax) - F(x*)l
+
The first factor is small if IIA 'I2x - A - 'l2x* I( is small by the continuity of the norm. The second factor is small by the continuity of F in 11.11 and (18.17) which implies that (IAx - x*II, I D ( I A 1 ' 2-~ A - ' 1 2 ~ * 1 1 2 ( 1 8.29) This verifies (18.28). Pick c so that (1 8.26) holds. Then on the set
s = {x 1 llA"2x
x * 112
-
P 7
llxllm I c>
we have that
+
+
+
Q,,(X) = H ( A ~ ) F ( A ~ x/&) - F(AX) +n-li2(x, X) 5 E + H(x*) + C , c n - " / 2 ' ~ ~ '+ + c 2 n - l / 2
18.
193
Laplace's Method
where C , is the constant in (18.1 l), we have used llx112 I llxllrnand R is first chosen so that IIAxll, < R / 2 for x E S [use (18.29)] and then s,o that cn-'I2 < R/2. Thus for n large:
+ ~ ( x *+) O(n-"J2)])E(S)l
~ ( e - " Q n2) exp(-n[~:
Since E ( S ) > 0, we have that h ( l / n ) l n E(e-"Qn) 2 is arbitrary we have the required lower bound. Upper bound: General strategy
-E
-
H ( x * ) . Since
E
Consider the three regions:
s, = Ix I IIAxll, I R , llxllm s,,= {x I IIxII, 2 R&J s,,,= { x I IIAxllrn 2 R',
R&J
We will control the contributions of S,, and S,,, for R sufficiently large and S, for any R. We suppose that (d') holds. The reader can check that the extra condition on p, Tr(A'-2A) < cc is only used (via Lemma 18.8) to control the extra c:IIKAxll: terms present i n case (d'). Thus case (d) goes through with minimal changes. Upper bound: Region I Q,,(x)
by using - x y
As in the lower bound, we have for x E S,:
H ( A x ) + F ( A x + x/&) 2 H(x*) - c2Rllxll2/& 2 H ( x * ) - +n-1'2C:, =
+ iy2 2
-)x'.
-
F ( A x ) + &'"(x,
+ +n-11211xII:
It follows that
I
E(e-"Qn x E s,)I e-nH(x*' exp(+n"2CfR)
yielding the required upper bound.
x)
194
VI.
Asymptotics
so Q,(x) 2 (a,/2)(Ax, x )
+ h,(x, x )
-
c2(&)- cn-'(x, K Z A x ) (18.30)
u, = ( 1 - c l / ~ A 2c1n-"2 ~ ~
hn
+
= L2 n - 1 / 2 [ 1 - 2 1 , . - I 1 2
-
2 c ~ ~ A ~ ~ l - 2 A ) (1 8.30~1)
-
4&llA I/ I
- 2 9
(1 8.30b)
One can choose I: small and then n so large that a,, h, > 0. It follows that E(e-"Qn
I s1,) I
p 2 ~ ( ~ ~ (A ' " .
enc2E(e2t(x. A
_< c
I SIJ
-2Ax)
1/2
E(Sl,)"2 exp(nc2 - $nyR2) )
where we used the Schwarz inequality in the second step and Lemmas 18.7 and 18.8 in the last inequality ( 6 may have to be decreased). So long as we choose R so large that c2 - f y R 2 I - H ( x * ) , we have the requisite bound on
s,,. Upper hound: Region 111 Use (18.30) and choose E so small and n so large that h, > 0 and u, 2 6 > 0 for some fixed 6. Notice that in region 111, llAxl12 2 D-IIIAxlj, 2 D - I R by (18.17). Thus, Q , ( x ) 2 (6/2)D-'R2
-
c2(.5) - & n - ' ( x , A - 2 A x )
so that
I
E(e-"Qn S,,,)
E(e'(X*A-X)
) exp(nc2 - ( 6 / 2 ) 1 7 D - ~ R ' )
So long as c2 - ( S / 2 ) D - 2 R 2 I- H ( x * ) , we have the required bound on
s,,,. I At the present moment, a complete proof of the Bender-Wu formula (1 8.8) has only been obtained by very different methods [ 123al. However, one can imagine a proof along the lines above; only one crucial step is missing. The passage from the asymptotics of b,(T) to those of a,(T) is not, hard. One only needs to use the Taylor series for In(1 x). The net result + O(l/n)) (this is that for T fixed and n large a,(T) n"cr(T)"nb(T)(y(T) assumes that one can push Theorem 18.1 to higher order for some very special degenerate x*), where a(T) + a,etc., as T -, co. The missing point in the proof is justifying the interchange of n and T limits. The subtlety of the missing step is indicated by the following example.
+
-
Example (Donsker and Simon, unpublished)
,/'(T,fi)
=
Let
-(l/T) In Tr(exp( - T ( L ,
+ $x2)))
18.
195
Laplace's Method
Thenf'(co, T=m:
p) = limT+mf ( T , p ) = $ [ ( p + I)'/' a,(co)
-
i)n+ln-3/2(1
++2(-
-
11. Thus, for n large and
+ o(I/~))
(18.31)
On the other hand, Tr(exp( - T ( L ,
"
+ i/3x2))) =
exp( - [(n I1
-
Using sinh(x)
=x
1 .f(T,/3>= -ln(l 2T with y,(T)
=
1
n:=
(1
=0
+ i>Jm -. $IT)
r"/2'Tf[sinh(( T / 2 ) K + T ) ]
+ ~ ~ / ( n nwe ) ~ obtain )
" + B) + T1 n 1 In (1 + =, -
A)
-
1
+1
' ln(2 sinh
+ T-2(2n7c)2.Thus for,fixed T as n + 00, n,(T) - ( 2 T ) - ' ( - I ) " + ' n - ' ( l + O(l/n))
( 1 8.32)
Comparing ( I 8.3 1) and (1 8.32), we see that the leading ( - 1)"terms agree but the nh and constant terms are wrong. Analytically, it is clear why this happens. The square root singularity for n = a is the limit of an infinite number of logarithmic singularities at -y,(T) which coalesce as T + co. For fixed T, only one log contributes as n + EI. Since (18.8) has been proven by other means [123a], the interchange is permissible for the anharmonic oscillator; the issue is to understand why.
*** The ideas of this section are useful for understanding the imaginary time analog of the classical limit for solutions of the time dependent Schrodinger equation. Let H,
=
( -h2/2m)A
+ V(y)
The classical limit question involves studying exp( - itH,/h)$ :is h will instead consider
4 L y, h;
$1 = [exp( - tfffi/W$I(Y)
(18.33) +
0. We
(18.34)
in the limit as h + 0. The quantity when = 1 will be denoted by a([, y , h) (see Section 25 for the definition of exp( -sH,) on L"). We will state one result and then discuss its significance and the kinds of extensions possible.
196
VI.
Asymptotics
Theorem 18.10 Let V be bounded from below and C’ on R. Let H, be given by (18.33) and u(r, y, h) by (18.34) with $ = 1. Then
where
(18.35)
ry,is the set of C’-paths with y(0) = y,.
Proof By scaling y and/or
one can (by changing V ) suppose that
I/
t = m = I. By translation covariance, we can suppose also that y , = 0. The operator A - with quadratic form A j y’(s)’ ds, with domain the closure of
’
the form domain To, is precisely A - I = - d2/dr2with boundary conditions y(0) = 0, y’(1) = 0 (see [217]). The integral kernel of A , the inverse of this A I , is just p(s, t ) = min(.s, t ) . Noticing that F(h) = V(b(s))ds obeys (18.1 1, 12) on account of the hypothesis on V , we see that Theorem 18.1 is applicable so that
yo
cl(1,O) = lim - h In[ii(l, 0, h)] 8-0
where d(l,O, h ) = [exp[ -h-’
~ O ’ V ( h ’ ~ 2 hds] ( . ~ )Db )
with Db the Brownian motion measure. Thus, by the Feynman-Kac formula,
where is the function which is identically one and V,(y) (Wq)(y)= q(h”’y). Then using
=
V(h’”y). Let
we see that Z(l,O, h ) = [exp(-H,h”)W,~](h-”’O) = a(l,O, h)
since W,$
=
$ and h-’I20
=
0.
I
A closely related theorem but with different methods and hypotheses has been proven by Truman [28Oc]. The applicability of the Pincus-Schilder methods to this context seems to be new. We examine the connection of the above result with “the classical limit” in a series of remarks.
18.
197
Laplace’s Method
Remarks 1. Standard calculus of variations arguments show that the minimum of
E(y) over y’s with y ( t ) obeys
= y,
=
s:
[ ( 2 m ) - ‘y’(.s)2
+ V ( y ( s ) ) ]ns
[by replacing y( .) by y ( t - .) this minimum is u(t, v,)]
y”(s) = m(VV)(y(s))(Note: not - V V )
(18.36)
with “boundary conditions,”
y’(0) = 0,
r ( t > = Y,
(1 8.37)
That is, if we let Y , ( y o ,s) denote the solution of (18.36) with initial condition Yl(y,, 0) = y o , dY,/ds(y,, 0), then the existence of the minimizing y guaranteed by Theorem 18.1 assures us that Yl(yo, t ) = y , always has a solution Yo(y,, t ) ; it must be that the minimizing y is just (1 8.38) Y,( Y d Y , , t), s> If there is more than one solution y 0 , we must pick one that minimizes E . We note that the study of “mixed boundary conditions” like (18.37) is characteristic of the Hamiltonian Jacobi theory. We also emphasize that the corresponding classical solution obeys Newton’s equation with F = V V ; i.e., in going from the solution of the Schrodinger equation, e - i r H $ , where the ti + 0 limit isformally given by ordinary classical mechanics to the solution of the heat equation C f H $ the, sign of V changes in the corresponding classical mechanics. This is a well-known phenomena. 2. l t is easy to extend the Pincus-Schilder theory to allow independent Brownian motions y,(s) and thereby to extend Theorem 18.10 to v-dimensional y’s. 3. If we replace $ = 1 by $ ( y ) = e-fSo(y)/h, then the formal minimum problem is to minimize yo(s)
=
subject to y ( t ) = y,. The solution to this problem obeys (18.36) but the boundary conditions (18.37) are now Y’(0)= (VSo)(?/(O));
Y(f) = Y,
which arise in the full Hamiltonian-Jacobi theory. The only problem with this kind of extension of Theorem 18.10 is that 1
F(b) =
V ( b ( s ) )ds
+ S,(b(l))
198
VI.
Asymptotics
does not obey condition (1 8.1 1); i.e., F is Holder continuous in I(. 11 but not in 11. 112. It may be possible to extend Theorem 18.1 to handle this case. Alternately, it may be possible to treat certain So’s (certainly So(y) = - u y 2 ) by writing exp( - h - ‘So) as a conditional expectation.
for suitable S,. 4. If there is a unique y in I-,” minimizing E and I/ has extra smoothness, then Schilder’s evaluation of the full asymptotic series for u(t, y , h ) is valid and one obtains the series in terms of the Gaussian approximation of E(y) near the minimizing y(s) given by (18.38).The covariance of the corresponding Gaussian process will enter. This covariance is just the integral kernel of the inverse of - ( d 2 / d t 2 ) ( d 2V/dy2)(yo(t))with suitable boundary conditions (the *‘Feynman Green’s function” of DeWitt-Morette). In particular, one finds that
+
e+hau-+ (dY,/ayo)-”2
(18.39)
5. If the minimum is unique, the method of Pincus shows that for suitable g’s,
b* being the unique minimum. Doing the scaling in Theorem 18.10 and using 18.31, one finds that for suitable $‘s,
6. The celebrated formula of Maslov is just the analog of (18.40) for e - i f H h i h $ . His formula follows from jbrrnal application of stationary phase
ideas to the formul Feynman integral. This is notoriously difficult to make rigorous; for some partial results, see Truman [28Oc], Yajima [288], and the papers of Fujiwara [97a,b] and Hagedorn [291].
19. Introduction t o the Donsker-Varadhan Theory
In this section, we wish to give a brief introduction to an important method of Donsker and Varadhan. Since these authors have seven papers [54-601 covering roughly 220 journal pages on the subject, we cannot hope to give a
19.
199
Donsker-Varadhan Theory
comprehensive overview; see especially [54] for more details. We intend to concentrate here on one aspect of the theory, namely the connection with the Gibbs' variational principle of statistical mechanics. To get the flavor of the results, we begin with a theorem of Kac [I1381which motivated the theory:
Theorem 19.1
Let I/ 2 0 be in L;,,,(R) and let Lo = -+((P/rix2
-
x2
+ 1)
as usual. Then
with
L
Proof The right-hand side of (19.1) is the bottom of the spectrum of = Lo + I/ by the Rayleigh-Ritz principle. The left-hand side is
-
- lim I
r
~
I
In(R,,, c>-'~R,)
a,
by the Feynman- Kac formula, so ( 19.1) follows from (1.19). The Donsker-Varadhan theory gives suitable limits as t -+ ac, of path integrals as an infinimum of a variational object. An example of the theory is the following.
Let W be a bounded positive continuous symmetric Theorem 19.2 function on R2. Then
200
VI.
Asymptotics
The proof of (19.2) is not nearly so easy as that of (19.1) because there is no Feynman-Kac formula for rewriting the left-hand side of (19.2). Below, we will settle for proving one-half of (19.2); namely,
(19.3) for everyfE C: with j J 2 dx = 1. We want to describe an analogy between (19.2) and the Gibbs' variational principle of statistical mechanics. The connection of (19.1) and this principle has already been noted by Guerra et. ul. [1201 who consider extensions to P ( 4 ) 2 Euclidean field theories (see [12O, 121,961). This work is partial motivation for our presentation here. We begin by describing the Gibbs' principle for classical lattice gases following Ruelle [225] (see also [226, 227, 133, 1193). Let Z' denote the lattice of points in R' with integral coefficients. Let A be a "box" Z'n [0, L )' and let I A I = L" be the number of points in A. Let J be a function on Z' which has bounded support (we restrict ourselves to pair interactions with finite range; were we to consider connections with tangent functions to the pressure [225, 1331, this would be a serious restriction). Define Z,(J)
c
=
e-Uh(J)(u)
o,=fl.ieA
uA(J)(o) =
c oiojJ(i
-j )
i, j e A
=
I A I - In
zA(J)
Then, we will show that lim p,,(J)
= p(J)
A+ m
exists for A + 00 in the sense of taking boxes all of whose sides go to infinity. Let p be a probability measure on { - 1 , l}"". If we restrict p to the variables {ui}isA ,we obtain a measure p,, on { - 1, 1) '. p is called translation invariant if p,, and pA+. are identical under the natural identification of { - 1, l}" and { - 1, I},,+'. One defines sA(p)
=
-
c
pA(0)[ln
o,= k 1 , i e A
where p,,(a) is the p A measure of {u}. We will also show that
19.
201
Donsker-Varadhan Theory
exists. Finally, we will show that lim
(c)
1 ~ 1 - 1 JliAdp = u ( J , ~ )
A+'X
exists. The Gibbs' variational inequality asserts that p ( J ) 2 s ( p ) - U ( J ,p )
(4
for any translation invariant p. The full Gibbs' principle asserts that
The above describes the interaction of a family of spins, p is the free energy (the letter p comes from the fact that in a lattice gas language, it is a pressure) and s an entropy per unit volume.
P r o o f o f ( a ) For simplicity take A to run through cubes of side 1. Fix 1, and let I = nl0 + a with 0 I a < I,. Decompose the I ' cube into n" cubes of volume I; and a leftover "strip" of volume I" - (nl,)'. Write (1 9.4)
where U[,\,is the interaction within the I" cube and ULlAl(i) is the interaction within the ith [I;] cube. R is what is left over, i.e., interaction between the cubes and within the strip. Now, suppose that J has range b ; i.e., J ( n ) = 0 if In1 2 b. Then R only involves spins in the strip and in regions within b of the boundary of a 1; cube. Thus,
so that
[+ nl0
I PI"
+ IJll{
Taking 1 + co,we see that
[I
-
(1
-
$1
+
[( + i)" 1
-
I]}
VI.
202 Now taking I,
+
Asymptotics
a, -
lim pI I
lim pI0
i.e., the limit exists. Finiteness of the limit follows from ( p A I I IJI,. I Remurk The above proof has the advantage ofextending to rather general interactions. For the case at hand, there is a simpler proof: Write U A l u A z = U,, U,, Then Eexp(-U,, - U A 2 ) l i \ , ; A 2 = 0, since I A I ; A 2 is linear in the spins in A , and we can separately invert the spins in A,. Thus, by Jensen's inequality:
+
+
zA1~A2
2
=
zA~zAle-o
'A1~,42
so In Z , is superadditive implying existence of the limit. Proqfof(b) We first claim that for disjoint A , and A,: 'AI(~)
sA~uA2(P)
+ sA,(p)
(19.5)
On { - 1, 1}"*uA2 define functions p 1 2 , p l , and p2 by PI(^,, 0 2 ) = ~ ~ , t a ~l 2)( ,0 1 , 0 2 ) = p A ~ ( ~ 2 )Then, . since Pl(01) =
c
p12
= P,,,.~,,~
and
P12(0lr 0 2 )
6 2
we have that SAIuA2
-
S,,
-
S,,
=
E pi2Cln pi + In
~2
-
In
~ 1 2 1
6
=
c
PI2
In[-]
PIP2 PI2
5 I.(; P I 2 [ Z ] )
=In1=0 proving (19.5). In the above, the inequality comes from Jensen's inequality if we note that p I 2 is a probability measure and that -In is convex. We have also used the fact that pIp2 is a probability density on A , u A 2 . From (19.5) and a standard argument lim I A l-'S,,(p) exists and equals inflAl-'S,(p). The finiteness of this infimum comes from the fact that s, 2 0. I
Proof of((.) Since p is translation invariant,
19.
203
Donsker-Varadhan Theory
where N(n, A) = # {(i, j ) I i - j = n ; i, j E A}. Since it is easy to see that lAl-lN(n, A) + 1 as A + 00 for each 17,
4.cJ ( n )
tlf’
SUA
n
SgOgn
dp
Proofof(d) By Jensen’s inequality and the fact that
exP(sA(P)
- I u A
I
1pA(o) = 1:
dp)
’, and taking A + co,(d) results.
Taking logs, dividing by I A 1-
I
Proofof(e) In order to understand both the proof and the content of the Gibbs’ principle, we note that to get equality in the argument in (d), we must have In p A =
-
UA
+ const
since exp is strictly convex. Thus, the finite volume analog of the Gibbs’ principle picks out the finite volume Gibbs’ distribution r-
11,
W/ZA
We will therefore try to take a measure, p, as much like this Gibbs measure as possible. Fix A. and cover L” with nonoverlapping copies of A, : A, with tl E Zv. Let PO
=
n 3F
[exp( - u4-) / z A o l
o\
p, is not translation invariant, but it is periodic. If z, is translation by i units,
then P1
c
= IAol-
TIP0 ~ e A o
is translation invariant. We must compute s(pl) and U ( J , pl). Since - x In x is concave, sA(P
1)
2
I
c
I
sA(z~ P O ) I G A ~
Since In x is monotone: sA(Pl)
lAO1-’
1
sA(T~PO)
I 6
AII
+ lnlAOl
204
VI.
Asymptotics
Moreover, it is easy to see that
The last three relations imply that
I 1- sAo(pO) = I A ; 1 ' In
s(pl> =
+
*
oa=
UAu(@)exp(-UAo(@))z,,,' 1
ieAo
Using the fact that p o is invariant under reversal of the 0's in any Am,we see that this last term is U ( J , p o ) and so, by linearity, it equals U ( J , p l ) . Thus, s(Pl> - U ( J , p1) =
I
I - In
It follows that SUPCS(P) - U ( J >P)1 2 P ( J ) P
which, given (d), proves (e).
I
Before turning to the path integral case, it is useful to consider changes of s, p , U under change of "reference measure," i.e., instead of summing over oi= & 1, i E A, suppose we pick a fixed probability measure p'") on { - 1 , 1 jZ'
and let @(J) =
s
lim IAIA- m
;(J)
=
-
lim I A 1A- w
dp(")
s
ln[pA/pt)] dp,
We have in mind that p'") is some kind of limit of e ~ " A ' J o ) / Z A ( JFormally o). one expects that in this case:
@(4 = P(J + J o ) .?((PI = S ( P >
-
- P(J0)
U(J0,P)
- P(J0)
and it should be possible to prove this using the methods of [96]. If these formulas hold, then the Gibbs' principle is equivalent to @(J) =
SUPC.0) -
W J ,/)I
P
The point of this reformulation is that the reference measure in the path integral case is Dq which, in lattice approximation, is not a product measure, but a nontrivial p(").
19.
205
Donsker-Varadhan Theory
All of the above with the exception of the change of reference measure is fairly standard and is further discussed in [226,227, 1331. One aspect that we have not considered is the connection with the tangent structure of the pressure [133, 226, 2271, a subject which could well be relevant to the Donsker-Varadhan theory. Let us begin by studying (following [120]) the relation between (19.1) and the Gibbs' principle. Fix V and consider the P(4),-process associated to Lo V as a measure dp, on C(R). Then (6.10) asserts that dp, 1 (which indicates the a-algebra generated by {4(s)la I s I b } ) is Dq absolutely continuous with Radon-Nikodym derivative
+
pLu,b]=
g ( 4 ( ~ ) ) g ( 4 ( b ) ) F h" )~e + E ( V ) ( h - o )
(1 9.6a)
+
V)= where h , = gQo is the ground state of Lo + V , E ( V ) = inf spec(l, h -limb+ b - In(j e - "((I h , Dq), and U ( u , = V(q(s))ds. Clearly - E( V ) is the analog of the pressure. Let us compute the analog of the entropy
s,
,,
'
s(p,) = - lim h - ' /(In Pro. b ] ) P [ O ,hj Dq h - a,
Using the above formula for prU,,,],the integral in question is easily seen to be
bE(V
-
M h v , V h v ) + 2(hv, (In g)hv)
where the inner products are in L2(R). Thus,
4P")
=
+ (h,,
-E(U
Vhv) = -(A,,
Lob)
so the entropy per unit volume is the "free Hamiltonian expectation." We claim that -E ( V ) =
SUP(S(P,)
-
U ( V , Pw))
( 19.6b)
W
with U ( V , p W > = lim hh-1 a,
I
s
U,,.,,(V) dp,
=
(h,, Vll,)
For (19.6b) is just the assertion that E ( V ) = inf(h,, L,hw)
+ (h,,
Vh,)
W
i.e., (19.1)! One can extend (19.6b) by allowing more general measures p on C(R) than the p,. All that is required [120] is translation invariance, absolute continuity of dp X[O,bl,and a weak growth restriction on the ,!,"-norm of
r
206
VI.
Asymptotics
d p 1 X L o ,,,/Dq ("weak-temperedness"). In that case, one can prove the existence of s(p), U ( V, p ) and the inequality in (19.6b). The proofs closely follow
those of (a)-(d) above. [Parenthetically, we note that one direction for extending (19.6b) is to the P(4)2 field theory of Section 24: the Gibbs' inequality is proven in [120], the Gibbs'principle in [ 1211, and in [96] it is shown that the supremum is actually realized for the p's connected with the field theory.] The formal connection of Theorem 19.2 and the variational principle should be clear by this point. That this connection extends to some proofs is illustrated by the following, patterned after the proof of (d) above.
Proof of (1Y.3) We prove the result for f's in Y(R) which are strictly positive. By a limiting argument, the result extends tof's which are positive, continuous, and piecewise C'. Since i 2 ( f ) = i2( 1 f I) and a ( f ) 2 o( I f I) (by "Kato's inequality"; see [153) and (1.1)); (19.3) then follows. Let I/ = * [ , f " j - - x 2 + 11. The P(+),-process, p, for Lo + I/ has Radon-Nikodym derivative (19.6a) with g = f f & . Fix t and let
'
'
ut
' L>s
= t-
5).
W(q(s),q(u>)
Then, by Jensen's inequality
s
D q e-"
=
5
pro.,]e
-
- I"
Dq (19.7)
As above,
'
t - SI(P)
+
-
4f)
Let p,(x, y ) be the joint distribution of q(s), q(s + t) with respect to dy. Then
By the ergodicity of dp (see Example 2 of Section 3), one has that lim s+ m
1
W(X7 Y ) dPs(X, Y ) =
so that
Thus (19.7) implies (19.3).
J W(X,Y ) I f ( x ) l2 I f(Y)12 dx dY
19.
207
Donsker-Varadhan Theory
Remarks 1. Although it appears different, the above proof is (approximately) identical to that in [54]. In [54], Donsker-Varadhan d o not talk about ”P(I$),-processes” but rather of “drift-processes.” By the discussion at the end of Section 16, the two notions are identical. 2. (suggested by Varadhan) There are two special aspects of the DonskerVaradhan theory that are not present in the general Gibbs’ variational framework : First, in the Donsker-Varadhan theory, the a-functional is a functional of a density (namely I .f 1)’ on [w, i.e., of the state at a single time, while in the Gibbs’ framework s is a functional of the state for all times. Secondly, the minimizing state in the case of the Donsker-Varadhan theory above is a P(I$),-process, i.e., a Markov process. If we consider the special family of onedimensional lattice systems with interaction J + J , where J , is a nearest neighbor coupling and J is ”mean field interaction,” we have a lattice analog of the kind of interactions treated in Theorem 19.2. We notice that for ergodic states p , one expects that U ( J , p ) is only a function iJ of p { , ) , the restriction of p to functions of a,. Thus sup[S(p)
-
U ( J , p)1
=
sup
([
sup
$(.)I
-
iJ(a))
( P I P { O J= 1 1
P
=
sup( - I ( . )
-
iJ(cc))
n
where
c(
denotes a probability distribution on
I(a) =
inf
k 1 and
(-.?(.)).
wlP(oJ=“I
This calculation “explains” the first special feature of the Donsker-Varadhan theory noted above. Next suppose that p is some state which gives equality in @ ( J ) 2 S(p) - U ( J , p). Let p be p,,, 1 1 , the measure on { - 1, 1)’ obtained by restricting to functions of a, and a l . Let us suppose that /?(so, s l ) > 0 for all four values of so and s I . By an argument below, there is a Markov process p’ with p;o,,, = fi and a “nearest neighbor interaction”, J’ so that p ( J ’ ) = s(p’) - U(J’, p ’ ) . Since U ( J , p ) is only a function of p { , ) , it is only a function of plo. and so U ( J , p) = U ( J , p‘). U ( J ’ ,p) is only a function of p { , , since J’ consists only of ”nearest neighbor interactions”; similarly U ( J , , p ) IS only a function of p { , , Thus from the Gibbs’ inequality
,!
s(p’) - U ( J ’ ,p‘) 2 s ( p ) - U(J’, p ) and the equalities (definition of S )
208
VI.
Asymptotics
we conclude that S(p') 2 .S(p)
and thus p' also gives equality in the Gibbs' principle. This explains the second special feature of the Donsker-Varadhan theory noted above. In the above we wanted to construct an interaction J' and Markovian state p' once we are given 8.This can be done as follows: Given J', we consider a matrix J ( x , y ) = e - J ' ( " , ' )The . basic transfer matrix formalism of statistical mechanics [120] shows that a state p' giving equality in the Gibbs' principle has n
P ; O . ..., n )
=
+(so)
n
T(si- 1,
i= 1
si)+(sn)
/ an C
where 4 is the (necessarily unique up to constant) eigenvector of T with positive components and CI is the corresponding eigenvalue. This p' is Markovian. Thus given 8, we seek J' with
(1,
B(x, Y ) ) " ~ and -J'(x, y) = In p(x, y ) This is easy; we let +(x) = - In 4(x) - In +(y). Then 4 is an eigenvector of T with eigenvalue one and 4 ( ~ =) ~1 and is the associated pio, !.
We close this section with a somewhat imprecise description of the general framework in which Donsker and Varadhan imbed Theorems 19.1 and 19.2. The oscillator local time is the probability measure Lt(q, .) defined for each path q by Lt(q74 = t-'l{s Iq(s>EAIl
where, as usual,
1. . .I indicates Lebesgue measure. Notice that
f J; SdW(y(r), d u ) ) ds du
=t
s
W(X> Y ) W i J ,dx)L,(q, d Y )
Thus, Donsker-Varadhan consider general maps @ from .U,the probability measures on R to R. The general form of their variational principle is
19.
209
Donsker-Varadhan Theory
where I ( P ) = a, =
(,L Lo f ) ,
if p is not absolutely continuous with respect to dx if p = f Z d x
Of course, there are technical hypotheses on @ for (19.8) to hold, and more general *.reference" processes than Dq are considered. Donsker-Varadhan express a related result which is almost equivalent to (19.8); namely, for suitable sets C in A: E(L,(q, .) E C)
- ( exp
-
1
t inf l ( p ) PCC
( 19.9)
Let us mention two explicit applications of the machinery, especially (19.9) extended to much more general Markov processes than Dq. (a) If one takes independent copies of a fixed Gaussian process and uses the fact that sums of independent Gaussian variables are Gaussian, (19.9) gives information on the distribution of the Gaussian process in suitable asymptotic regions. In this way, Donsker--Varadhan [57] recover the result of Fernique-Marcus-Shepp quoted in the remark following Lemma 18.7 and also some versions of Theorem 18.1. (b) In Section 22, we will discuss the Wiener sausage Wa(t) which is a set-valued random function of the Brownian path, b(s), defined by VVs(t)= {x I dist(x, b(s)) I 6, some s E [O, t ] } .
There we will prove that the volume of W,(t )obeys t - ' I W,(t)I + 271
(19.10)
(in v = 3 dimensions) as t -+ cx, for almost every path b. This suggests that E( W' , ( l ) l 1 e - 2 a a r as f -+ a.In fact this is false! Donsker-Varadhan prove that [58] -
21
-
(19.11) for an explicit constant p E (0, a).(Without the explicit value of /?,this result had been conjectured by Kac and Luttinger [1441 on physical grounds.) (19.1 1) says that E(e-alW1'L)I) goes to zero more slowly than one would expect on the basis of (19.10). The reason is that I W,(t)I is large on sets of measure going to zero as t -+ co but not as fast as would be necessary for the e-2nu' behavior. Let us sketch the first step in the argument of [SS] which shows the relation to Theorems 19.1 and 19.2. As we will show in Section 22, the scaling covariance of b(s) implies that in v-dimensions
I WI(t)l
=
AVi2I W , h W ' ) I
210
VI.
Choosing ;I= rZivt2 (chosen so that ;Ivi2= t ; I - ' ) and letting s see that lim t 1-m
v/v+2
log E(exp(-'aIK(r)l))
=
Asymptotics = fviv+2,
we
lim s-l log E(exp(-txs( Ws-l&)I)) .5-
m
Except for the "mild" s- I i V dependence of 6, we are precisely in the situation of (19.8) since I W I is only a function of the image of b and thus of the support of L,(b, .). See [58] for the remainder of the argument.
VI I Other Topics
20. Perturbation Theory for the Ground State Energy Perturbation theory for discrete eigenvalues goes back to Rayleigh’s classic The Theory ofSound [213]; it was rediscovered by Schrodinger in his famous series [232] written at the dawn of the “new” quantum theory. These Rayleigh-Schrodinger series were placed on a firm mathematical footing by Rellich [218] with further developments by Kato [149] and Sz-Nagy [271]. All these authors dealt primarily with the regular case where the series are convergent. A typical case where the series are not convergent is L(p) = Lo + fix4 where as usual Lo = -f(d2/dx2) fx2 - ). Indeed, the BenderWu formula (18.8) says that the series is divergent. One can also prove this divergence directly (see [7] and below). In this case, Kato [149] and Titchmarsh [277] proved that the Rayleigh-Schrodinger series were asymptotic. See [SS, 152,217,2193 for reviews of these results. In the forties and early fifties, Tomonaga, Schwinger, Feynman, and Dyson (see [233] for a collection of relevant papers) developed some systematic series for certain objects in quantum field theory. One of special interest is the “energy per unit volume.” If one specializes to one space-time dimension (i.e., zero space dimensions), the Hamiltonian of the quantum field is just L(B)and its ground state energy is just this energy per unit volume. While the coefficients of this series can be shown to be equal to those of the RayleighSchrodinger series for this case, they appear quite different and more compact. One goal in this section is to present the Feynman series for E(P) = inf spec(L(B)) and prove that it is asymptotic. Our arguments below are essentially a specialization of those of Dimock [52] from two space-time dimensions to one.
+
21 1
212
VII.
Other Topics
Even though the perturbation series for E(P) is divergent, the function can be recovered from the series by a summability method known as Borel summability. Using the Rayleigh-Schrodinger theory, this was first proven by Graffi et al. [113]. By working a little harder on the estimates below, one could prove Borel summability using the path integral realizations. Indeed, this is just the specialization from two space-time dimensions to one of some work of Eckmann et al. [72]. Throughout this section E(P) denotes the lowest eigenvalue of L(p) = Lo + Px". Our goal is first to establish that E(P) has an asymptotic series aspas P 1 0 and then to obtain explicit expressions for the a,. Recall first the following definition.
c
Definition A function f(P) on (0, a) (a > 0) is said to have asymptotic series, as J.10, written
f(P)
-c
bsB"
c b,p as
(P 10)
if and only if, for each N , N
lim [f(& PlO
1 b,p s=O
A functionfhas at most one asymptotic series but the existence of functions like exp( - p- ')with zero asymptotic series implies that two distinct functions may have the same asymptotic series. Lemma 20.1
'
(a) I f f (p) is C" on (0, a) and for each n, b, = ( n !)- limb d"f/dp" exists, then b,P is an asymptotic series for$ In this case we say thatfis C" on [0, a). (b) IffN(p) is a sequence of C"-functions on [0, a) andf(P) = 1irnN+" fN(p) exists and for each n, there is a C , with
1
thenfis C" on [0, a) and (20.1) (including p
=
0).
20.
21 3
Perturbation Theory
Proof that
(a) Taking Po 1 0 in Taylor's theorem with remainder, one sees
Since d N +tf/dBN" has a limit as B 10, the last multiple integral is easily seen to be bounded by CNPN l . (b) Suppose that we know the limit on the right-hand side of (20.1) exists. Then by using the dominated convergence theorem in Taylor's theorem with remainder, one easily sees thatfis C" on [0, a ) and that (20.1) holds. By induction, we only need to show that if I d2fN/dB2 I I C2 andfN(P) converges, then dfN/dpconverges. Fix B > Po and let A N = dfN/dP(Do). Then +
1 [fN(b>- fM(8)1 -
(fl
-
-
flO>LAN
by Taylor's theorem. Taking N , M
l hAN so taking j
-+
[fN(BO>
-
Po we see that AN
-+
-f d p 0 ) l
rx, we
c2(P
-
PO)'
see that
b AN( 5 C,lP
-
BOl
converges.
The basic strategy of proof that E(P) has an asymptotic series is the following: By (1.9) and the Feynman-Kac formula: (20.2)
Each EN(B)is C" on [0, a)so we will prove a bound
(20.4) and thus conclude that E ( P ) is Cm on [0, 1) by Lemma 20.l(b) and therefore that E(P) has an asymptotic series by Lemma 20.l(a). To prove (20.4), we use a compact expression for d"EN/dp" in terms of the N the Ursell function Ursell functions of (12.14). Let ( X I , . . . , X n ) T , p , denote un(X1,.. . ,X,) with respect to the measure exp( - p J N N q4(s) ds) Dq/Normalization. Then, by (12.14) and the linearity of u,:
214
VII.
Other Topics
Now we claim that for s1 < .. . < s, and 0 I p I 1, . . * ,q 4 ( s n ) > T , p . N
(q4(sl)r
QnexP(-DnIsn - S l I )
(20.6)
From this, one easily sees that the integral in (20.5) is bounded as N varies so that (20.4) holds. Clearly (20.6) follows from (q4(S1),
. . ., q 4 ( S n > ) ~ , 8, N I Qn exP( - D' I s j
-
sj- 1 I
(20.7)
allj, for take D, = D'/n and I sj - sj- I maximal. [At this point we are being very crude and losing all hope of a careful estimate on the growth of C, with n ; more careful analysis ([72]) shows that one can take D, rather than D' independent of n. Our n-' dependence of D, leads to an extra factor of n" in estimating (20.5).] To prove (20.7), we use Cartier's formula, (12.11), which says that (q4(Sl),
where
(.)p,N,
.. .
3
q4(sn)>T,p,N
=
<Xl(sl)
*
"8n(sn))8,N,
-
- is an expectation with respect to n independent copies of
( . ) P , N and [w = exp(2ni/n)]
c wjqj(si)4
n- 1
8(Si) =
j=O
+
+
Now, let z(p) = Ll(B) . . . L@) on L_2(R") where Lidenotes L as an operator on the ith variable and let L(B) = L(p) - nE(B). Let
do = Finally, let 8 =
zy;,'
n
R0(Xi)
i= 1
d x ? . Then, by a Feynman-Kac formula:
(X(sl) . . . X,(s,)> =
(ao,e-'JX
. . . Xe-z~~ido)Z-l
(20.8)
+
to = s1 N , t l = s2 - sl, . . . , t,-l = s, - s,,.-~, with Z = (do,e-2Nido), t, = N - s,. Let d be the ground state for L. Then, as in the proof of (12.16), for any i = 2,. . . , n,
( ~ o , e - f ~ i " . . e - " - l ~ d=) 0 and thus in (20.8), we can replace e - f i - l Lby e-fi-l'(l - P ) where P = - E is nonvanishing and continuous for all /?in [0,1] so E = info s p s ( E , - E ) > 0 and
(d, .)d.Let E 2 ( p ) be the second lowest eigenvalue of L(B). Then E , lle-tt-li(1 - P>II
e-Efv-1
It follows that
(Z(sl). . . X(s,)>
~ - 1 ~ - & f ~ r l l X e - t ~. .rdoll z i . llXe-tl'..
. doll
20.
215
Perturbation Theory
Each 11...11 can be rewritten using the Feynman-Kac formulas as a Dq integral for which Holder's inequality can be applied to bound
px,
' .'
X" Dq
n (1
by e-"I 3,Im Dq)"". Thus, using the fact that 2 2 bounded for P E [O, 11, we see that:
[I
(ao,
is uniformly
+ (ao,e - ~ ~ 1 8 1 m e - ~ ' f i ~ > ]
1sm
so (20.7) follows if we show this last supremum is finite. This can be accomplished by considering three cases, s, t i 1, t 2 1, s 2 1. Case 1 s, t I 1.
which can be controlled since q4 2 0 and
I f(q(0)) 1" Dq < co.
Cuse 2 t 2 1. It suffices to show that I?lrnFLis bounded. By the Feynman-Kac formula and q4 2 0,
I (e - i f ) ( q ) I
I I (e- L'lf)(q)I r E ( D ) +
so I r?I"e-' bounded follows from the fact that I 21me-Lois bounded. s 2 1. By Case 2, e - i
Cuse 3
12 Irn
=
(1f
I"C-~)*
is bounded.
We have thus proven (20.7) and therefore we have the following. Theorem 20.2
E(P) has an asymptotic series
a,
as 10.
Now we want to identify a,. Looking at (20.5), the following is the obvious guess. Theorem 2 0 3
u,
= (~
The coefficients a, of Theorem 20.2 are given by
"+
n!
1-
(q4(0), q4(s2),. . . , q4(s,))Td"-'s
m<s2<m
-m
<S,,< m
where { ., . . . , . ) T is an Ursell function with respect to Dq.
(20.9)
216
VII.
Other Topics
Remark The same proof as below shows that the analog of (20.9) with the left-hand side replaced by (n!)-' d " E / d p and the right-hand side by (. . . ) T , a holds for all p if (. . . ) T , p denotes Ursell functions with respect to the P($),-process associated to L(p). Proof By the proof of Theorem 20.2, a, = IimNdma,(N) where
N , = (44(sl), . . . q 4 ( S , ) ) T , N , / J = 0 (for p = 0,fis independent of N but we write the proof so as to be applicable to fi # 0). As in the proof of Theorem 20.2, in the region s1 < . . . < s,; If(sl, .. ., s,; N ) - f ( s l , . .., s,; .c)l I C [ e - E ( N + S 1 ) e- &(N- S J J ~ - D ~ ~ I SSI I ~ ~ f(S1?
..
' 7
sn;
9
+
so that
s
[
a,(N) = ( - ly"(2N)-
ds1
...dsNf(s,,
. . . , s,;
CO)
- N < S I< ~ ~ ~ < s , , < N
+ O(N-')
1
Now, the right-hand side of (20.9) is equal to
J
(-l)n+l
dsz...ds,f(O,S2,...,Sn;
'JO)
0 < s2 < ... < s,,
= ( - 1 y +' ( 2 N ) - 1
J
dS1
... d S N f ( s 1 , . . . , s,; a)
-N<si
-
SI < S 2 < ' " < S , ,
= a,
sincef(. . .; m) only depends on the successive time differencesafter reordering. Thus,
20.
217
Perturbation Theory
Ursell functions are very complicated sums so the expression (20.9)is of limited value. To go beyond it we must use the fact that Dq is a Gaussian measure. As a preliminary, we note two facts: Let X I , . . . , x,k be jointly Gaussian random variables.
Lemma 20.4 Then (xl
' ' '
*2k>
=
' ' ' (xikxjk>
(xilxjl>
pairings
xpairings denotes the sum over all (2k)!/2,k! ways of breaking (1,. . . ,2k)
where into k pairs.
Proof Both sides are multilinear and symmetric in the X's so it suffices to = X Z k .That is, we need to show that for Gaussian check the case X , = X: 2k! (X'k) = (X2)k 2kk ! ~
But this follows from (exp(aX)>
=
exp(+a2(X2))
I
Lemma 20.5 Let X , , . . . ,X , , be Gaussian with respect to some expectation, (.). Let Y, = X I . . . X I 1 Y2 , = X I ,+ . . . Xlz,.. . ,Y, = X I , , -,+ . . . XI,,, with 1, = 2k.Then the Ursell function with respect to ( .) is given by: um(Y1, . . . , Ym) =
1
( X i , X j , > . . <XikXjk>
(20.10)
connected
pairings
where a pairing is connected if and only if { 1, . . . ,2k}is connected after one joins (1, . . . , 11) together, . . . , (1,+ 1, . . . , I,) together, and then i , to j , , i , to j,, etc.
,
Example Take k = 3; Y, = X 1 X 2 X 3 ,Y2 = X , , Y3 = X , X , . Of the 15 pairings, only the three with 5 paired to 6 are not connected.
Proof We first claim that for any random variables
(Y1 . . . Y m >
=x
C
n
(20.1 1)
4.i)
Pe.9 a , c P
where 9 is the family of all partitions P = {q, . . . , q} of (1, . . . , m} and u(ni) = uk( TI,. . . , qk)with x i = { j , , . . . ,j k } .(20.11) is equivalent to um(Y1,
. . ., Ym> = < Y, . . ym>
-
1 P€9 #(P)tZ
n
u(ni)
(20.11')
218
VII.
Other Topics
This may be proven inductively by checking Percus’ axioms (see Proposition 12.11) for u, defined by (20.11’) assuming that (20.11) holds for fewer than m Y’s. On account of (20.11’), if we are given (xlyi2... yi,) for any subset { i l , . .. , il} of (1, . . . , m } and candidates a(.) for the ~ ( nand ) if the a(n) satisfy (20.11) for any ... then a(n) = u(n).In this way, we can verify (20.10). For given Gaussian X’s, ( yi, . . XI) is a sum over pairings by Lemma 20.4.Any pairing induces apartition of {il, . . . ,il} by looking at the connected subsets induced by joining ( 1, . . . , l i } , etc., and then the pairs. Clearly
(x, x,),
+
pairings of Ill
so that the candidates for u(n) given by (20.10) obey (20.11).
I
We are now ready to give the Feynman rules for the coefficients a, of (20.9). By a labeled n-graph, r, for q4 we mean n labeled points 1, . . . , n and some lines joining the points with the property that exactly four lines come out of each point. Moreover, we label from 1 to 4 those lines associated to each point. A typical labeled three-graph is shown in Figure 2. Clearly there is a one-one correspondence between labeled n-graphs and pairings that enter in (q4(s1), . . . ,q4(s,))T.We define the value, u(I‘), of a graph as follows : For each line 1 joining i and j let g(I) = exp( - I si - sjI). Then
+
n g(l) dsz . . . ds, -
c s 2 c m,
(20.12)
..., - m c s , , < 00
In general, the integral in (20.12) will be convergent if and only if r is connected. In colloquial usage, one does not distinguish between and u(T) and calls them both, Feynman graphs, Feynman diagrams, or Feynman integrals. The last term is unfortunate since the integral in (20.12) is very different from the formal object of (1.8) which is also called a Feynman integral. (We note that the Feynman integral of Homology and Feynman Integrals [131] is of the type of (20.12)-more precisely, it is of the more
Figure 2. A typical labeled three-graph.
20.
219
Perturbation Theory
general type associated below to Green’s functions-the object of [131] is to study the analyticity properties of o(T) as the external parameters defining the Green’s function are varied.) Clearly (20.9) and Lemma 20.5 imply that the a, of Theorem 20.2 is given by a, =
{o(T) I connected labeled n-graphs
r}
(20.13)
Notice that the value of a graph is independent of any of the labeling. For this reason, it is useful sometimes to consider unlabeled graphs y. We set u(y) = v(T) for any r E y, and s ( y ) is the number of labeled T‘s which become y after removing all labels. We leave it to the reader to show that thesymmetry numbers, s(y), are given by:
where mij(y) is the number of lines joining points i and j in some r E y and M ( y ) is the number of ways of labeling the points leading to identical pointlabeled graphs. For example, if n = 2, there are two connected unlabeled graphs, shown in Figure 3. Moreover, s(yl) = 4! = 24 [ M ( y ) = n ! here;
Y!
7-2
Figure 3.
Unlabeled connected two-graphs.
a typical y with n = 3 and with M(y) = 2, not 3 ! is shown in Figure 41 and s ( y 2 ) = 4!4!/2!2.2 = 72. Moreover,
Figure 4.
A graph with M b ) # n !
VII.
220
Other Topics
so using a,
=
1 {s(y)u(y) 1 y an unlabeled connected graph}
(20.14)
we see that =
-21/8
Let us use these graphs to show that n!AB" I (-l)n+lan I n!CD"
(20.15)
(see also [7,241]). Lemma 20.6 Then,
Let N ( n ) be the number of connected labeled n-graphs.
(n - 1)!(2n)!/(n!2") I N ( n ) I (4n)!/[(2n)!P]
Proof The upper bound is just the number of ways of pairing 4n objects which is the total number of labeled diagrams, connected or not. To get the lower bound, pick one of the (n - l)! orderings of 2,. . . ,n, say, i,, . . . , inLet line 2 from point 1 be joined to line 1 from point il, line 2 from il to line 1 from iz, etc., and line 2 from in to line 1 from point 1. Pair the remaining 2n lines in an arbitrary way. Each of these ( n - 1)!(2n)!/n!2" graphs is connected.
,.
The point is that the lemma implies that (n!)'ab" I N ( n ) I (n!)'c d" Lemma 20.7
(20.16)
For some fixed p, 6 and all I-:
p
I (-i)n+lu(r)n!I 6"
(20.17)
Proof To get the lower bound we consider points si with 0 I si I the integral in (20.12). Then g ( / ) 2 (412- in that region so ( - I>,+
& in
u(r)2 ~ ( 4 ~1)1 "-
To get the upper bound, we first claim that one can pick (n - 1) lines I,, . . . , 1,-1 E r which connect the graph. Assuming this fact, we let xk = si - sI for the line l k . The fact that 11, . . . , 1,- connect the points implies that any si is a linear combination of xk's so the coordinate transformation
20.
221
Perturbation Theory
si + x,, is nonsingular. Moreover, it is easy to see that the siare combinations of x's with integral coefficients and vice versa. Thus the transformation has Jacobian 1. Since Ig(l)l I
4,
,.
This leaves the proof of the claim about I,, . . . ,I,For n = 2, the result is obvious for an arbitrary graph, even without the four lines per point no - 1 for such an arbitrary condition. Suppose we know the result for n I graph. Given an n point connected graph pick some line, I,, joining i and j. Consider the graph obtained by fusing i and j. It can be connected by no - 2 lines by the induction hypothesis. Adding the line 1,, we connect the original graph with no - 1 lines. I (20.16) and (20.17) clearly imply the following theorem. Theorem 20.8
(20.15) holds.
Actually, one can estimate the quantity N ( n ) of Lemma 20.6 for n large rather exactly. Let G(n) = (4n)!/[(2r1)!2~"]= (4n - 1)(4n - 3)...(1)
be the number of graphs connected or not. Then lim N ( n ) / G ( n )= 1 n-
(20.18)
(c
Indeed, for n 2 2 IG(n) - N(n)l
< CG(n)n-'
(20.19)
which is even stronger than (20.18). Our proof of (20.19) is motivated by ideas of Cvitanovic, Lautrup, and Pearson [39a]. We begin by considering the functions: a(t) = (277-
'I2
c ( t ) = In a ( [ )
exp( -fx' - t x 4 ) d x
222
VII.
Other Topics
Then a(t) has a diagramatic perturbation expansion identical to that of the y4 oscillator except that now every graph has value 1. Thus (with G(0) = 1, N O ) = 01, m
- C C(n)( 03
c(t)
-t)"
n=O
with A(n) = G(n)/n!;C(n)= N(n)/n!. Equating coefficients of t" in a(t) = exp(c(t)), we see that n
A(n)
=
C(n)
+ C (k!)-' C k=2
C ( j , ) . . .C(j,)
ji 2 1
(20.20)
j,t...+j,=n
(20.20) can also be obtained by realizing that an arbitrary labeled graph breaks into k connected pieces. The number of graphs in k pieces is just
Now let
Noting that A(n - j ) A ( j )is decreasing in j until j one sees that for n 2 2:
(4n
-
2n 1)(4n
-
-
3)
=
[n/2] and then increasing,
- 1) + n ( (4n - 1)(4n -n(n3)(4n - 5)(4n
5 Dn-'A(n)
-
7)
(20.21)
for a suitable constant D. In the above, the first contribution comes from the - 1 terms in the sums and the second by bounding the remaining sum by nA(2)A(n - 2). Now
j = 1 and j = n
n-k+l
B(k, H ) =
C
A(jl)B(k - 1 ; n - j , )
jl=l
From this formula and (20.21) one easily proves inductively in k that
B(k, n) 5 D'n-'A(n)
(20.22)
20.
223
Perturbation Theory
Using (20.20), (20.22), and the trivial bound CCj) 4 A ( j ) , we obtain (20.19) with C = e”. By working with these same ideas, one can obtain fairly easily explicit terms in an asymptotic expansion N(n)/G(n)
-I+
h1n-l
+b2K2+.
*** We close this section with a discussion of two further aspects of the Feynman graphical analysis: “graphs in p-space,” and graphs for Green’s functions. In (20.12) introduce a variable p , for each I and write
Doing this involves choosing a direction (from i to j ) on each line. Proceeding formally, we do each s, integration as follows: The s,-dependence is
IOU,
p i is over those lines pointing from i to some j and where those lines pointing from some j to i. Then
xi,, is over p,
While the above manipulations are formal, it is not hard to justify the final result. In essence, the last integral is a generalized convolution and its equality to u(T) is just a generalization of the usual Fourier transform formula relating products and convolutions. Graphically, the above can be described as follows: Each line has a momentum p, with conservation of momentum enforced at each vertex i = 2,. . . ,n (it then follows automatically at vertex 1). The number of free momenta to integrate over (usually called loop momenta since they can be chosen to “flow” in loops) is thus not 2n but only n + 1. This “momentum space” analysis is common for two reasons: In quantum field theory, it is most usual to discuss renormalization theory in “ p-space ” (e.g., Weinberg [284]) although more recently x-space analysis has become more common (e.g., Speer [262]); see Hepp [125] for a review of renormalization theory. In discussing scattering theory in quantum field theory, the Fourier transforms of the Green’s functions we are about to discuss are natural and these have a p-space analysis.
224
VII.
Other Topics
Finally, one often considers the (truncated) Euclidean Green's functions defined by ...,sn)
= (~(s~),...,s(s">>T,N=~.~
It is not hard to show by mimicking the proof of Theorem 20.2, that
=
I
( - l)k
-m
. . . q4(lk))T,p dkt 7
Thus G , has an asymptotic series given by a sum of a suitable kind of connected Feynman graph.
21. Dirichlet Boundaries and Decoupling Singularities in Scattering Theory
We have already seen in Section 7 (see Lemma 7.10) that Dirichlet boundary conditions are closely connected to the exclusion of Wiener paths from certain regions. However, because we dealt with Wiener paths with a fixed initial point, i.e., with Brownian motion, we required a rather strong regularity condition to establish the connection. In this section, we begin by establishing a weaker kind of connection under the most general circumstances; we will then describe some ideas of Deift-Simon [45] on the connection of those exclusion formulas and a problem in scattering theory. The following involves the general operator HDisdefined in Section 7; i.e., the operator on L2(R"\S) which is the Friedrich's extension of - A on C,m(RV\S). Theorem21.1
Let S be an arbitrary closed set in
[w".
Then for any
f , g E L2(RV\S):
(f,e - r H D ; s 9 )=
I,
f(o(O>)s(o(t>) dPL,(O)
(21.1)
~
where Q is the set of paths:
Q
=
{o)o(s).$S,O I
s I t}
Proof([252]) Extend the form of HDiSto L2(R") by viewing it as a nondensely defined quadratic form.
21.
225
Dirichlet Boundaries
Write R‘\S = Choose Define
K, c K,+,.
K , where each C$ so that 4,
+,E
{
I V4,(x)12
K, is =
compact, K , is open, and 1 on R, and supp 4, c K,,,.
+ dist(x, S)-3,
x$S
V(x> = , ; X€S
Let Q(V) = { , f L~ ’ 11 V(x)lf(x)12 dx < .c}. We first claim that Q ( V ) n Q( -A) c Q(H&, for . f Q( ~ V ) implies that lI(V4,)fl1’ < m so that if f~ Q ( V ) n Q(-A), then (V4,)f- 0 in L’. Thus 4 , f - L V ( 4 , f ) + Vfi i.e., f~ Q(HD;s). But clearly, C;(K\S) c Q(V) n Q(-A). Thus, by the monotone convergence theorems for forms [ 152,253,2543, e-rHv;s
-
s-lim exp[ - t( -$A
+ AV)]
(21.2)
210
By approximating V with V, ( f , expC-t(-tA
=
min(m, V ) , one sees that
+ 2V)lg)
But since V has a dist(x, S)- term and Wiener paths are Holder continuous of order i, fo V(m(s)) ds = cc if m(s) E S for some s. Conversely fo V(m(s)) ds < 00 if w never hits S since o then lies in some K , and moreover, dist(m(s), S ) - 3 is bounded. Thus
Q
= {m
so that (21.2, 3) imply (21.1).
I
I
J:V(m(s)) ds <
00
1
Before turning to our main application of (21.1), we remark upon some other connections: (1) (21.1) gives an expression for P D ; s ( ~y,; t ) holding almost everywhere such as in x, y. It is an ideal tool for studying rather subtle properties of whether it approaches zero as x + S. (2) Intuition connected with (21.1) is useful in deciding what kinds of singularities in V can destroy simplicity of the ground state for - A + V (see [SO]). (3) Motivated by (21.1), Klauder [159] has noted an interesting phenomenon in singular perturbation theory. Let Lo = -f d2/dx’ fx’ - t, as usual, and let V = x-’ ( a > 2). Then (e.g., [215]) L(p) = Lo + PV is
+
226
VII.
essentially self-adjoint on C$(R\{O}) for state energy of L(p).Then
Other Topics
> 0. Let E(P) denote the ground
lim E(P) = 1 # inf spec(,!,,) = 0 PlO
For, if one writes a Feynman-Kac formula for L@), j b V(o(s))ds = 00 for any (Holder continuous) path crossing zero. Thus, taking D 10, we recover the Feynman-Kac formulas for “Lo” with a Dirichlet condition at zero, call it LDand not Lo. We have inf spec(,!,,) = 1. In fact, our proof of (21.1) is motivated by [1591. See [43,246] for a discussion of the Klauder phenomena using purely quadratic form ideas; see [160, 114, 1231 for more detailed information on E(P) for small; see [79] for a more thorough analysis of the Wiener paths and the Feynman-Kac formula for ,!,(p). We now turn to the application to scattering theory. Given an operator, H , which is somehow like H o = -+A at spatial infinity, the two fundamental “foundational questions of scattering theory are the existence of the wave operators Q + ( H ,H,) = s-lim ei‘f’e-itf’~] ”
t-TC.2
and their completeness Ran Q * ( H , H,)
=
Ran P,,(H)
where P,,(H) is the projection onto all vectors whose H-spectral measures are absolutely continuous. An extensive literature has developed to study the case H = H o + V mainly based on perturbation formulas like
(H - z)-l
-
(H,
- z)-l =
-(H,
- z)-lV(H -
z)-1
which do not distinguish between positive and negative singularities of V . (See [216] for a review.) Motivated by an example of Pearson [200], described further below, Deift and Simon [45] tried to see what local singularities of V had no effect on the existence and completeness questions; i.e., they sought theorems which asserted that R’(H, + V , H,) existed and were complete if R*(H, W , H,) existed and were complete and if W - V had compact support (earlier Kupsch-Sandhas [1631 had studied the analogous existence question). For simplicity, we suppose that W = 0; i.e., we consider here V’s of compact support; it is easy to extend the results to the more general setting. Deift-Simon exploited the semigroup e - r H , which is more sensitive to the sign of V , Feynman-Kac formulas, and some rather elaborate machinery. Some small simplificationsof the case V 2 0 occur in [252]. Using different machinery (and in particular, neither path integrals nor Dirichlet boundaries) Combescure-Ginibre [38] recovered the Deift-Simon results with fewer
+
21.
227
Dirichlet Boundaries
technicalities (see also [216] for an exposition of their work); below we use some of the ideas of [38] to simplify the details of the Deift-Simon approach. Independently of Deift-Simon, related results were obtained by Pearson [201] and Semenov [236]. Suppose first that V 2 0 has compact support. We will suppose that V E L’([w’\G) where G is a closed set of measure zero. In that case, H , + I/ is a densely defined closed form on Q ( H , ) n Q( V )and so defines an operator H . By Theorem 6.2, e-tH is given by a Feynman-Kac formula. Let S be a sphere which surrounds supp( V ) .Let H b be the Dirichlet Laplacian H D ; and let H’ = Hb I/. The strategy is to prove that e - H - e-H(lis trace class by proving that eCH’- e-”, eCH0- e-H6,and eCH‘- eFH6are trace class and then use the following (for a proof, see, e.g., [216]).
+
Theovem21.2 Let A , B be self-adjoint operators which are bounded from below. Suppose that e - * - e - B is in Yl, the trace class. Then
R’(A, B ) = s-lim eirAe-i‘BP,,(B) t-gm
exist and have the same range as P,,(A). The following illustrates the use of (21.1). Lemma 21.3 Let Hb be HD:swhere S is the boundary of any compact convex set. Then, for all x, y E R, the unbounded component of W\S:
0 I (e-tHo- e-‘Hb)(x, y) I (2nt)-”’ exp( - [dist(x, S)’.
+ dist(y, S)’]/4t) (21.4)
In particular, is Hilbert-Schmidt for any integers, k and m. Proof By symmetry, it sufficesto prove (21.4) with 4t replaced by 2t and with the dist(y, S ) term replaced by zero. Given x E R, let zo be the point in S closest to x and let n be the plane through zo perpendicular to x - z,. By some simple geometry, .n separates S and x. Thus, any Wiener path joining x and y which passes through S must pass through n;i.e., since (21.1) implies for almost every x, y ( e - t H ~ ~- , - t H o : \
dPo, x , y ; t ( 0 )
(21.6)
228
VII.
Other Topics
we have 05
- e-'Hh)(X,
(,-tHu
y) 5
(e-tHi~
-
e - t H ~ .
"XX, Y)
= e-'H"(X, y')
(21.7)
where y' is the image of y under reflection about 71. In obtaining (21.7), we use the method of images formula for e-lHD:m. (21.7) and the trivial
I (e-'",,
-
e-'Hil)(x, y) I I e-tH(l( x , y)
imply that -
l(e-tH[l
e-lHi)))(x,y)I 5 min[e-tH"(x, y), e-tHo(x,y')]
Since either or
dist(x, y) 2 dist(x, 2,)
dist(x, y') 2 dist(x, 2,)
we have proven (21.4). The formula (21.4) implies the kernel of ( 2 1 . 5 ) is square integrable for x, y E R. A similar proof yields a
( 2 ~ t ) - "exp( ' ~ - dist(x, R)2/2t) bound for x E R, y $ R and a (27ct)- v2 bound for x, y $Q.
1
Theorem 22.4 If V 2 0 has compact support [ V E L'(R'\G), H as above] then e - H - e -"(I is trace class and, in particular, R*(H, H , ) exist and are complete. Proof Let A = e - H c - e - H O B = e - H - e-H' C = e-HL prove A , B, and C are trace class. Write A = PQ Q'P', where
+
e-H',
We
p = e - H o / 2 (1 + X2)-k Q = (1 + ~ 2 ) k ( ~ - H 0 / 2 - ,-Hb/2) Q'
= (e-H0/2
- e-Hi1/2)(1
+
~ 2 ) k
p' = (1 + x 2 ) - k e - H b / 2
By the lemma, Q, Q' are Hilbert-Schmidt for any k and it is easy to see that P, P' are Hilbert-Schmidt for k large. Thus A is a trace class. Since the Feynman-Kac formula [including (21.4)] and I/ 2 0 show that the integral kernels of P, Q, P', Q' dominate the analogs with H,, Hb replaced by H , H', we see that B is trace class. Finally if R, R' are the unbounded and bounded components of R'\S, then e-Hb= e - K 0 e - L e-H'
= ,-CK+V)
oe-L
on L'(sz') 0L ~ ( R )
21.
229
Dirichlet Boundaries
Now e-" has a bounded kernel and thus, since R' is bounded, e-IK is HilbertSchmidt and so e - Z r Kis trace class. Again, by the Feynman-Kac formula, - r(K + V ) is also Hilbert-Schmidt and so trace class. Thus C = e - K 0 0 - e - " '1 0 0 is also trace class. I +
Remark By very different methods, Davies and Simon [41] have proven that H's of the above type also have empty singular continuous spectrum.
We now want to extend this result to allow V to have a negative part. The idea, also important in Section 25, will be to exploit Holder's inequality in the form:
which will lead to a bound on an integral kernel K ( x , y) by L(x, y)"M(x, y)'-". Thus, we will need the following lemma.
I
Lemma 21.5 Let p = 2" (ti 2 1). Suppose that A , B E Y,,( = { A I A I' with positive integral kernels L(x, y ) , M ( x , y ) [on some measure space L 2 ( X ,dv)]. Let C be an integral operator with kernel K obeying
E Y,})
I K ( x , y)l
(21.8)
I L(x, y Y M ( x , Y>'-"
f o r O s a I l.ThenCE9,. Proof (by induction on n ) n = 1 is easy, since A E Y2 if and only if its integral kernel is in L 2 ( X x X , dv @ dv) and L , M E L z implies L"M' -" E L2 by Holder's inequality. Suppose the result is known for n < n o . (21.8) implies that (by Holder's inequality)
IK*K(x, y )I I Jl K ( z , x)l I K ( z , Y)l W ) I (L*L)(x,y)"(M*M)(x,y)'
so by the induction hypotheses K * K Lemma 21.6
Let H ( a ) = H , (1
--(I
1
EYPi2.
+ a V . Let p
= 2"(n
2 1). If
+ X 2 ) - k exp(-tH(1 + 6))
is in Y pfor all large k (both 6 > 0 and t fixed), then (1
+ x2)'[exp(-tH)
is in Y pfor all I , m. Here H of a convex compact set.
=
-
+ x2y = HD;s+ V for S the boundary
exp(-t~')](l
H(1) and H'
230
VII.
Other Topics
Proof Let P be the family of paths hitting S . Then
where y - l
+ (1 + 6)-'
= 1. Let
+ x2)'(e-IH- eCrH')(x,y)(l + y2)" ~ ( xy), = (1 + x2)a(e-1H(l- e-IH;I)(x,y)(l + y2)' ~ ( xy), = (1 + x 2 ) - k e - t H ( ' + d ) (y)x , CI = y [ l + (1 + 6 ) - ' k ] ; p = ym. Then IKI I L1/yM1'('+").By K ( x , y ) = (1
with hypothesis, M is an 9 pkernel, and by Lemma 21.3, L is Y P so , K is 9 pby the last lemma. I Lemma 22.7 Let V- I 0. If H , + (1 + 6)V- is bounded from below for some 6 > 0, then H' = Hb + V _ obeys (1 + X 2 ) - k e - ' H 'is in 9,for all large k .
Proof
As in the proof of Theorem 21.4: - ,-It/,;
e-rH'
=
(e-t(K+V-)
-
e W t K0 ) 0
+
Moreover, (1 X 2 ) - k is bounded so we need (1 + X 2 ) - k e - t H €9, d and + V - ) ,e F t KE 9,. The former is true by considerations used in the proof of Theorem 21.4. eCrKE Y2 is proven in Theorem 21.4. Finally, since H , + (1 + 6)V- 2 a, we have that ,-t(K
K+V-=-
1 l + d
(K
+ (1 + 6)V-) + 1 K -
Y
2
1 K y
-
+ l +1 d U ~
so that the eigenvalues of e - ' ( K + V -are ) dominated by (constant times) eigenvalues of e - s K so €9,. I
+
Theorem 21.8 ([45]) Let V have compact support and let V = V+ V-, V+ 2 0, V- I 0 with V+ EL~,,(R"\G) and with H o (1 6)V- bounded I/. Then eKtH - eKtHc1 is trace from below for some 6 > 0. Let H = H , class for all t > 0 and, in particular, Q'(H, H , ) exist and are complete.
+
+ +
231
22. Wiener Sausage
Proof Consider A , B, C as in Theorem 21.4. A is trace class as it is there. C is trace class as in the last lemma. Moreover, as in Theorem 21.4, we can
eliminate the V+ and need only show that the analogs of P, Q, P', Q' with H , replaced by fi = H o + V-, etc., are Hilbert-Schmidt. This we do by induction, showing they lie in 9 pp, = 2". By hypothesis, ( H , Vc)- l" < a(H, + 1 ) - l I 2 so (1 + X y ( A + C ) - q l + X y E $ ~ , , for some k andp, = 2"~1.1tfollowsthat(l+ X2)-k(fi + ~ ) - " ~ a n d t h u s ( + l X2)-ke-'H X 2 ) m is in .azp,, lie in $ 2 p , , . By Lemma 21.6, (1 + .X2)'(e-'H - e-'H')(l and by Lemma 21.7, (1 + X2)-ke-"" is in Using the P, Q factorization of e-" - e-'"', we see that this difference lies in -apc,. Since (1 + X2)-ke-ffi' is also in .ap,,(by Lemma 21.7), we see that (1 + XZ)-ke-'His in 9,,,,. We can now repeat the above argument (no - 1) times and conclude that all the required operators lie in 9,.I
+
+
+
Remark Some condition like H , + ( I + 6)V- 2 --a is needed for completeness since Pearson [200] has constructed a potential V for which V is C" away from zero, supp V is compact, H o + V is bounded below and essentially self-adjoint on Cr(R3\O), R'(H, H , ) exist, but Ran 51' # Ran R - !
22. Crushed ice and the Wiener Sausage
The title of this section suggests a cocktail party and, indeed, our formal goal will be to understand the first law of cocktail-dynamics: crushed ice is a more effective cooling agent than block ice. That is, n small balls of ice each of volume 4nr3/3 are a more efficient cooler than one chunk of volume 4nnr3/3. The folk wisdom is that this is due to surface area; i.e., the n balls are roughly comparable to a chunk with surface area 47cnr'. In fact this intuition is wrong; the relevant factor is nr: explicitly n coolers of radius r,, will make no difference if nr, + 0 as n -+ 00 and will be infinitely efficient if nr, -+ co and the spheres are distributed "throughout" the region to be cooled in a suitable sense. Thus, crushed ice is even better than the folk think ! The model we consider is a bounded, open region R in R3 and n balls, B,,. . . , B,, of radius r, about points x,, . . . , x, E R. We consider the lowest eigenvalue El(n) of -+A on L2(R) with Dirichlet boundary conditions on 82and on B , , . . . , B,. The solution of the heat equation ic = ~ A with U zero boundary conditions behaves asymptotically as t + co as ce-fE'(n)with c > 0 if u(x, t = 0 ) > 0 so that the size of Elis a measure of the efficiency of cooling. Putting Dirichlet boundary conditions on dR is somewhat unnatural, corresponding to placing the "pitcher," R, in an ice bath; Neumann
232
VII.
Other Topics
boundary conditions would be more natural but more complicated for both the analytic and probabilistic approaches described below. This problem (without the picturesque cocktail analogy) was first considered by Kac [141] in relation to some related work by Kac and Luttinger [1441. Kac used Wiener integral methods. Rauch and Taylor [210] considered the problem analytically. We will reverse the historical order and first describe the analytical results following a lecture of Rauch [208] and then describe the probabilistic methods. The latter are more involved but give much more detailed information. In line with the methods, the analytical approach interprets “distributed throughout R ” as “uniformly distributed” while the probabilistic approach as “randomly distributed”-due to fluctuations these are very different notions. The crushed ice problem is one of a number of closely related problems, some of them quantum mechanical; these are discussed in [141, 144, 20821 11; here for intuition, we mention one other: Consider optical scattering off n spherical absorbers of radius r, distributed uniformly through R; i.e., the wave equation ii = Au where the Laplacian has vanishing boundary conditions on an. Then ([209,210]) as n -+ m, the absorbers become transparent (i.e., the scattering approaches that with no absorbers) if nr, -, 0 and opaque (i.e., the scattering approaches that with zero boundary conditions on an)if nr, -, co.At first sight this is surprising, for in the limit of ray optics, cross-sections (and so nr,”) should count. The point is that ray optics is only good for wavelengths short compared with the dimensions of the problem, so as r , -, 0, for no wavelengths. What is relevant is Brownian paths-the occurrence of nr, rather than nr,”is an expression of the wigglyness of Brownian paths-indeed it is, in some sense, a restatement of the fact that the hitting probability for a sphere of radius r is proportional to r and not r in v = 3 dimensions.
’
We first discuss the analytical approach. Fix R and define E,(n) as above and let E,(O) be Theorem 22.1 ([208]) the lowest eigenvalue of -An on L2(R) with Dirichlet conditions on aR. Suppose that I xi - x j I 2 4r, for all i, j . Then for nr, 5 d ,
for suitable nonzero constants c, d depending only on R (and independent of n, r, and xi). In particular, E,(n) -+ E,(O) as n --+ cc if nr, 0. -+
Remark Using more sophisticated methods, Rauch [208] proves a stronger result.
22.
233
Wiener Sausage
Proof E,(O) I E,(n) is easy. Let $ be a normalized eigenvector for -An with eigenvalue E,(O). For any “piecewise C’ ”function 4 with 4 = 0 on each ball Biand 0 I 4I 1, we have that
by the variational principle. Now clearly as 4 I 1
and J14$12dx 2 1 -
J
I*12dX
(Xl$(X)<
2 1-
11
ll$llf I{x (4(x)< 1>1
Thus (22.1) follows if we find I$ with
llv4ll; I any, I {x I 4(x) < 11 I I bnd with a, b independent of xi,n, r,. Let xibe the center of ball B , and write r for r , . Let
=
and take 4
=
1:
Ix - xi[- r , r
Ix - x i /I r r I Ix - xil I 2r
Ix
-
xi[2 2r
nl=, 4i. Then the supp( 1 - &) are disjoint so
and
Definition We say that the Biare uniformly distributed of degree m if and only if there exist R , so that if Bi are the balls of radius R , about x i , then (a) R c Biand (b) each y EL?is contained in at most m different Bi.
u
234
VII.
Other Topics
If the Biare uniformly distributed of degree m,
Theorem 22.2 ( [ 2 0 8 ] ) then
(22.2) where E is a function of n113rngoing to zero as n113r,.+ 0. In particular, if nr, .+ 00, n113r,.+ 0, then E,(n) -, co. Remark If nr, -+ co,we can always decrease r, [which will only decrease E , ( n ) ] so that nr, + co,n1I3r,+ 0. [Notice that uniform distribution is only a property of the xinot of the r,’s.]
Proof’ Let c(R, r ) be the lowest eigenvalue of -+A in L2({xlr < 1x1 < R } ) with the boundary conditions 4(x) = 0 if 1x1 = r, a$/& = 0 if 1x1 = R . Then c(R, r ) = R - y ( r / R ) by scaling. We first claim that lim y - t f ( y )
=
(22.3)
1.5
YlO
For choose R = 1, let 4(x) be the required eigenfunction and note that, by a partial wave expansion, $(x) = I x 1- ‘ u ( I XI) where - u” = k2u, f ( r ) = $k2, u(r) = 0, and u‘ - (l/x)u = 0 at x = 1 . Thus u(x) = sin k(x - r ) where k is the smallest solution of
-
k cos(k(1 - r)) = sin(k(1 - r ) )
+
+
-
Using tan x x 3x3 O(x5) one easily sees that k2 3r as r .+ 0. This proves (22.3). Thus, using the fact that Neumann boundary conditions correspond to no condition on the form domain [217], for any trial function vanishing on
Bi : (22.4) where E 1 0 as r,/R,
+ 0. Using
and m I R I 2 $ m R : 2 I R 1, one obtains (22.2) from (22.4).
I
235
22. Wiener Sausage
Remark While the above uses no Wiener integrals, it implies something about Wiener paths; namely, if {Bin'}:= is a family of uniformly distributed spheres with nr, + 00, then almost every Wiener path w has the property that for all large n, w hits some By'.
*** Next we turn to the probabilistic approach. Let x = ( x l , .. .) be an infinite sequence of points in R. We put the product measure dy ==;() I R I - dx, on these sequences; i.e., the x's are independently distributed "uniformly" through R (intuitively, they are randomly placed). Fix a sequence r l , r 2 , . . . of positive numbers once and for all. Given x in R" and n we let H(x, n ) be -$A on L2(R) with Dirichlet boundary conditions on 8R and on the n balls of radius r, about xl,.. . ,x,. Let &(x, n ) be the kth eigenvalue of H(x, n). We dy). To state the main result view E,(x, n ) as random variables, Ek(n), on (Ra, we need a new notion: Definition Letf,,fbe random variables. We say that probability if and only if lim &If, ,+"
-fl
> E)
=
converges tof'in
0
for each E > 0. Remark It is easy to see that Ilf, - f l i p + 0 implies convergence in probability and also thatf,(x) +f(x) for almost every x implies convergence in probability. From the former fact one easily sees that if E(f,) + a and E(,f:) + u2, thenf, -+ a in probability. Let nr,
Theorem 22.3 ([141]) Then
Ek(n)
+ a (a
Ek(0)
may be zero or infinity) as n + co.
+ 271a I
I
~
(22.5)
in probability for each k. Thus random impurities produce a constant shift of the spectrum. Since &(n) is monotone increasing in r,, it is easy to see that the 0 < a < co results imply the a = 0, co results also, so henceforth we suppose that 0 < a < co. The proof of Theorem 22.3 requires some elaborate preliminary machinery. First, we need a Feynman-Kac formula for Tr(e-'H'")), a formula already used in Section 10.
236
VII.
Other Topics
Let Q be an arbitrary open bounded set in R' and let HR Theorem 22.4 be the operator -+A on L2(R) with Dirichlet conditions on 82.Then, for any t
> 0, Tr(e-'"")
=
(2nt)-'"
J,
d'x E(x
+ b(s) E R, 0 Is IfI b(t) = 0)
(22.6)
Proof Let I/ be the potential used in the proof of Theorem 21.1 and let H(A, rn, E ) = -+A + A min(V, m) + &x2.Then, e-tH(a,m*c) is trace class and its trace is given by (9.5). We take rn -+ 00, E + 0 and then A + 0 and recover the required result. I Thus Tr(e-'"("'")> = (27ct)- 312 1 d 3 y E(y y
+ b(s) E R;
+ b(s) 4 Bi(x, n), i = 1, . . . , n, 0 I:s s t 1 b(t) = 0)
(22.7)
where Bi(x, n) are the balls excluded in the definition of H ( x , n). Let (.) denote expectation with respect to dy on 0". Our goal is to show that (Tr(e-tH(x;n))) ( I Tr(e-fH(x;n))
12)
Tr(e-fH(0))
+
,-2ztalnl-'
+
I e- 2ntalfil Tr(e-tH(0))12
(22.8) (22.9)
so that for each t, e-lEk(")converges to C e-'(Ek(o)+ZnalRII ) in probability. To prove (22.Q we interchange the dy integration with the JJ and Db integration and do the dy integration first. Thus we will have a fixed path b and want to know whether a randomly placed sphere of radius 6 is hit by b. This clearly depends on the volume of a &neighborhood of b. We therefore make the following definition.
1
Definition W,(a, c) is the set-valued random variable on Brownian paths defined by
Wa(a, c)(b) = {x I I x - b(s) I I 6 for some s E [a, c ] } Wa(t) = W,(O, t) is called the Wiener sausage for obvious geometric and punnish reasons. We work in v = 3 dimensions throughout. We are interested in 1 W,(t)I as 6 + 0. Because of scaling
I Wa,(A2t) I = A3 I Wa(t>I
237
22. Wiener Sausage
for any i> 0 so we can consider I W,(r)l as t 2 d3 I W,(6- 2 t ) l as d + 0. behavior of I Wa(t)I
+ rx?
and then deduce the
Theorem 22.5 (Spitzer [266])
+
E(I Wa(t)I) = 2 x 6 ~ 46’(2nt)”*
+ (4n/3)a3
(22.10)
and, in particular, (22.1 1) (22.12) Proof Without loss take 6 f(x, t ) = E(x
1. Let
=
+ b(s) E B , for some 0 5 s 5 t)
where B , is the ball of radius 1 about 0. Now E(l WI(t>I) = JDb Jdx {x I x =
E
Wl(t)I
+ b(s) E B , for some 0 I s 5 t}
I
Jdx S D b { x - x
Next notice that j ( x , t) has the following properties: (a) for 1x1 > 1, lim, l o f ( x , t) = 0, (b) for t > 0, l i m l x , ~f(x, l t ) = 1, (c) in 1x1 > 1, t > 0,fobeys
(d) for 1x1 I 1, t > O , f ( x , t ) = 1. (d) is trivial and (a), (b) follow from the kind of hitting probability considerations in Section 7 (see especially the Aside following Lemma 7.21). (c) follows from (7.12) which says that 1-f=
s
P d x , Y ; t ) dy
238
VII.
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and the formula
(using the convergence of the integral, one gets the differential equation forfa priori in distributional sense). Sincefis spherically symmetric, we take I x I f(x) = g( I x I) and find that
with the boundary conditions g(r, t) 0 as t 0 for r > 1 and g(1, t ) = 1. The equation and first boundary condition suggest we try -+
-+
0
g(r, t ) =
S_rn(2at)-l/2e-(r-y-l)'~2'
and the second boundary condition gives d p
=
d.4Y)
2 dy. Thus
for 1x1 2 1. We thus compute f(x, t ) d 3 x = (4n)2(2~t)-"~Jlmr dr
Jome-(z+r-
1
~
dz ~
1
XI21
= (4~)2(2nt)-'/~ Jlmr dr
re-(Y-l)'/2f
dY
(u2 + 2 ~ ) e - " " ~du ' = 2xt
Since flxl
+ 4(2~t)"~
,f(x, t) d3x = 4x/3, we have (22.10). I
Remarks 1. Spitzer [266] considers a more general problem, namely Q(t) = j x 9 Kf(x, t) d3x where B , is replaced by a general set K in defining f(x, t). He finds the asymptotics of Q(t) to O(t'/2).In particular, the 2x6 in (22.10) enters as the Newtonian capacity of the ball. Of course, Spitzer's considerations in general are more sophisticated than those above; (22.10) is mentioned as an aside.
239
22. Wiener Sausage
2. (22.11) is a quantitative expression of the “fatness” of Wiener paths. For a smooth curve of length 1, the volume V, of the neighborhood goes as nh21 (note d2, not 6) as 6 + 0. In some kind of sense, (22.1 1) and its analog in higher dimensions say that the image of the Wiener path has “dimension 2.”
cj”=l
It is not quite true that I W,(nT)I = I W,((j - l)nT, jnT)I since the successive pieces of sausage overlap, but if T is large this should be negligible. The pieces in this last sum are independent. It is therefore reasonable that some kind of strong law of large numbers holds and that I Wl(t)I/It + l271 for almost every b. The first half of the following theorem is due to Kesten, Spitzer, and Whitman (quoted in Spitzer [265]). The second half is stated by Kac [141] without proof. The proof we give is based in part on that of Kesten, Spitzer, and Whitman, and in part on unpublished remarks of Varadhan. Theorem 22.6
With probability one, (22.13a)
for fixed 6 and (22.13b) for fixed t . Proof Without loss, we can take 6 = 1 in (22.13a) and t = 1 in (22.13b), since the other results then follow by scaling. Now let A , = I W,(l)l/S and B, = I W,(6-2)1/6-2. For eachjxed 6, A , and B, have the same probability distribution, so that L pconvergence theorems like E(A,) + 27c for one imply the analogous result for the other. However, the joint distributions as 6 varies are not the same: For example, 2A2, 2 A , with probability one while B2,I 4Bdwith probability one but these inequalities (which rely on the fact that I W,(t)Iis increasing in t and 6) are not true almost everywhere if A and Bare interchanged. Thus (22.13a) and (22.13b) are not equivalent statements. We will prove below that for 6 < 1, E([B,
-
2711’) I C61’2
(22.14)
Assuming (22.14), let us prove (22.13). Fix p E (0, 1). Then, by the standard Borel-Cantelli argument we have used often before (see the proof of the strong law of large numbers), lim B,, n-m
=
271
240
VII.
Other Topics
with probability one. Moreover, since W l ( T )is monotone in T , if p"" 6 Ip", then 1)' B p n IBa I11- 2Bpn+ 1
I
so that almost everywhere 271p2 I
B6 5
lim B6 < 2np-'
Since p is arbitrary in (0, l), (22.13a) follows. To prove (22.13b), we first note that since (22.14) involves a single 6, the same inequality holds for A6 so that, as before, lim A,,
=
271
n-rm
with probability one. Using now the inequality pApn+l
I Ad I p-'A,n
which comes from the monotonicity of W6(1) in 6, we obtain (22.13b) as above. This leaves the proof of (22.14). Let rT." = I Wl((n - l)T, nT)I. For fixed T , the rT,"are identically distributed and they are independent by the basic property of Brownian motion of starting afresh. By (22.10):
I E(I*,")/T - 2711 Ic1T - 1"
(22.15a)
for all T 2 1. Clearly (22.15b) j= 1
which implies, taking T
=
1 and changing t to T, that
E(6.n) I (T
+ 1)2E(l W1(1>1')
(22.1%)
In this last inequality, E(I Wl(1)l') < co, for if maxo,,,l Ib(s)l I m + 1, then Wl(l) is less than or equal to the volume of a sphere of radius in + 2, so m= 1
Now use (22.15b) again and the independence to note that
so using (22.15a, c) we see that (taking T
= t1/2)
E([I W'(t)I - 2nt]') I
for t > 1. This is just a restatement of (22.14).
I
22.
241
Wiener Sausage
In (22.7) it is not arbitrary Wiener paths that enter but paths with o(0) therefore need the following corollary.
= o(t).We
Corollary 22.7 Brownian bridge
For almost every path for the (three-dimensional)
lim F11 W6(l)( = 271 610
Proof By considerations similar to those in Theorem 5.5, the Brownian bridge cc restricted to [0, t ] is absolutely continuous with respect to Brownian motion so long as t < 1. Thus for any t < 1 and almost every Brownian bridge path :
lim 6 - l I W6(t)I = 271r 610
Since I W,( 1) I 2 I W,(t)I we see that ~ & ' l W a ( l ) 12 2n 610
Since cc(1 - t ) G a(t) we see that I W6($)I = I W& 1)l so that 6-' I W,
we have that
We now have the tools for proving (22.8). To prove (22.9) we must consider a somewhat more complicated problem. Namely, fix x and y (perhaps equal) and let b, b' be independent Brownian motions with sausages M! W'.Then we need to know that 6-'I(x
+ W,(l)) u (y + Wi(l))l
-+
471
i.e., that the overlap of the two sausages is negligible relative to the total size as 6 -+ 0. A moment's reflection will convince the reader that this should be true and that stopping times are the right tools for its proof. Theorem 22.8 Let W and W' be the Wiener sausages for two independent Brownian motions and let x, y be fixed. Then, with probability one 6-'I(x
+ Wa(t)) u (y + Wi(t))I
+
4nt
VII.
242
Other Topics
as 6 LO. The same result is true for the (three-dimensional) Brownian bridge and t I 1. Proof Without loss take y = 0. Consider first the case of Brownian motion. Clearly, we need to show that
6-'I(x
+ W,(t))n W,(t)(-,0
(22.16)
Fix the b path and E and define stopping times for the b-path, tl, o1, t2,. . . , inductively by t1 =
inf{s I b(s)
+ x E W2,(t))
I b(s) + x $ W;,(t)) ti+ = inf{s 2 ciI b(s) + x E W2,(t)} oi = inf{s 2
ti
with the proviso that the stopping times are t if no such s exists. Clearly for 6I E, x + W, and Wl intersect only for h-times belonging to the intervals (ti,ci),so that m
I(x + W(t))n We(t)I I By the Dynkin-Hunt theorem, each b(s so with probability one
CI
W s ( t i , 0i)I
i= 1
+ t i )- b(ti) is a Brownian motion,
m
lim 6-' 610
I(X
+ ~ , ( t )n) ~ : ( t )Il 271C (ai - ti) I 271){s I x + b(s) E W;,(t);s I t } I i= 1
5
2719,
Thus, (22.16) follows from lim gE= 0 E l 0
with probability one. But g, decreases as I: does so, by the monotone convergence theorem, it suffices that lim E(g,) = 0 (22.17) E l
0
This we can compute by Fubini's theorem: E(gJ
=
[Db f d s {(s, b) 1 x 0
=
[ j s E(x + b(s)
+ b(s)
E
%,(t))
E W;,(t))
For each s # 0, the probability distribution of x + b(s) is absolutely continuous so that E(x + b(s) E W;,(t))10 as E .-, 0 since I W3,1 -, 0 by Theorem
22.
243
Wiener Sausage
22.6. Thus (22.17) follows by the monotone convergence theorem. This proves the theorem in the Brownian motion case. For the Brownian bridge, we note first that for t < 1, the result follows from the Brownian motion case as in Corollary 22.7. Thus, for t = 1
+ W,) u WLl 2 sup(4nt) = 4n
n i!J 6-'I(x 610
1<1
On the other hand, by Corollary 22.7
lim 6-'1(x + w,) u
W ~I Ilim 6-'(1 W,I
+ I W,I)= 471 I
610
We now have the tools for proving (22.8) and (22.9). To complete the proof of (22.5) we need one more fact: Lemma 22.9 Suppose we are given a doubly indexed family of random variables Ej(n) and two sequences a j and bj of numbers so that
in probability as n
-+ co
for each t > 0. Then, as n -+ co Ej(n) + bj
in probability. Proof It sufficesto prove that El(n) + b , in probability, for once we have e - l E J c will n ) converge in probability to CjtZe-IbJso by induction that, we will obtain E j ( n ) + b j . Let us show that the probability that E , ( n ) 2 h , + 6 goes to zero for each 6 > 0. The proof of the other inequality is similar. Suppose that E l ( n ) 2-bl + 6. Pick j o so that ajo+l2 bl + 6. Then for
xjz2
t 2 l
1
e-lEJ(n)
j
=
1 j= 1
e-lEJ'") f
m 1
,-lEJ(n)
jo+ 1
244
VII.
Other Topics
Thus, for suitable t large and some E > 0,
C ,-tE,(n)
5
-
e-tbl
Thus condition (v) implies that E ( E l ( n ) 2 b ,
E
+ 8) -+
0 as n -+ co. I
Proof of Theorem 22.3 Fix b. Then
since the center xi must lie outside W,,(t). Since the xi's are independent, we have by (22.7) that (Tr(e-'H(";n)))= ( 2 7 ~ t ) - J~d 3/ Y~ N 1 - I Q 1 - l y
+ b(s)
EQ
l(Kn(t)+ y) n QII";
I b(t) = 0)
+
+
For almost every path with b(t) = 0 and y b(s) E R, y K,(t) c R for r, small and thus we can replace I(y w,(t))n RI by I W,.(t)Ifor n large. For almost every path
+
[I - IRI-'IW,~(~)I]~ -+ e-2ntalnl-'
by the compound interest formula and
on account of Corollary 22.7. Thus, by the dominated convergence theorem and Theorem 22.4, lim(Tr(e-'H(x;n)))= ( 2 7 ~ t ) - ~e /-2ntalRI-' * n-t
Jd3y E ( y
+ b(s) E R I b(t) = 0)
00
- e - 2ntalRI
I
Tr(e-tH(0))
Thus, we have (22.8). The proof of (22.9) is similar, except we need two independent Brownian motions to write out Tr(.)' and then doing the ( -)average first,we need information on the volume of two independent sausages. This information is provided by Theorem 22.8 giving (22.9). By the Remark preceding Theorem 22.3, (22.8) and (22.9) imply that Tr(e-'H(";n))converges in probability to Tr(e-"H'o)+2na1RI')) whence (22.5) follows from Lemma 22.9. [Choose aj = E,(O).] I
Remark Rauch and Taylor [210] apply operator theoretic methods to avoid some of the more involved probabilistic considerations above, especial-
23.
245
Statistical Mechanics
ly Theorem 22.8. For example, as in the proof of Theorem 22.3, we can use the Feynman-Kac formula and Theorem 22.6 to see that (22.18) where A
=
H(0)
+ 27m I IR I
-
Since (22.18) holds for t and 2t we have that
lim (II(e-*"(*;n)- e-")fil')
=
0
fl'W
This "strong convergence in probability," which is neither stronger nor weaker than (22.5), is often physically useful. Similarly, we are able to prove (22.5) directly from (22.8) and (22.18) without Lemma 22.9 or (22.9): For (22.18) and (22.8) for t and 2t imply that (Tr[(e-'f"";n) - e - ' A ) 2 ] ) = 0
(22.19)
Moreover, a theorem of Lidskii (see Kato [152, Section 11.6.53) implies that
1(CLfl(C) - Pfl(DN25 Tr((C - D)') for any bounded positive operators C and D with eigenvalues pl(C) 2 p 2 ( C ) 2 . . . so that (22.19) yields (22.5).
23. The Statistical Mechanics of Charged Particles with Positive Definite Interactions
Most of the applications described so far have involved the nonrelativistic quantum mechanics of at most a few particles and Gaussian measures on continuous functions (Db, Dq, and Dcr) or their perturbations. In the next two sections we want to briefly indicate some applications to statistical mechanics and quantum field theory. These applications will require us to utilize some more complicated Gaussian processes than those considered thus far including some which cannot be naturally realized on the space of continuous functions. In this section, we consider some statistical mechanical models. There has been considerable application of the Wiener process itself to study quantum statistical mechanics. We will not discuss this here but we refer the reader to Edwards-Lenard [73] and Siegert [239] for two of the earliest discussions, and to Ginibre [99, 1001 and Brydges-Federbush [27] for some deep applications to the construction of correlation functions at low 0 and/or z. Here we want to discuss a Gaussian realization of the basic objects of the
246
VII.
Other Topics
classical statistical mechanics of a special but very interesting class of interactions. This realization has been described by various authors, e.g., [239, 1, 931. By essentially passing first to a Wiener integral realization and then to this Gaussian realization inside the Wiener integrals, one can extend this formalism and the results below to quantum statistical mechanics with Boltzmann statistics (i.e., no symmetrization or antisymmetrization) (see [239,94]). After presenting this Gaussian realization we will describe some beautiful results of Frohlich and Park [94] controlling the thermodynamic limit for this special class of interactions at all P, z. We want to consider v-dimensional particles coming in two charges k 1 interacting with a bounded potential V(x - y) obeying ZiZjV(Xi - Xj) 2 0
(23.1)
i, j = 1
for all zl, . . . , z, E C and xl, . . . , x, E R"; that is, two particles with equal charges at points x and y have interaction V(x - y) and with opposite charges - V(x - y). The grand canonical partition function in volume A c R" is defined by
j = 1,
...,N
where (23.3) The pressure in region A is given by (23.4) and the correlation functions (essentially giving the probability density of finding particles of charge E~ at points xi) are given by
j = 1 , ....N
(23.5) The results below depend critically on the somewhat artificial restriction to a single fugacity z for both charges. For a discussion of the physics of the grand canonical formalism see Huang [129]. Our goal in this section is to prove the following.
23.
Statistical Mechanics
Theorem 23.2 (Frohlich-Park [94]) have that
247
Under the hypothesis (23.1) we
(a) lim,,l+mP,(z, p) = P(z, p) exists for all positive z and /3 where the liml,,-m is intended in the sense of choosing A to run through hypercubes and taking all the sides to infinity. (b) lim,,,,, [$)(xi, c i ; z , p) = /)(“)(xi, e i ; z, p) exists for all positive z and fi, and g i = 1, xi E Iw” where the lim,,,,, is intended in the sense of the net of all regions ordered by inclusion. The p(”)are translation invariant [i.e., p(”’(xi a) = p(”’(xi)(same a)], are charge reversal invariant (i.e., under c j .+ - c j , all j ) , have any symmetry of V [i.e., if V ( T x )= V ( x ) for some linear T ,then ~‘“’(Tx,)= p(”’(xi)] and are monotone increasing in z.
+
Remarks 1. Since V is supposed bounded, this result does not include the Coulomb potential 