Penalising Brownian paths

Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris 1969 Bernard Roynet...

Author: Roynette B. | Yor M.

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Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris

1969

Bernard Roynette · Marc Yor

Penalising Brownian Paths

ABC

Bernard Roynette

Marc Yor

Université Nancy I Institut Elie Cartan Faculté des Sciences Département de Mathématiques B.P. 239 54506 Vandoeuvre-les-Nancy CX France [email protected]

Université Paris VI Laboratoire de Probabilités 175 rue de Chevaleret 75013 Paris France

ISBN: 978-3-540-89698-2 e-ISBN: 978-3-540-89699-9 DOI: 10.1007/978-3-540-89699-9 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008942371 Mathematics Subject Classification (2000): 60J65, 60F99, 60J25, 60G44, 60G30, 60J55 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com

Contents

0 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 The Penalisation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 Asymptotic Study of the Normalisation Factor . . . . . . . . . . . . . . . . . 0.3 From the Family (ν x , x ∈ R) to a Penalisation Theorem . . . . . . . . . 0.4 Penalisation and Conditioning by an Event of Probability 0 . . . . . . 0.5 Penalisation and Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . 0.6 Penalisation as a Machine to Construct Martingales . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 8 13 23 27 33

1 1.1 1.2 1.3

35 35 36 40

Some Penalisations of the Wiener Measure . . . . . . . . . . . . . . . Introduction : A Rough Idea about Penalisation . . . . . . . . . . . . . . . . Some Meta-Theorems Leading to Penalised Probabilities . . . . . . . . Case I : Γt = St := sup Xs , or Γt = L0t (X) . . . . . . . . . . . . . . . . . . . . . . t s≤t 1.4 Case II : Γt = ds q(Xs ), with (Xs ) : BM(Rd ), d = 1, 2 . . . . . . . . 0

45

(δ)

1.5 Case III : Γt = L0t (Rt ), with R(δ) := BES(δ), 0 < δ < 2 . . . . . . . . 48 (),() 1.6 Case IV : Γt = Σt where Σt := sup (u − gu ), u≤gt

or Σt := sup(u − gu ), or Σt := sup (u − gu ) . . . . . . . . . . . . . . . . . . . . u≤t u≤dt 2 1.7 Case V : Γt = sup θs (d = 2) or Γt = 1TC >t exp γ2 Ht ts≤t ds where Ht = ......................................... 2 R 0 s 1.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

56 62 63

2 Feynman-Kac Penalisations for Brownian Motion . . . . . . . . . 67 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.2 On the Solutions of Sturm-Liouville Equations and Associated Brownian Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

v

vi

Contents

2.3 A Direct Proof of Point 1) of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . 2.4 Absolute Continuity Relationships between the Probabilities (q) Wx (q ∈ I) and Deﬁnition of the σ-ﬁnite Measures Wx (x ∈ R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 An Extension of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2 . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Penalisations of a Bessel Process with Dimension d (0 < d < 2) by a Function of the Ranked Lengths of its Excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Some Prerequisites about Bessel Processes of Dimension d = 2(1 − α), 0 < α < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Penalisations of a BES(d) Process, Involving One Ranked Length of its Excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) 3.3 Penalisation by (Vgt ≤ x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) (1) 3.4 Penalisation by (Vt ≤ x) and (Vdt ≤ x) . . . . . . . . . . . . . . . . . . . . . .

75

81 103 113 130

3

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1 . . . . . . . . . . . . . . . . . 3.6 Weak Convergence of the Penalised Laws Q(n,x) , as n → ∞; a Commutative Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (n) (n) 3.7 Penalisations by (Vt ≤ x) and (Vdt ≤ x), for n ≥ 2 . . . . . . . . . . . Penalisations of a BES(d) Process, Involving Several Ranked Lengths of its Excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions of a BES(d) Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) (n) 3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn ) . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A General Principle and Some Questions about Penalisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . t) 4.2 Asymptotic Studies of (Mth , t ≥ 0), Exh(Γ [h(Γt )] , t ≥ 0 t) and of Mth − Exh(Γ [h(Γt )] , t ≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t) 4.3 Asymptotic study of Ex 1Λ Exh(Γ (h(Γt )) for Λ ∈ F∞ . . . . . . . . . . . . . . . 4.4 Convergence in Law, as t → ∞ ofthe Family of Processes (t) X tu (Xu ; 0 ≤ u ≤ 1):= √t ; 0 ≤ u ≤ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Some Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131 131 131 135 148 148 162 165 181 183 185 185 205 222

4

225 225 232 247 250 259 260

Contents

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Some Commutative Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Index of Main Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D Classiﬁcation of Rigorous Results and Meta-theorems in this Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

261 262 266 269 274

Preface

Since roughly 2002, we have been interested in establishing a number of penalisation results for Brownian motion. • Let us ﬁrst explain the term “penalisation”: by putting an adequate weight Γt on the Wiener measure W , constructed on (Ω = C(R+ , R), (Xt , Ft )t≥0 ), where Xs (ω) = ω(s), ω ∈ Ω, s ≥ 0, denotes the canonical process, and (Ft = σ{Xs , s ≤ t}, t ≥ 0) its natural ﬁltration, we wish to show that : (P1) WtΓ := Γt •W when restricted to Fs , for any ﬁnite s, converges as t → ∞, i.e. : Γ ∀s > 0 , ∀Fs ∈ Fs , WtΓ (Fs ) −→ W∞ (Fs )

(P2)

t→∞

Γ Assuming that this holds, it is not diﬃcult to show that W∞ induces a probability on (Ω, F∞ ), such that : Γ ∀s > 0 , W∞|F = Ms •W|Fs s

(P3)

for a certain martingale (Ms ) with respect to (W, (Fs )). We then say that we have penalised W with the weight process (Γt , t ≥ 0); for example, taking Γt = ϕ(sup Xu )/W (ϕ(sup Xu )), for bounded ϕ’s with comu≤t

u≤t

Γ under which sup Xu is ﬁnite. pact support, we obtain the existence of W∞ u≥0

Γ has a radically diﬀerent behavior than unThus, the process (Xt ), under W∞ der W , namely the supremum process has been penalised so that it becomes Γ . ﬁnite valued under the new probability measure W∞ We have been looking systematically for such alterations of the Wiener measure, by taking weight processes involving the supremum, or the local time at 0, or..., or in dimension 2, the winding process of planar Brownian motion...

ix

x

Preface

Besides these examples, the most natural penalisations of W , consist to take : t t q(Xs )ds /EW exp − q(Xs )ds Γt = exp − 0

(P4)

0

for some integrable function q : R → R+ . We call these Feynman-Kac penalisations. • We now explain about the organisation of these Lecture notes : - Chapter 0, is a detailed Introduction to the whole volume, including some discussion comparing penalisation and other topics, e.g. : the small ball problem... - Chapter 1, a version of which has been published in Japanese Journal of Mathematics (2006) develops a number of cases of penalisations, together with the presentation of the relevant prerequisites for Brownian motion, e.g. : the deﬁnition of Brownian local times, and so on... Each of the cases presented there has been the subject of a fully developed paper, a series of which have appeared in Studia Math. Hung. (see [RVY, i], i = I, II, . . . VII and [RY, j], j = VIII, IX, in the Bibliography, p. 34). - In Chapter 2, we take up the study of Feynman-Kac penalisations, but there, we look for a global approach, as follows : we show the existence of a σ-ﬁnite measure Λ on C(R, R+ ) such that, for conveniently integrable functionals Γ(Lyt ; y ∈ R), we have : √ t W (Γ(Lyt ; y ∈ R)) −→ Λ(Γ) . t→∞

Our aim then is to show that, associated with these functionals, there is a penalisation result, with limiting martingales (Ms ) as in formula (P3), expressible in terms of the measure Λ. - In Chapter 3, we consider another general framework of penalisations of W with, for example, functions of the sequence : (1)

(2)

(n)

(Vt , Vt , ..., Vt

, ...)

of lengths of Brownian excursions, away from 0, up to time t, ranked in decreasing order, as in the studies of Pitman-Yor related to the PoissonDirichlet distributions (see [PY5 ], references at the end Chapter 3). (1) Again, a preliminary study, involving only Vt has been made in [RVY,VII] (see the end of Chapter 1), whereas here we show the existence of a σ-ﬁnite measure Π much as in Chapter 2, from which we shall be able to describe the general penalisations obtained from these ranked lengths of excursions.

Preface

xi

- In Chapter 4, we question in broad terms the validity of our discussion in the following sense : in our asymptotic studies, we always restricted ourselves to ﬁxed ﬁnite intervals [0, s], i.e. : we looked for the existence of the limit of WtΓ (Fs ) , as t → ∞ , for Fs ∈ Fs . Γ and of Γt •W|Ft , as In this Chapter 4, we ask about the closeness of W∞|F t t → ∞, and we are able to show that they are far apart, in that the total variation of the diﬀerence converges to a positive constant. A number of related questions are also examined.

************************** As a temporary conclusion of our penalisation studies, let us look at what has or has not been achieved : • we have shown the existence of a penalised limiting measure for WtΓ , as t → ∞, for a large class of interesting and “natural” weight processes. On the other hand, we have not been able to ﬁnd general criterion on such processes that would ensure the existence of this limit; • even if one is not a priori interested in penalisations per se, these studies are a source of “creation” of Brownian martingales, which may be of interest by themselves; see in particular the discussion in Chapter 0. • more complicated weight processes have been considered in the probabilistic - statistical mechanics oriented - literature, involving instead of the simple integrals t q(Xs )ds 0

e.g :

double integrals : [0,t]2

q(Xu − Xs )du ds

(P5)

In fact, Symanzyk’s program - which is closely related to Edwards’ model consists in looking for the existence of limiting results as n → ∞, for ﬁxed t, for the normalized weights : exp −

[0,t]2

qn (Xu − Xs )du ds

where : qn (x) = nd q(nx) (x ∈ Rd ), with q integrable on Rd , and W the law of d-dimensional Brownian motion. We hope that, in some near future, some of the techniques we have developed throughout this monograph may be of some use for these more complicated penalisations. **************************

xii

Preface

A few notable features of this volume i) Certain σ-ﬁnite measures, with inﬁnite total mass, play an important role in the description of our results. These measures are always denoted in bold character, in order to “separate” them from our more common probability measures, indicated in plain character; examples : Wx , Λx , (t) in contrast to Λ± , Px ,... ii) We have made sure that each of the ﬁve chapters may be read independently from the others; this is quite natural, as each topic can be developed from the use of adequate tools. Thus, the reader may browse through the volume easily, and be attracted ﬁrst by Chapter 4 say, then come back to Chapter 3 and so on, ..., without diﬃculty. Consequently, each Chapter ends with its own bibliography. However, throughout the volume, all references have been homogenized, i.e : reference [R] in Chapter 1 is also reference [R] in Chapter 3, if it plays some role there... We thank Jim Pitman for asking us to complete our references, with respect both to “classical works” and to our more recent understanding of penalisations, e.g. : in Chapter 2, we have sketched some results obtained between April and June 2007 jointly with J. Najnudel. iii) A priori, penalisation studies might be developed for a fairly large class of stochastic processes; however, in this volume, our processes of reference are either Brownian motion, or Bessel processes of dimension d ∈ (0, 2). The reader shall ﬁnd the main properties which are shared/enjoyed by these processes and are of interest in our penalisation studies discussed in Chapter 1 in the form of speciﬁc Items . Acknowledgement : We feel very much indebted to three very special persons, without whom this monograph could not have been produced : • Pierre Vallois, who undertook a large part of these penalisation studies with us, as our “Roman Number” series clearly shows; • Joseph Najnudel, through his meticulous analysis which allowed him in his thesis (June 2007 - Paris VI) to transform many of our “metatheorems” into rigorous statements; deep thanks also to Joseph for his very careful proof reading, his wonderful lectures during the Warwick Crash Course (October 2005) and Torgnon Summer School (July 2006); let us not forget either a number of memorable piano, and even organ(!) performances; • Kathleen Qechar, for her very professional TeXpertise and relentless searches for technical solutions, always done with impressive calm and good humour! Many thanks again to the three of you!!

This Monograph is dedicated to Frank Knight (1933-2007).

His study of Taboo Processes in 1969 is a beautiful example of penalisation.

Chapter 0

Introduction

Abstract As an Introduction to this monograph, we present, in a Brownian and Bessel framework, the general problem of penalisation which will be discussed throughout this volume. We sketch a number of examples, the study of which constitutes the diﬀerent Chapters of the monograph. Finally, we make a list of the martingales which occur as Radon-Nikodym densities between the Wiener measure and the penalised limiting measures. Keywords Penalisation problem · Small ball problem · meta-theorem · Statistical mechanics

0.1 The Penalisation Problem

0.1.1 Let Ω = C [0, ∞[→ R denote the space of continuous functions from [0, ∞[ to R and (Xt , t ≥ 0) denote the coordinate process on this space: Xt (ω) = ω(t), ω ∈ Ω. We denote by Ω, (Xt , Ft )t≥0 , F∞ = ∨ Ft , Px (x ∈ R) , t≥0

a canonical process which will be, most of the time the canonical brownian motion or a canonical Bessel process. (Ft , t ≥ 0) stands for the natural ﬁltration of this process. 0.1.2 i) Let (Δt , t ≥ 0) be a process deﬁned on (Ω, F∞ ), not necessarily (Ft ) adapted, taking values in R+ and such that : 0 < Ex [Δt ] < ∞

for all t ≥ 0 and x ∈ R.

To penalise the probabilities Px (x ∈ R) by the process (Δt , t ≥ 0) consists in studying the limit, in a sense to be made precise, as t → ∞, of the (t) probabilities Px deﬁned by : Px(t) :=

Δt · Px Ex (Δt )

B. Roynette, M. Yor, Penalising Brownian Paths, Lecture Notes in Mathematics 1969, DOI 10.1007/978-3-540-89699-9 0, c Springer-Verlag Berlin Heidelberg 2009

(0.1) 1

2

0 Introduction (t)

Thus, the probabilities Px are the probabilities Px modiﬁed with the (t) “weight” Δt . We note that, in general, the family (Px , t ≥ 0) does not constitute a projective system of probabilities, as : Px(t) |Fs = Px(s) |Fs

(s ≤ t)

(0.2)

On the other hand, and this is one of the interesting aspects of penalisations, the projective property holds as t → ∞. ii) Very often - and this is what we shall do essentially in this Introduction and throughout this monograph, we work in the set up of “functional penalisations”, a term which we now explain. Let (Γt , t ≥ 0) denote a process, not necessarily (Ft ) adapted, taking values in a measurable space (E, E) and let h : E → R+ measurable such that :

0 < Ex h(Γt ) < ∞ for all t ≥ 0 and x ∈ R.

We then penalise with the process Δt = h(Γt ), t ≥ 0 i.e. we study the (t,h) deﬁned by : limit, as t → ∞, of the family of probabilities Px Px(t,h) :=

h(Γt )

· Px Ex h(Γt )

(0.3)

(t,h)

is an important element of the problem. In The dependency in h of Px fact, one of the reasons - but it is not the only one (see Section 0.2 below) for which we work in this set up is the following. We shall show below (see Examples 0.1 and 0.2 further on) that, in a great many situations, if the process (Γt , t ≥ 0) takes values in R and is increasing, then : P (t,h) −→ Q(h) t→∞

and the probability Q(h) , deﬁned on (Ω, F∞ ) satisﬁes : Q(h) (Γ∞ ∈ dy) = h(y)dy. Thus, we solve a “Skorokhod-type problem”, i.e. : the density of probability h being ﬁxed, we ﬁnd a probability Qh on (Ω, F∞ ) such that, under Q(h) , the r.v. Γ∞ admits h as density. 0.1.3 We shall now work systematically in the set up of penalisation by

(t,h) h(Γt ), t ≥ 0 . In order to study the limit, as t → ∞, of Px , the ﬁrst step consists in

studying the asymptotic behavior of the “normalisation factor” Ex h(Γt ) also called, in Statistical Mechanics - see Section 0.5 below partition function. Let us begin this study by showing, via the examples presented below, some links which exist between penalisation and the notion of “small balls” in probability theory.

0.2 Asymptotic Study of the Normalisation Factor

3

We ﬁrst recall what is the problem of small balls. Let (Yt , t ∈ I) denote a process with continuous paths, for simplicity. Let us endow the space C(I → R) with a norm, denoted by || · || (e.g. : the uniform norm, or a H¨ older norm). Let ψ ∈ C(I → R) and let : B(ψ, r) = {ϕ ∈ C(I → R) ; ||ψ − ϕ|| < r} denote the ball with center ψ and radius r. The study of a problem of small

balls consists in ﬁnding, as ε → 0, the asymptotic study of P Y• ∈ B(ψ, ε) , where we denote by Y• the generic continuous function : I → R, t → Yt . This problem has been solved classically for many processes (Yt , t ∈ I) (see [SZ]).

0.2 Asymptotic Study of the Normalisation Factor In this Section, we shall describe some results about the asymptotic behavior of the normalisation factor ; these results will be established in the following Chapters of this Monograph. Example 0.1. (Xt , t ≥ 0) denotes here the canonical Brownian motion and : (0.4) (Γt , t ≥ 0) = (St := sup Xs , t ≥ 0) s≤t

Thus (St , t ≥ 0) is the one-sided supremum process of Brownian motion. For every h : R → R+ , Borel and integrable, one ﬁnds : ∞

πt Ex h(St ) −→ h(y)dy (0.5) t→∞ x 2 Indeed : ∞ ∞ 2

2 2 − y2t Ex h(St ) = E0 h(x + St ) = h(x + y) e dy ∼ h(y)dy t→∞ πt 0 πt x (0.6) from the dominated convergence Theorem. Thus, we have obtained the existence, for every x ∈ R, of a measure ν x carried by R, namely : ν x (dy) = 1[x,∞[ (y)dy such that, for every h ≥ 0, belonging to L1 (dy) :

πt Ex h(St ) −→ h(y) ν x (dy) t→∞ R 2

(0.7)

(0.8)

We shall see, throughout all our examples, that the existence of such a family of measures (positive and σ-ﬁnite) ν x is a general phenomenon (see [RY,IX]).

4

0 Introduction

Now, if we choose for h the function : h(z) := 1]−∞,y] (z) the quantity which we just studied equals, for x = 0 :

y E0 h(St ) = P0 [St ≤ y] = P0 S1 ≤ √ t

by scaling

(0.9)

Thus, it is the probability, for the r.v. S1 , to belong to the small ball centered y at the origin, and with radius √ · t Example 0.2. (Xt , t ≥ 0) still denotes the canonical Brownian motion and : (Γt , t ≥ 0) = (Lt , t ≥ 0)

(0.10)

where (Lt , t ≥ 0) is the local time at level 0 of (Xt , t ≥ 0). We have (see Chap. 2), for every h ∈ L1 (dy) :

πt Ex h(Lt ) −→ t→∞ 2

R+

h(y)ν x (dy)

(0.11)

with : ν x (dy) := |x|δ0 (dy) + 1[0,∞[ (y)dy.

(0.12)

Taking now h(z) := 1[0,y] (z), with y > 0 ﬁxed, the formula (0.11) becomes, with x = 0 : 2 y P0 [Lt ≤ y] = P0 L1 ≤ √ ∼ y (0.13) t→∞ πt t Thus, this formula (0.13) gives the asymptotic behavior of the probability, for the r.v. L1 , to belong to the small ball centered at the origin, and with y radius √ · t Example 0.3. Here again, (Xt , t ≥ 0) denotes the canonical Brownian motion. Let a and b two reals, with a < b. Let (Dta,b , t ≥ 0) denote the process of downcrossings on the interval [a, b], before time t. More precisely, let : σ1 := inf{t ≥ 0 ; Xt > b} ; σ2 := inf{t ≥ σ1 ; Xt < a} σ2n+1 := inf{t ≥ σ2n ; Xt > b} ; σ2n+2 := inf{t ≥ σ2n+1 ; Xt < a} (0.14) Then : Dta,b :=

n≥1

1(σ2n ≤t)

(0.15)

0.2 Asymptotic Study of the Normalisation Factor

Let h : N → R+ such that :

5

h(n) < ∞. It is shown in [RVY,II] that :

n≥0

1 |x − b| πt a,b Ex h(Dt ) −→ 2(b − a) + h(n) + h(0) t→∞ 2 2 2(b − a) n≥1

(0.16)

In other terms :

πt Ex h(Dta,b ) −→ t→∞ 2

with : ν x (dn) := 2(b − a)

δp (dn) +

p≥1

N

h(n)ν x (dn)

1 |x − b| + δ0 (dn) 2 b−a

(0.17)

(0.18)

Here again, formula (0.17), with : h(n) = 1[[0,··· ,p]] (n) (p ∈ N) provides us with an equivalent for the probability of a small ball since : a−x b−x √ , √ p a,b a−x,b−x t t ≤ p] ∼ P0 D1 ≤ Px [Dt ≤ p] = P0 [Dt by scaling t→∞ t

2 ν x [[0, · · · , p]] (0.19) ∼ t→∞ πt Example 0.4. Here again, (Xt , t ≥ 0) denotes the canonical Brownian motion. Let q denote a Radon measure on R, which is ≥ 0 (and diﬀers from 0) such that :

1 + |x| q(dx) < ∞ (0.20) 0< R

and let : (q)

At :=

R

Lyt q(dy)

(0.21)

where (Lyt , t ≥ 0, y ∈ R) denotes the jointly continuous family of the local times of Brownian motion (Xt , t ≥ 0). When q admits a density with respect to Lebesgue measure - and we still denote this density by q - the occupation formula yields : t

(q)

At =

q(Xs )ds.

(0.22)

0

We show, in Chapter 2 of this Monograph (see also [RY,IX]) the existence, (q) for every x ∈ R, of a measure ν x carried by R+ , ≥ 0 and σ-ﬁnite, such that : for every ϕ : R+ → R+ , with sub-exponential growth at inﬁnity : √ (q) t Ex ϕ(At ) −→ ϕ(y)ν (q) (0.23) x (dy) t→∞

R+

6

0 Introduction (q)

In Chapter 2 of this Monograph, many examples of measures ν x are explicitly computed. In particular, the above example 0.2 corresponds to the choice of q, taken to be the Dirac measure in 0. There again, taking this time q := 1]−∞,0] and h := 1[0,y] (y > 0), we get, y y 1 (q) (q) (q) ν 0 (dz), (0.24) P0 [At ≤ y] = P0 A1 < ∼√ t t 0 (q)

i.e. an equivalent of the probability of a small ball for the r.v. A1 . (see [RY,IX], for a detailed study of this example). Example 0.5. Here again, (Xt , t ≥ 0) denotes the canonical Brownian motion. Let E = C(R → R+ ) and (Γt , t ≥ 0) the process, taking values in E, and deﬁned by : (0.25) Γt = {Lyt ; y ∈ R} where, again, (Lyt , t ≥ 0, y ∈ R) denotes the jointly continuous family of the local times of (Xt , t ≥ 0). In (0.25), the notation (Lyt , y ∈ R) denotes the element of E deﬁned by : y → Lyt

(y ∈ R)

(0.26)

There again, we show (see Chap. 2) the existence, for every x ∈ R of a measure Λx carried by E = C(R → R+ ), ≥ 0 and σ-ﬁnite, such that : - for every h : E → R+ satisfying a growth condition at inﬁnity (see Chapter 2 for a precise formulation) : √

t Ex h(L•t ) −→ h(l) Λx (dl) (0.27) t→∞

E

Here again, we note that formula (0.27) provides the asymptotic probability of a quantity which resembles a small ball since, with x = 0, h(l) = 1{l ; l(y)≤g(y), ∀y∈R} where g : R → R+ is a given function ; indeed, by scaling :

P0 Lyt

y √

1 g(y) t ≤ g(y) ; ∀y ∈ R = P0 L1 ≤ √ ∀y ∈ R ∼ √ 1l≤g Λx (dl). t→∞ t t E (0.28)

We also remark that the results of Example 0.4 may be obtained as a particular case of Example 0.5 by choosing for function h the function hq deﬁned by : ∞ hq (l) := ϕ(< q, l >) = ϕ l(y) q (dy) (0.29) −∞

(q)

and where the measure ν x is the image of the measure Λx by the application ∞ iq : l →< q, l >= l(y)q(dy). −∞

0.2 Asymptotic Study of the Normalisation Factor

7

Example 0.6. (Xt , t ≥ 0) now stands for the canonical Bessel process with dimension d, with d = 2(1 − α) and 0 < d < 2, or 0 < α < 1, starting from 0. Thus, this process is recurrent and 0 is a regular point for itself. Let, for every t ≥ 0 : gt := sup{s ≤ t ; Xs = 0} (0.30) (1)

(2)

(n)

g := (Vgt , Vgt , · · · , Vgt , · · · ) denote the sequence of lengths of excurand V t sions above 0, before gt , ranked by decreasing order. Let also :

gα , t ≥ 0) = (Vg(1) )α , (Vg(2) )α , · · · (Vg(n) )α , · · · (0.31) Γt := (V t t t t (Γt , t ≥ 0) is a process taking its values in E = S, with : S := σ = (σ1 , σ2 , · · · , σn , · · · ) ; σ1 ≥ σ2 · · · ≥ σn ≥ · · · ≥ 0

(0.32)

In Chapter 3, we prove the existence of a measure Π(= Π0 ) on the set S such that, for every h : S → R+ , with compact support in the ﬁrst coordinate on S, one has :

α) = lim tα E0 h(V h(σ) Π(dσ) (0.33) gt t→∞

S

The measure Π is ≥ 0, σ-ﬁnite, and is entirely described in Theorem 3.16, Section 3.8, Chap. 3. Thus, taking for h the function : h(σ1 , σ2 , · · · σn , · · · ) = 1(σ1 ≤x1 ,σ2 ≤x2 ,··· ,σn ≤xn ,··· ) where x = (x1 , x2 , · · · xn , · · · ) is a ﬁxed element of S, we get, by scaling, from (0.33) :

P0 (Vg(1) )α ≤ x1 , · · · , (Vg(n) )α ≤ xn , · · · t t

1 x1 xn α (n) α ) ≤ , · · · , (V ) ≤ · · · ∼ 1(σ1 ≤x1 ,σn ≤xn ,··· ) Π(dσ) = P0 [(Vg(1) g 1 1 t→∞ tα S tα tα

(0.34) and we notice that formula (0.34) resembles very much the asymptotic behavior of the probability of a small ball. We note that the measure Π does not depend on α and that one can also, in a similar manner to (0.33), obtain the existence of a measure Πx , for every x ≥ 0, carried by S, ≥ 0 and σ-ﬁnite, such that :

α ) −→ h(σ)Πx (dσ) tα Ex h(V (0.35) gt t→∞

S

As a summary, throughout all the preceding examples, we have seen that there exists : i) a function λ : R+ → R+ , such that : λ(t) −→ +∞ t→∞

1

(0.36)

(λ(t) = t 2 in the Examples from 0.1 to 0.5, λ(t) = tα in Example 0.6).

8

0 Introduction

ii) For every x, a measure ν x carried by E, positive and σ-ﬁnite, such that, for a large class of measurable functions h : E → R+ , one has :

h(y)ν x (dy) (0.37) lim λ(t) Ex h(Γt ) = t→∞

E

We note that if, in (0.37), we take the function h “too large”, e.g. : if h ≡ 1, then the two sides of this relation are both inﬁnite, and (0.37) is no longer of interest. Deﬁnition 0.7. Let (Γt , t ≥ 0) be a process taking values in (E, E) such that (0.36) and (0.37) are satisﬁed. We call : i) λ the normalisation function of the process (Γt , t ≥ 0) ; ii) (ν x , x ∈ R) its family of normalisation measures. We shall now show, heuristically, how the existence, for a process (Γt , t ≥ 0), of a normalisation function λ and of a family of normalisation measures (ν x )x∈R yields a penalisation theorem.

0.3 From the Family (νx, x∈R) to a Penalisation Theorem 0.3.1 Most of the penalisation theorems which we shall prove in this monograph are of the following kind. Meta-Theorem of penalisation. For every function h : E → R+ measurable and belonging to a large class - in general, the set of functions which (h) belong to L1 (ν x ) for every x - there exists a (Fs , Px ) martingale (Ms , s ≥ 0) (this martingale depends on x, but we omit this dependency in our notation) such that : h(Γt )

Fs −→ Ms(h) f or every s ≥ 0, Ex (0.38) t→∞ Ex h(Γt with this convergence taking place Px a.s. and in L1 (Px ). An immediate consequence of this meta-Theorem of penalisation is the following : Meta-Theorem of penalisation. (bis) i) Under the same hypothesis, for every s ≥ 0, and every Fs ∈ b(Fs ), the space of bounded Fs measurable r.v.’s : h(Γt ) = Ex [Fs · Ms(h) ] (0.39) lim Ex Fs

t→∞ Ex h(Γt )

0.3 From the Family (ν x , x ∈ R) to a Penalisation Theorem

9

ii) The formula : (h) Q(h) x (Fs ) := Ex [Fs Ms ]

induces a probability

(h) Qx

(0.40)

on the canonical space (Ω, F∞ ).

Deﬁnition 0.8. The canonical process (Xt , t ≥ 0), considered under the h probabilities (Qhx , x the Q -process, or the process obtained by

∈ R) is called penalisaton w.r.t. h(Γt ), t ≥ 0 0.3.2 How to prove the (meta)-Theorem of penalisation ? Although, to our knowledge, there does not exist a canonical procedure to prove the (meta)-theorem of penalisation, starting from λ and from (νx , x∈R), it is very often possible to show the a.s. convergence of h(Γt )

Fs towards Ms(h) in a very simple manner. E Ex h(Γt ) a) We adopt, momentarily, the following notation (see [Gal], [CP]). For every s, we denote by ωs the part of the trajectory ω before s, and by ω the part of the trajectory after s, i.e. : ω = θs (ω) : u → ω(u + s). We may write : t (ωs , s, ω ) Γt (ω) = Γ t (ωs , s, ω ) is Fs ⊗ F∞ measurable. The where the application (ωs , ω ) → Γ Markov property implies :

t (ωs , s, •) Ex h(Γt )Fs = EXs (ωs ) h Γ where, in this expression, ωs is frozen, and the expectation is taken with respect to •. Thus :

t (ωs , s, •) λ(t) EXs (ωs ) h(Γ h(Γt )

Fs ∼ Ex t→∞ Ex h(Γt ) h(y)ν x (dy) E

t (ωs , s, •) has an asympWe now assume that, as t → ∞, the quantity Ex h(Γ totic behavior which is also given by (0.37). Then, we obtain : Γ(ω ,s,•) ν Xs s (dy)h(y) h(Γt )

Fs −→ := Msh Ex (0.41) t→∞ Ex h(Γt ) ν x (dy)h(y)

b) In the particular case where the couple (Xt , Γt ), t ≥ 0 is Markovian, with transition semi-group (Θt , t ≥ 0), the LHS of (0.41) is even more explicit, as : h(Γt ) Θt−s (h) (Xs , Γs )

Ex (0.42) Fs = Θt (h) (x, Γ0 ) Ex h(Γt )

10

0 Introduction

with

Θt−s (h)(b, γ) = Eb,γ h(Γt−s )

(0.43)

But then, in all our examples, the asymptotic behavior of Θt (h)(b, γ), as t → ∞, may be expressed in terms of the measures (ν x , x ∈ R). We illustrate this with two examples. (q)

i) We come back to the above example (0.24), where (Γt , t ≥ 0) = (At , (q) t ≥ 0). Since (At , t ≥ 0) is an additive functional, it satisﬁes : (q)

(q)

At = A(q) s + At−s ◦ θs where θs denotes the usual time translation operator ; the relation (0.42) becomes : (q) (q) Eb h(a + At−s ) h(At ) Fs = (s < t) Ex

(q) (q) Ex h(At ) Ex h(At ) (q)

with : a = As and b = Xs . Thus, from (0.23) :

(q) h(At ) F ∼ Ex

s (q) t→∞ Ex h(At )

√ t

∞

h(A(q) s + y)ν Xs (dy) √ t−s

· h(y)ν x (dy) 0 ∞ h(A(q) s + y)ν Xs (dy) (h) 0 −→ Ms := ∞ t→∞ h(y)ν x (dy) ∞

0

(0.44)

0 (h)

Thus, the martingale (Ms , s ≥ 0) is expressed in an extremely simple manner as a function of h and of the family of measures (ν x , x ∈ R). The same arguments, which we now develop in the context of Example 0.5 lead to : h(L•s + l)ΛXs (dl) h(L•t ) C(R→R+ ) (h)

Fs −→ Ms = Ex t→∞ Ex h(L•t ) h(l)Λx (dl) C(R→R+ )

with now : h : C(R → R+ ) → R+

(0.45)

ii) We come back this time to Example 0.1 : Γt = St = sup Xs . Since : s≤t

St = Ss ∨

sup Xu = Ss ∨ (Xs + St−s )

s≤u≤t

(0.46)

0.3 From the Family (ν x , x ∈ R) to a Penalisation Theorem

11

where St−s = sup (Xs+v − Xs ) is independent from Ss and Xs , the relation v≤t−s

(0.42) becomes : Ex

E0 h σ ∨ (b + St−s ) h(St )

Fs =

Ex h(St ) Ex h(St )

with : σ = Ss and b = Xs ∼

t→∞

πt 2

∞

0

h Ss ∨ (Xs + y) dy π(t−s) 2

h(y)dy x

−→

t→∞

·

∞

∞

1

h(Ss )(Ss − Xs ) +

from (0.8)

∞

h(y)dy

(0.47)

Ss

h(y)dy x

Thus, here again, the martingale Ms(h) :=

∞

1 h(y)dy

h(Ss )(Ss − Xs ) +

∞

h(y)dy

(0.48)

Ss

x

is obtained very simply in terms of the measures (ν x , x ∈ R). 0.3.3 We note however that, in the preceding computations, we only obtained h(Γt )

Fs toward Ms(h) , but we did not obthe a.s. convergence of Ex Ex h(Γt ) tain the convergence in L1 . In general, this convergence is more diﬃcult to establish (than the a.s. convergence). However, there exists a simple criterion which we use a number of times, and which allows to obtain this convergence in L1 . Indeed, let us assume that : h(Γt )

Fs −→ Ms(h) Px a.s. i) Ex t→∞ Ex h(Γt ) (h) ii) Ex [Ms ] = 1 for every s ≥ 0

then, from Scheﬀ´e’s lemma see [M], T. 21, p. 37 , the convergence of h(Γt )

Fs towards Ms(h) takes place in L1 (Px ). Ex Ex h(Γt ) 0.3.4 A digression. In all the above examples 0.1 to 0.6,

the normalisation function λ has a “polynomial” behavior at inﬁnity λ(t) = tα , with 0 < α < 1). However, in some penalisation set-ups a little diﬀerent from those we have just described, the normalisation function may have an exponential behavior at inﬁnity. Here is an example, lifted from [RVY,II]. Let (t,μ) the family of probabilities deﬁned by : μ > 0, ﬁxed, and let Px

12

0 Introduction

Px(t,μ) :=

eμ(St −Xt ) h(St )

· Px Ex eμ(St −Xt ) h(St )

(0.49)

It is shown, in [RVY,II] that :

μ2 t Ex eμ(St −Xt ) h(St ) ∼ e 2 t→∞

R

ν (μ) x (dy)h(y)

(0.50)

(μ)

(see Lemma 3.10 in [RVY,II]), where (ν x , x ∈ R) is a family of ≥ 0 and σ-ﬁnite measures on R. In order to pass from (0.50) to the penalisation Theorem : eμ(St −Xt ) h(St )

Ex Fs −→ Ms(h,μ) Px a.s. and in L1 (Px ) (0.51) t→∞ Ex eμ(St −Xt ) h(St ) the procedure presented in point 0.3.2 b), ii) remains essentially valid. What is precisely being changed ? The fact that the normalisation function is here exponential - and not polynomial - leads, in the computation ending up in

2 (0.47), to the introduction of the new factor exp − μ2s . Thus, the limiting (h,μ) (h,μ) martingale (Ms , s ≥ 0) is such that Ms is a (deterministic) function (h) of Ss , Xs and s, whereas Ms , deﬁned in (0.48) is a (deterministic) function of Ss and Xs uniquely. We then obtain (see [RVY,II], Prop. 3.3, and Theorem 3.9) :

∞ sinh μ(Ss − Xs ) μ2 s (h,μ) μXs −μz +e = h(Ss ) h(z)e dz e− 2 (0.52) Ms μ Ss Formula (0.48) is then the particular case of (0.52), where we take μ = 0. 0.3.5 Let us now leave aside the above digression, and come back to the penalisation situation described in point 0.1.2 ii) i.e. to the study as t → ∞, (t,h) of the limit of probabilities (Px , t ≥ 0, x ∈ R) deﬁned by : Px(t,h) :=

h(Γt )

· Px Ex h(Γt )

(0.53)

As a temporary conclusion, the study of the penalisation of probabilities h(Γt )

consists in : (Px , x ∈ R) with the weight Ex h(Γt ) Finding the normalisation function λ, and the family (ν x , x ∈ R) of normalisation measures ; h(Γt )

Fs −→ Ms(h) , Px a.s. and in L1 , and comii) showing that Ex t→∞ Ex h(Γt ) (h) puting explicitly the martingale (Ms , s ≥ 0) ;

i)

0.4 Penalisation and Conditioning by an Event of Probability 0

13

iii) Studying the Q(h) canonical process Xt , t ≥ 0 ; Qhx (x ∈ R) where the (h) probability Qx is induced by : (h) Q(h) x (Fs ) = Ex [Fs Ms ]

(0.54)

(∀s ≥ 0, Fs ∈ b(Fs ), the space of bounded Fs measurable functions) (t,h)

We remark that, although the original set of the probabilities (Px , t ≥ 0) was not projective, this set became projective, after passing to the limit as (h) t → ∞, since, with Px∞ = Qx : Q(h) x |Fs = Mt · Px|Fs = Ms · Px|Fs

(s < t)

(0.55)

The aim of this Monograph is to develop the work indicated in point i), ii) and iii) above, in the set-up of many examples. Unfortunately, we have not been able to present completely uniﬁed proofs of our results. In other terms, the methods which we use vary from one example to another. It is precisely for this reason that we have attempted, in Chapter 4, to develop a set of the most generic results, i.e. : those which are “independent” from the examples to which we apply them. The study of penalisation has many relations with : - conditioning of a process with respect to a set of probability 0 ; - a number of considerations made in Statistical Mechanics and the notion of phase transition. We shall now illustrate these two aspects.

0.4 Penalisation and Conditioning by an Event of Probability 0 0.4.1 We begin with the observation that there does not exist, in general, any canonical procedure to condition a process by a negligible set. Let us take the example of the standard Brownian bridge, starting from 0, and ending with the value 0 at time 1. This Brownian bridge may be deﬁned in several ways :

i) For every F : C [0, 1] → R → R+ continuous and bounded, one deﬁnes :

E0 F (Xt ; 0 ≤ t ≤ 1) 1−ε<X1 <ε (0.56) E F (bt ; 0 ≤ t ≤ 1) = lim ε→0 P0 (−ε < X1 < ε)

ii) E F (bt ; 0 ≤ t ≤ 1) = E0 F (Xt − tX1 ; 0 ≤ t ≤ 1) (0.57)

14

0 Introduction

iii) (bt ; 0 ≤ t ≤ 1) is distributed as the solution of the SDE : ⎧ t Ys ⎨ ds 0≤t<1 Yt = Xt − 0 1−s ⎩ Y1 = 0

(0.58)

iv) Consider a disintegration with respect to X1 of the law of (Xt ; 0 ≤ t ≤ 1) : ∞

x2 1 E0 F (Xt ; 0 ≤ t ≤ 1) = E0 F (Xt ; 0 ≤ t ≤ 1)X1 = x) √ e− 2 dx 2π −∞ As the family of probabilities P0 (•|X1 = x) may be chosen to be weakly continuous with respect to x, the law of (bt , 0 ≤ t ≤ 1) may be deﬁned without ambiguity as being P0 (•X1 = 0). v) Let (Gt , t ≤ 1) be the ﬁltration (Ft , t ≤ 1) initially enlarged with the r.v. X1 , so that Gt = Ft ∨ σ(X1 ) (0 ≤ t ≤ 1). In that ﬁltration t , t ≤ 1), independent (Gt , t ≤ 1), there exists a Brownian motion (X from X1 , such that : t X1 − Xs ds (0.59) Xt = Xt + 1−s 0 It then suﬃces, in order to deﬁne the law of (bt ; 0 ≤ t ≤ 1) to choose the solution of (0.59) which corresponds to the “initial value” X1 = 0. 0.4.2 Conditioning by S∞ < ∞. i) Let (Xt , t ≥ 0), P ) denote the canonical Brownian motion, starting from 0, and St = sup Xs . We would like to condition this process by the negligible s≤t

event (S∞ < ∞). One way to proceed, which is due to F. Knight (see [K]) consists to show that, with a > 0 being ﬁxed, the conditional law, for every 0 < t1 < · · · < tn of (Xt1 , · · · , Xtn ) given (St < a) converges weakly, as t → ∞ ; the induced limiting law Q(a) will then be that of Brownian motion, conditioned by (S∞ < a). The process obtained in this way is the diﬀusion ∂ 1 1 ∂2 , that is : a “3-dimensional Bessel process − with generator 2 ∂x2 a − x ∂x below level a”. In the same paper, F. Knight gives a construction of Brownian ∗ ∗ motion “conditioned by (X∞ < a), where : X∞ := sup |Xs |, by considering s≤∞

Brownian motion conditioned by sup |Xt | < a , where τl := inf{t ≥ 0 ; Lt > t≤τl

l} denotes the right-continuous inverse of the local time at 0, and the letting l → +∞. ii) We now present another manner, which we think is also natural, to “condition Brownian motion by (S∞ < ∞)”. For this purpose, we look for a

0.4 Penalisation and Conditioning by an Event of Probability 0

15

probability Q on (Ω, F∞ ), which is “close to Wiener measure P ”, in the following sense : C) there exists a (Ft , P ) positive martingale (Mt , t ≥ 0), with M0 = 1 such that, for every t ≥ 0 : (0.60) Q|Ft = Mt P|Ft and we also demand that : Q[S∞ < ∞] = 1

(C, ∞) or, more precisely : (C, a)

for some a > 0, ﬁxed, Q[S∞ < a] = 1

(0.61)

or, in a diﬀerent manner again; h being a probability density on R+ : Q[S∞ ∈ dy] = h(y)dy

(C, h)

(0.62)

We note that a positive martingale (Mt , t ≥ 0) and the associated probability Q, obtained via relation (0.60) satisfy (C, a) as soon as : E[MTa ] = 0 or equivalently : MTa = 0, P a.s.

(0.63)

where Ta := inf{t ≥ 0 ; Xt = a}. Indeed, from Doob’s optional stopping Theorem, we obtain, if (0.63) is satisﬁed : Q[S∞ > a] = Q[Ta < ∞] = E[MTa ] = 0

(0.64)

In the same manner, a positive martingale (Mt , t ≥ 0) and the probability Q which is associated to it via (0.60) satisfy (C, h) as soon as, for every a ≥ 0, ∞ E[MTa ] = h(y)dy (0.65) a

since, from Doob’s optional stopping Theorem : ∞ h(y)dy ≡ Q[S∞ > a] = E[MTa ]

(0.66)

a

iii) Solving problems (C, a) and (C, h) (a) Given (0.64), it is clear that the martingale (Mt , t ≥ 0) which is deﬁned by : (a)

Mt

Xt Xt∧Ta = 1St ≤a 1 − = 1− a a

(0.67)

induces a probability Q(a) via : (a)

(a)

Q|Ft = Mt

· P|Ft

(0.68)

16

0 Introduction (a)

which satisﬁes (C, a), since MTa = 0. More generally, let μ denote a proba(μ)

bility measure on R+ and deﬁne the martingale (Mt , t ≥ 0) by : ∞ ∞ ∞ μ(da) (μ) (a) Mt μ(da) = μ(da) − Xt Mt = a 0 St St It is easy to see that, if we deﬁne h(μ) by the formula : ∞ μ(da) h(μ) (y) := (y > 0) a y

(0.69)

(0.70)

then h(μ) is a probability density on R+ and that : ∞ (μ) Mt = h(μ) (St )(St − Xt ) + h(μ) (y)dy.

(0.71)

St

Moreover, the probability Q(μ) on (Ω, F∞ ) induced by : (μ)

Q(μ) |Ft = Mt

· P|Ft

(0.72)

satisﬁes the condition (C, h) since, from (0.66) and (0.71) ∞ (μ) Q(μ) (S∞ > a) = E[MTa ] = h(μ) (y)dy. a

We have just associated to the probability measure μ the density of probability h(μ) via the relation (0.70), then we obtained the martingale (0.71). More generally, for any probability density h on R+ , let : ∞ (h) Mt := h(St )(St − Xt ) + h(y)dy (0.73) St (h)

(h)

Then (Mt , t ≥ 0) is a positive martingale such that : M0 = 1, and the (h) probability Qh which is induced by : Qh|Ft = Mt · P|Ft satisﬁes (C, h). iv) After reading the preceding paragraph, one may think that the martin(μ) gale (Mt , t ≥ 0) which we just considered has been obtained in a somewhat artiﬁcial manner one could refute this idea thanks to several arguments, in particular, as shown in Obloj [O2 ], the only martingales of the form f (St , Xt ) St k(x)dx, for some constant C, and are given by : C + k(St )(St − Xt ) + 0 some locally integrable function k . However, “the penalisation approach” which we now present, has a “mechanical character” which allows to obtain the martingale M (h) in a rather natural and automatic manner. We wish to “create” a probability Q such that Q(S∞ < ∞) = 1. For this purpose, we shall put a weight upon the Wiener measure P which favors the Brownian trajectories which do not “climb too

0.4 Penalisation and Conditioning by an Event of Probability 0

17

high”. Let h denote a probability density on R+ . Thus, h is “small” at inﬁnity. h(St ) shall favor the trajectories such that St Consequently, the weight

E h(St ) is small. This is the intuitive content of the following penalisation Theorem. Theorem 0.9. Let h : R → R+ be Borel, and integrable, 1) For every s ≥ 0 and Fs ∈ b(Fs ) : h(St )

= Ex [Fs Ms(h) ] := Qhx (Fs ) lim Ex Fs t→∞ Ex h(St )

(0.74)

(h)

where (Ms , s ≥ 0) is the positive martingale deﬁned by : ∞

1 (h) Ms := ∞ h(y)dy h(St )(St − Xt ) + h(y)dy St x

(0.75)

(h)

2) Under the probability Qx induced by (0.74) : h(y) 1(y≥x) dy Q(h) x [S∞ ∈ dy] = ∞ x h(z)dz

(0.76)

In Chapter 2 of this Monograph, or in [RVY,II], the reader will ﬁnd a more complete statement and a proof of Theorem 0.9. We saw, in 0.2.3 b) ii) how 1 (h) 1[0,a] (y) the martingale (Ms , s ≥ 0) appears. The case where h(y) = a (a) corresponds to the martingale (Ms , s ≥ 0) deﬁned by (0.67). We conclude in this direction that the penalisation study of the Wiener measure by a function of St yields some solutions to the question of conditioning with respect to the negligible set (S∞ < ∞). v) Let us emphasize that there is no canonical manner to deﬁne this conditioning since there are - at least - as many ways to proceed as there are

probability densities h, and we also note that the process (Xt , t ≥ 0), Q(h) depends in an important manner on h. In particular, the probability Q(h) solves “Skorokhod’s problem” as indicated in Section 0.1 ii) : Q(h) (S∞ ∈ dy) = h(y)dy Finally, we remark that the conditioning procedure which we have just described is “richer” than the procedure of F. Knight which we recalled above. Indeed, the probability Q(a) obtained by F. Knight is such that, under Q(a) , S∞ is uniformly distributed on [0, a] whereas, via our procedure, we are able to obtain for the law of S∞ any absolutely continuous probability on R+ . In 1 particular, if we take for h the function h(z) := 1[0,a] (z), then the law of a S∞ under Q(h) is the uniform distribution on [0, a], as in the procedure of F. Knight.

18

0 Introduction

0.4.3 Conditioning by {L∞ < ∞}. Here again, (Xt , t ≥ 0) denotes the canonical Brownian motion and (Lt , t ≥ 0) its local time process at level 0. L´evy’s Theorem (see Chap. 1, Item B )

(law)

(St , St − Xt ), t ≥ 0 = (Lt , |Xt |), t ≥ 0

(0.77)

allows to translate the results obtained in the preceding paragraph when St is replaced by Lt . Here is this translation : (h) Let h be a density of probability on R+ , and let (Mt , t ≥ 0) denote now the positive martingale deﬁned by : ∞ (h) Mt := h(Lt )|Xt | + h(y)dy (0.78) Lt (h)

Then, the probability Q0

on (Ω, F∞ ) which satisﬁes : (h)

Q0|F = Mth P0|Ft

(0.79)

t

is such that :

(h)

Q0 (L∞ ∈ dy) = h(y)dy

(0.80)

(h)

In particular, Q0 (L∞ < ∞) = 1. The statement of Theorem 0.9 now becomes : Theorem 0.10. Let h denote a probability density on R+ . Then : 1) For every s ≥ 0, and Fs ∈ b(Fs ) : h(Lt )

= E0 [Fs Ms(h) ] := Q(h) (Fs ) lim E0 Fs t→∞ E0 h(Lt )

(0.81)

(h)

where (Ms , s ≥ 0) is the positive martingale deﬁned in (0.78). (h) 2) Q0 (L∞ ∈ dy) = h(y)dy There again, there is no canonical manner to condition the process (Xt , t ≥ 0) by the negligible set {L∞ < ∞} since there are - at least - as many ways to proceed as there are probability densities on R+ . In fact, more generally, if h+ and h− are two ∞Borel functions from R+ to R+ , 1 + − h(y)dy = 1, we may deﬁne : such that, denoting h = (h + h ) and 2 0 ∞ + − h(y)dy (0.82) Mth ,h := h+ (Lt )Xt+ + h− (Lt )Xt− + Lt

and +

−

+

Qh|Ft,h = Mth

,h−

· P|Ft

0.4 Penalisation and Conditioning by an Event of Probability 0 +

Then, (Mth

,h−

, t ≥ 0) is a martingale and : +

Qh

,h−

(L∞ ∈ dy) = h(y)dy

(0.83)

Indeed : +

Qh

,h−

19

+

(L∞ > l) = Qh

,h−

+

(τl < ∞) = E[Mτhl

,h−

∞

]=

h(y)dy

(0.84)

l

from Doob’s optional Theorem, where we have denoted : τl := inf{t ≥ 0 ; Lt > l} 0.4.4 Conditioning Brownian motion by

(0.85) ∞

q(Xs )ds < ∞ .

0

i) Our notation here is that of Example 0.4 : (Xt , t ≥ 0, Px , x ∈ R) denotes the canonical Brownian motion, and q : R → R+ is a Borel function from R to R+ such that :

0< 1 + |x| q(x)dx < ∞ (0.86) (q)

We deﬁne At

:=

R

t

q(Xs )ds.

Since Brownian motion is recurrent, then :

0

(q) A∞

= ∞ a.s. We would like to condition this Brownian motion by the (q) negligible set {A∞ < ∞}, i.e., we would like to ﬁnd a positive martingale (q) (q) (q) (Mt , t ≥ 0) such that, if : Qx|Ft = Mt · Px|Ft then : (q) Q(q) x (A∞ < ∞) = 1

(0.87)

An answer to this problem is provided by the following penalisation Theorem, which will be proven, and much enriched, in Chapter 2. Theorem 0.11. 1) Let q satisfy (0.86). There exists a function ϕq : R → R∗+ such that, for every s > 0, and Fs ∈ b(Fs ) : lim Ex Fs

t→∞

t exp − 12 0 q(Xu )du = Ex [Fs Ms(q) ] := Q(q)

x (Fs ) t Ex exp − 12 0 q(Xu )du

(0.88)

(q)

where (Ms , s ≥ 0) denotes the martingale deﬁned by : 1 s ϕq (Xs ) · exp − q(Xu )du Ms(q) = ϕq (x) 2 0

(0.89)

2) If q˜ is another function which satisﬁes (0.86), there exist two strictly positive constants C1 and C2 such that : C1 ϕq (x) ≤ ϕq˜(x) ≤ C2 ϕq (x)

for every x ∈ R

(0.90)

20

0 Introduction

We now show that the probability deﬁned by (0.88) satisﬁes : (q) Q(q) x (A∞ < ∞) = 1.

Indeed, one has, from (0.88), for λ ∈]0, 1[ : ϕq (Xt ) λ t 1−λ t EQ(q) exp q(Xu )du = Ex exp − q(Xu )du · x 2 0 2 ϕq (x) 0 ϕ(1−λ)q (Xt ) ϕq (Xt ) 1−λ t · q(Xu )du · = Ex exp − 2 ϕq (x) ϕ(1−λ)q (Xt ) 0 t ϕ(1−λ)q (Xt ) ϕ(1−λ)q (x) 1 (1 − λ) Ex exp − · ≤ q(Xu )du · C1 (λ) 2 ϕ(1−λ)q (x) ϕq (x) 0 (by applying (0.90) with q˜ = (1 − λ)q) ϕ(1−λ)q (Xt ) 1−λ t q(Xs )ds · ≤ C (λ)Ex exp − 2 ϕ(1−λ)q (x) 0 (by applying (0.90) once more) (0.91) ≤ C (λ) t ϕ(1−λ)q (Xt ) 1−λ since exp − q(Xs )ds , t ≥ 0 is a martingale, from ϕ(1−λ)q (x) 2 0 (0.89). Letting t → ∞ in (0.91), we obtain : λ ∞ q(Xs )ds ≤ C (λ) EQ(q) exp x 2 0 which, of course, implies (0.87). (q) We have just conditioned Brownian motion by the negligible set {A∞ < ∞}. There again, there is no canonical way to operate : indeed, we may replace the martingale M (q) by the martingale M (μ·q) where μ a positive constant, (q) which will yield (obviously) the same result Q(μq) (A∞ < ∞) = 1. In fact, we shall even show, in Chapter 2, that if q and q˜ satisfy the condition (0.86), ˜ are equivalent on F∞ , so that : then the probabilities Q(q) and Q(q) ˜ Q(q) (A(q) ∞ < ∞) = 1.

ii) Let us deﬁne, for every x ∈ R, the σ-ﬁnite measure Wx on (Ω, F∞ ) via : 1 (q) A∞ (0.92) ·ϕq (x) Wx (Z) := EQ(q) Z · exp x 2 where Z is a generic positive F∞ -measurable variable. The following remarkable fact is proven in Chapter 2 : the measure Wx does not depend on q.

0.4 Penalisation and Conditioning by an Event of Probability 0

21

We are now able to deﬁne the positive and σ-ﬁnite measure Λx , (of which Example 0.5, Section 0.2, asserts the existence) as the image of Wx by the “local time” application : Θ : C(R+ → R) → C (R → R+ ) deﬁned by : Θ(Xt , t ≥ 0) = (Ly∞ , y ∈ R) iii) We have developed a number of other examples of penalisation which allow to condition a process by a negligible event. We refer the interested reader to Chapter 2, or to [RVY,II]. Here

is now an example which is not found in the preceding references. Let Ω, (Rt , Ft )t≥0 , F∞ , Px , x ∈ R+ d denote a canonical Bessel process with dimension d, and index ν = − 1, 2 with ν > 0. Thus, as is well-known, this process is transient. Let r > 0 and t (r) Ut := 1[0,r] (Ru )du (0.93) 0

Thus, we know that : (r) U∞ <∞

Px

a.s.

(0.94)

We shall now condition the Bessel process by the negligible event by using a penalisation procedure.

(r) (U∞

= ∞)

Theorem 0.12. i) For every s ≥ 0 and Fs ∈ b(Fs ) : lim

t→∞

Ex [Fs 1U (r) >t ] ∞

(r)

exists

a.s.

(0.95)

Px [U∞ > t]

ii) This limit equals : Ex [Fs Ms(r) ] := Q(r) x (Fs )

(0.96)

(r)

where (Ms , s ≥ 0) is the positive martingale deﬁned by : 2 z1 (r) Ms(r) := h(r) (Rs ) exp U (0.97) 2r2 s

(r) h and z1 are made precise below ; see formula (0.100) . (r) iii) Formula (0.96) induces a probability Qx on (Ω, F∞ ). With respect to the (r) family of probabilities (Qx , x ∈ R+ ), the canonical process is a diﬀusion process with generator L(r) deﬁned by :

L(r) f (x) =

1 2ν + 1 h(r) f (x) + f (x) + (r) (x) f (x) 2 2x h

(0.98)

In particular, for x ≥ r : L(r) f (x) =

1 1 − 2ν f (x) + f (x). 2 2x

(0.99)

22

0 Introduction (r)

The process (Xt , t ≥ 0 ; Qx , x ≥ 0) is recurrent. In the case ν < 1, the process converges, as r → 0 towards the Bessel process with dimension δ = 4 − d, with index ν = −ν = δ2 − 1 < 0 (with ν < 1). The real z1 and the positive function h(r) which appear in (0.97) and (0.98) are deﬁned by : •

•

z1 is the ﬁrst positive zero of the Bessel function Jν−1 ⎧ ν z z ⎨ 2r 1 Γ(ν + 1) z −ν Jν if 0 ≤ z ≤ r (r) h (z) = z1 r ⎩ −2ν if z < r Cz

(0.100)

where C is a constant such that h(r) is continuous and of C 1 class. 0.4.5 Relation with Doob’s h-transforms. Roughly speaking, in its simplest form, a Doob h-transform of a Markow process (Xt , t ≥ 0) consists in replacing the original laws (Px , x ∈ E) of that Markov process by laws (Pxh , x ∈ E) associated with a R+ -valued harmonic function h, i.e. (h(Xt ), t ≥ 0) is a Px -martingale, such that: Pxh |Ft =

h(Xt ) •Px|F t h(x)

see, e.g., Chapter X of Doob’s book [D] about conditional Brownian motions. The penalised probabilities which we just discussed in 0.4 may all be considered as such Doob’s h-transforms. For example, we have associated to the original Brownian motion (Xt , t ≥ 0) the 2-dimensional Markov processes : (q)

(Xt , St , t ≥ 0) , (Xt , Lt , t ≥ 0) and (Xt , At , t ≥ 0) and we have constructed, by means of penalisations, harmonic functions h of two variables, for which the Radon-Nikodym densities : (q)

h(Xt , St ) , h(Xt , Lt ) and h(Xt , At ) between the original Wiener measure, and the penalised probabilities, are obtained. When studying the behavior of an h-transform process, the following applications of Doob’s optional stopping Theorem are crucial: if Q|Ft = Mt •P|Ft

(for every t ≥ 0)

then for any stopping time T such that P (T < ∞) = 1: Q|Ft ∩(T <∞) = Mt •P|Ft

0.5 Penalisation and Statistical Mechanics

23

Consequently: • if MT = 0, then T = ∞ Q a.s. • if (At , t ≥ 0) is a continuous increasing process, (Ft , t ≥ 0) adapted, such that A∞ = ∞ P a.s. and MTa = m(a) for Ta := inf{t ≥ 0; At = a}, and m a deterministic function, then: Q(A∞ ≥ a) = m(a) i.e. (m(a), a ≥ 0) is the tail probability of A∞ under Q.

0.5 Penalisation and Statistical Mechanics 0.5.1 Traditionally, in Statistical Mechanics, one studies a sequence of probabilities Pn of the form : e−λH(ω) Pn (ω) := e−λH(ω)

(0.101)

ω∈Bn

where Bn denotes the set of all conﬁgurations in a box of size n, where H is a Hamiltonian, and λ a parameter (: temperature, or its inverse). Then, one studies (the existence of) the limit of the probabilities Pn as n, the size of the box, goes to +∞. The existence of diﬀerent kinds of limits, as a function of λ, constitutes a phase transition phenomena. We now come back to our penalisation problem (see 0.1.2) by the process h(Γt ). We study (the existence) of the limit, as t → ∞, of the probabilities (t) Px deﬁned via : h(Γt )

· Px Px(t) := (0.102) Ex h(Γt ) This problem is close to the preceding one in Statistical Mechanics, with the diﬀerence that we do not let the “size of the box” grow to inﬁnity, but rather it is the time parameter which increases to inﬁnity. Thus, our penalisation problem is a kind of “time statistical mechanics”, rather than a classical “spatial statistical mechanics” problem. 0.5.2 We shall now illustrate, by means of examples, the appearance of phase transition phenomena in our penalisation problems.

Example 0.13. Here again, Ω, (Xt , Ft )t≥0 , F∞ , P0 is Brownian motion, starting from 0, which we penalise by a function h of the pair (Xt , St ). In fact, the function is an exponential : h(x, σ) = exp(μx + λσ) which depends on the two real parameters μ and λ so that : h(Xt , St ) = exp(μ Xt + λ St ) The following theorem has been established in [RVY, III].

(0.103)

24

0 Introduction

Theorem 0.14. 1) For every s ≥ 0 and Fs ∈ b(Fs ) :

Ex Fs exp(μ Xt + λ St )

lim = Ex [Fs Msλ,μ ] := Qλ,μ x (Fs ) t→∞ Ex exp(μ Xt + λ St ) where (Msλ,μ , s ≥ 0) is a positive martingale described in [RVY, III]. 2) The probabilities Qλ,μ on (Ω, F∞ ) may be described diﬀerently, whether (λ, μ) ∈ Ri , i = 1, 2, 3 with : R1 = (λ, μ) ; λ + μ < 0, μ ≥ 0 R2 = (λ, μ) ; λ + 2μ ≥ 0, λ + μ > 0 R3 = (λ, μ) ; λ + 2μ < 0, μ < 0 More precisely : i) If (λ, μ) ∈ R2 , Qλ,μ is the law of Brownian motion with drift (λ + μ). In particular, this process is Markovian. ii) If (λ, μ) ∈ R3 , Qλ,μ is absolutely continuous with respect to P (μ) , the law of Brownian motion with drift μ (and μ < 0), with Radon-Nikodym density : λ + 2μ λS∞ e 2μ iii) If (λ, μ) ∈ R1 , Qλ,μ may bedescribed as follows : under Qλ,μ , the canonical process Ω, (Xt , Ft )t≥0 , F∞ satisﬁes : •

S∞ < ∞

a. s. and S∞ is exponentially distributed with density : λ,μ

fSQ∞ (x) = −(λ + μ)e(λ+μ)x 1x≥0 •

(0.104)

Let g := inf{t ; St = S∞ }. Then : Qλ,μ (g < ∞) = 1 and the processes (Xt , t ≥ g) and (Xg+t − Xg , t ≥ 0) are independent. Furthermore, (Xg − Xg+t , t ≥ 0) is a 3-dimensional Bessel process starting from 0, and, conditionally on S∞ = y, the process (Xt , t ≤ g) is a Brownian motion, stopped the ﬁrst time it reaches level y. In particular, this process is not Markovian.

Thus, whether (λ, μ) belongs to R1 , R2 or R3 , the probabilities Qλ,μ are very diﬀerent from each other, and we may say that there are “three diﬀerent phases”. We also note that, when (λ, μ) belongs to R3 or to R1 , then the preceding Theorem provides yet some new ways to condition with respect to (S∞ < ∞). Example 0.15. This time, we penalise Brownian motion by a function of Lt , where (Lt , t ≥ 0) is the local time at level 0. This function is taken to be an exponential, depending on the parameter λ : h(x) := eλx

(x ≥ 0)

We have proven (see Chap. 1 and 2, or [RVY,I], [RVY,II]) the following :

0.5 Penalisation and Statistical Mechanics

25

Theorem 0.16. 1) For every s ≥ 0 and Fs ∈ b(Fs ) : Ex [Fs eλLt ] → E[Fs Ms(λ) ] := Q(λ) (Fs ) t→∞ Ex (eλLt ) t→∞ lim

(λ)

where (Ms , s ≥ 0) is the martingale given by : ⎧ (λ) M =1 ⎪ ⎪ if λ = 0 ∞ ⎪ s ⎪ 1 ⎨ (λ) eλy dy if λ < 0 Ms = eλLs |Xs | + |λ| L s ⎪ ⎪ ⎪ λ2 s (λ) ⎪ ⎩ Ms = exp −λ|Xs | + λLs − if λ > 0 2 2) Under Q(λ) , the canonical process (Xt , t ≥ 0) is : i) if λ = 0, Brownian motion ; ii) if λ > 0, the “bang-bang” process, which solves the SDE ; t sgn(Xs ) ds Xt = Bt − λ

(0.105)

(0.106)

0

where (Bt , t ≥ 0) is a Brownian motion. In particular :

(law) (λ) (λ) |Xt |, t ≥ 0 = (St − Bt , t ≥ 0) (λ)

(0.107) (λ)

where (Bt , t ≥ 0) denotes Brownian motion with drift λ and St sup Bu(λ) . This process is Markovian.

=

u≤t

iii) if λ < 0, the canonical process (Xt , t ≥ 0) under Q(λ) satisﬁes the following, where g := sup{t ≥ 0 ; Xt = 0} : • • • •

Q(λ) (0 < g < ∞) = 1, Q(λ) (L∞ ∈ dy) = |λ|eλy dy ; the

processes (X t , t ≤ g) and (Xg+t , t ≥ 0) are independent ; |Xg+t |, t ≥ 0 is a 3-dimensional Bessel process starting from 0 ; Conditionally upon L∞ = y, (Xt , t ≤ g) is a Brownian motion stopped the ﬁrst time its local time at 0 reaches level y. In particular, this process is not Markovian.

there again, we have seen the existence of three “phases”, which diﬀer very much from each other, whether λ = 0, λ > 0 or λ < 0. 0.5.3 Other authors have considered problems close to those we have just listed. Let us mention: • M. Cranston and S. Molchanov [CM] Let P0 denote the law of the symmetric random walk (Xt , t ≥ 0) starting from 0, indexed by R+ , and valued in Zd . Let β > 0 and denote by δ0 the Dirac measure in 0. The authors study the limit, as t → ∞, of the probabilities:

26

0 Introduction

t −1 Pβ,t := Zβ,t exp β δ0 (Xs )ds •P0 0

where Zβ,t := E0

t δ0 (Xs )ds exp β 0

and show a phase transition phenomenon (depending on d) when β is smaller or larger than a critical value βcr . Then, they study the behavior of the penalised process, in particular for β = βcr . • M. Cranston, L. Koralov, S. Molchanov and B. Vainberg [CKMV] Now, P0 is the law of a d-dimensional Brownian motion (Xt , t ≥ 0) starting from 0. Let β > 0, and v : Rd → R+ a regular function. The authors study the asymptotic behavior, as t → ∞, of: t −1 Pβ,t := Zβ,t exp β v(Xs )ds •P0 0

where Zβ,t := E0

t v(Xs )ds exp β 0

There again, the authors show a phase transition phenomenon depending on the position of β with respect to a critical exponent βcr . We note that this study is close to, although it is diﬀerent from, that developed in [RVY,I] (see Chap. 1 below, Case II). Indeed, in [RVY,I], it is the limit, as t → ∞, of the probabilities: −1 Pβ,t := Zβ,t exp −β

t

v(Xs )ds

•

P0

0

which is being studied, now with the factor (−β), instead of (+β) in [CKMV]. • H. Osada - H. Spohn [OS], V. Betz - J. L¨ orinczi [BL], M. Gubinelli [Gub] Consider 3 functions: ξ : R → R, v : R → R, and w : R × R → R and denote by Wt,ξ the law of the Brownian bridge (Xs , −t ≤ s ≤ t) indexed by the time interval [−t, t] such that X−t = ξ(−t) and Xt = ξ(t). Deﬁne: t 1 v(Xs )ds + w(s − u, Xs − Xu )dsdu Ht (X, ξ) = 2 |s|≤t,|u|≤t −t w(s − a, Xs − ξ(a))dsda + |s|≤t≤|a|

Then, deﬁne the probability μt,ξ on C(R → R) as: −1 μt,ξ = Zt,ξ,β exp(−βHt (X, ξ))•Wt,ξ (t ≥ 0)

with Zt,ξ,β = Wt,ξ (exp(−βHt (X, ξ)))

0.6 Penalisation as a Machine to Construct Martingales

27

The authors study the set of limit laws, so-called: Gibbs measures, of the probabilities μt,ξ , as t → ∞. Depending on the functions ξ, v, w, the authors establish the existence, uniqueness, or non-uniqueness of such Gibbs measures, and show phenomena of phase transition. This work diﬀers from our penalisation studies by the following features: • •

•

time is the whole real line R, not only R+ the penalisation functional Γt = exp(−βHt ) involves a self-interaction between the trajectories, arising from the process w(s − u, Xs − Xu ) the penalisation functional Γt = exp(−βHt ) also depends on time outside of the interval [−t, t] arising from the term: w(s − a, Xs − ξ(a))dsda |s|≤t≤|a|

• G. Giacomin [Gia] In his book ([Gia]), this author also studies penalisation problems, but this time in a discrete set-up. For instance, on p. 7 of [Gia], the law of a random walk (Sn , n ≥ 0), taking values in Z, and issued from 0, is denoted by P , and the behavior, as n → ∞, of the probabilities Pn,β (β > 0) deﬁned as: n −1 Pn,β = Zn,β exp β 1(Si =0) •P

i=1

is studied. A phase transition phenomenon is also established.

0.6 Penalisation as a Machine to Construct Martingales The (meta)-Theorem of penalisation (see Section 0.3, formula (0.38)) is stated (h) in terms of a positive martingale (Ms , s ≥ 0). Thus, each time we are able to prove such a theorem, we obtain a martingale. Here is a - non-exhaustive list of the martingales we have “unearthed” so far (via this procedure). Some of them are classical, others less so. Example 0.17. (see Chap. 2)

(Xt , t ≥ 0) denotes Brownian motion, which we penalise with h(St ), t ≥ 0 (h a probability density, and St = sup Xs ). s≤t

Ms(h) = h(Ss )(Ss − Xs ) +

∞

h(y)dy

(0.108)

Ss

These martingales are the so-called Az´ema-Yor martingales (see, e.g. [AY1 ], [O1 ], [O2 ], [OY1 ], [OY2 ]· · · ).

28

0 Introduction

Example 0.18. (see Chap. 1 and [RVY,V])

i) (Xt , t ≥ 0) denotes Brownian motion, which we penalise by h(Lt ), t ≥ 0 with h a probability density, and (Lt , t ≥ 0) the local time at 0 ∞ (h) Ms = h(Ls )|Xs | + h(y)dy (0.109) Ls

ii) (Xt , t ≥ 0) denotes Brownian motion. Let h+ , h− : R+ → R+ , be two 1 Borel functions R+ → R+ such that (h+ + h− ) is a probability density. 2

When we penalise with h+ (Lt )1Xt ≥0 + h− (Lt )1Xt <0 , t ≥ 0 we obtain the limiting martingale : ∞ 1 + h+ ,h− + + − − Ms (h + h− )(y)dy = h (Ls )Xs + h (Ls )Xs + (0.110) Ls 2 iii) Now, (Xt , t ≥ 0) denotes a Bessel process with dimension d = 2(1 − α), where 0 < d < 2, or 0 < α < 1. Let h be a probability density on R+ and the choice of the local time at 0, such that : (Xt2α − Lt , t ≥ 0) is a martingale (0.111)

(see [D-M,RVY]). We penalise this Bessel process by h(Lt ), t ≥ 0 . Then, we obtain the limiting martingale : ∞ Ms(h) = h(Ls )Xs2α + h(y)dy (0.112) Ls

1 . 2 iv) (Xt , t ≥ 0) denotes the Bessel process

withλXdimension d = 2(1 − α), with 0 < d < 2, which we penalise with h(Lt )e t , t ≥ 0 (λ > 0, h is a probability density on R+ . The limiting martingale is then : The Brownian case corresponds to α =

α ˜ s ) 2 Γ(1 + α)Iα (λXs ) Xsα h(L λ ∞ −α 2 ˜ Γ(1 − α)I−α (λXs ) + h(y)dy λ Ls

Ms(h) = e−

λ2 s 2

with :

˜ h(y) := h(y) − σλ eσλ y σλ =

2α λ Γ(1 − α) 2 Γ(1 + α)

y

∞

h(z)e−σλ z dz

(y ≥ 0)

(0.113)

0.6 Penalisation as a Machine to Construct Martingales

29

where Iα , (resp. I−α ) denotes the modiﬁed Bessel function with index α, (resp. −α). Example 0.19. (see [RVY,II]) (Xt , t≥ 0) denotes Brownian motion. Let λ > 0 and h : R+ → R+ such ∞

that :

h(z)e−λz < ∞.

We penalise Brownian motion with h(St )eλ(St −Xt ) , t ≥ 0 . Then, the limiting martingale is : ∞ λ2 s sinh λ(Ss − Xs ) + eλXs h(z)e−λz dz e− 2 (0.114) Ms(h) = h(Ss ) λ Ss 0

This martingale already appeared in the literature, and is known as a Kennedy martingale. It may also be written as : ∞

sinh λ(Ss − Xs ) − λ2 s (h) ˜ ˜ Ms = cosh λ(Ss − Xs ) · h(y)dy +h(Ss ) · e 2 λ Ss (0.115) with ∞ ˜ h(y) := h(y) − λeλy h(z)e−λz dz (y ≥ 0) y

Example 0.20. (see Chap. 2 or [RVY,I]) i) (Xt , t ≥ 0) denotes Brownian motion, and we penalise it with the mul 1 t tiplicative functional : exp − q(Xu )du , t ≥ 0 with q : R → R+ , 2 0

1 + |x| q(x)dx < ∞. 0< R

Ms(q) =

1 s ϕq (Xs ) · exp − q(Xu )du ϕq (x) 2 0

(0.116)

where ϕq denotes the unique solution of ϕ = ϕ · q, ϕ (+∞) = −ϕ (−∞) = 2 . π t ii) (Xt , t ≥ 0) denotes Brownian motion. Let At := 1(Xs <0) ds and h : 0 ∞ dy R+ → R+ be such that 0 < √ h(y) < ∞. We penalise Brownian motion y 0

with the weight process h(At ), t ≥ 0 . Then : ∞ √ dy − (Xs− )2 (h) + Ms = 2πh(As )Xs + (see [RY,VIII] ) √ e 2y h(As + y) y 0 (0.117)

30

0 Introduction

Example 0.21. (see [RY,VIII]) i) (Xt , t ≥ 0) denotes again Brownian motion. Let gt := sup{s ≤ t ; Xs = 0} and h : R+ → R+ such that ∞

h(y)dy = 1. We penalise Brownian motion with h(Sgt ), t ≥ 0). Then : 0

Ms(h)

1 = h(Sgs )|Xs | + h(Ss )(Ss − Xs+ ) + 2

∞

h(y)dy

(0.118)

Ss

ii) (Xt , t ≥ 0) denotes again Brownian motion. Let Xt∗ := sup |Xs |, and s≤t

h : R∗+ → R+ a probability density . We penalise Brownian motion with : h(Xgt ), t ≥ 0 . Then :

Ms∗h = h(Xg∗s )|Xs | + h(Xs∗ ) Xs∗ − |Xs | +

∞

h(y)dy

(0.119)

Xs∗

Example 0.22. (see Chap. 2 or [RVY,II]) (Xt , t ≥ 0) denotes again Brownian motion, and Dta,b the number of down crossings on the spatial interval [a, b] before time t (see 0.2.3, Example 0.3 for the deﬁnition of Dta,b and associated notation). Let h : N → R+ such that : h is decreasing, lim h(n) = 0 and h(0) = 1. We penalise by n→∞

(h) a,b Δ (Dt ), t≥ 0), with Δ(h) (n) := h(n) − h(n + 1), n ≥ 0; then : h(n) b − Xs h(1 + n) Xs − a Ms(h) = 1[σ2n ,σ2n+1 [ (s) 1+ + 2 b−a 2 b−a n≥0 h(1 + n) b − Xs h(n) Xs − a +1[σ2n+1 ,σ2n+2 [ (s) 1+ + 2 b−a 2 b−a (0.120) Example 0.23. (see Chap. 3 or [RVY,VII]) i) (Xt , t ≥ 0) denotes Brownian motion, gt := sup{s ≤ t ; Xs = 0}, dt := inf{s ≥ t ; Xs = 0} ; Σt := sup{ds − gs ; ds ≤ t}· Thus, Σt is the length of the longest excursion before gt . Let ∞ h : R+ → R+ such that h(z)dz = 1 0

√ We penalise Brownian motion with h( Σt ), t ≥ 0 . Then : Ms(h)

=

|Xs | 2 ! h( Σs )|Xs | + h1 As Φ ! π (Σs − As )+ √ |Xs | (Σs −As )+ v2 2 X2 h1 As + 2s e− 2 dv + π 0 v

(0.121)

0.6 Penalisation as a Machine to Construct Martingales

where As := s − gs , and h1 (x) :=

Φ(x) :=

√

√ x h( x) +

2 π

31

∞

e−

u2 2

du

x

∞

√ x

h(z)dz

ii) (Xt , t ≥ 0) denotes Brownian motion, and we keep the same notation as above. Let A∗t := sup As = Σt ∨ At . Let x > 0 be ﬁxed, and penalise with (1A∗t ≤x , t ≥ 0).

s≤t

Ms∗x = eλ0 x 1(A∗s ≤x) · s

x−As

0

2 λ0 u Xs |X | √ s e− 2u + x du 3 2πu

(0.122)

where λ0 denotes the ﬁrst strictly positive zero of the function : λ → γ(λ) where : 1 λz e dz √ · γ(λ) = 1 − λ e−λ z 0 iii) The two preceding examples are extended, in Chapter 3, to the setting where Brownian motion (Xt , t ≥ 0) is replaced by a Bessel process with dimension d = 2(1 − α), with 0 < α < 1. Example 0.24. (see [RVY,II]) (Xt , t ≥ 0) denotes Brownian motion, St := sup Xs , It := − inf Xs , (Lt , t ≥ 0) s≤t

s≤t

is the Brownian local time at level 0. Let ν(da, db) denote a probability on R+ × R+ and deﬁne :

1 1 1 e− 2 a + b l 1(s≤a, i≤b) ν(da, db) hν (s, i, l) := R2+

We penalise Brownian motion with hν (St , It , Lt ), t ≥ 0), then : Msν

= R2+

X+ 1− s a

X− 1− s b

1(Ss ≤a, Is ≤b)

1 1 1 + exp Ls ν(da, db) 2 a b (0.123)

In the particular case where ν is carried by the diagonal of R+ × R+ , (Msν ) takes the form : ∞ Ls |Xs | ν∗ Ms = (0.124) 1− e a ν ∗ (da) a ∗ Xs with ν ∗ denoting here a probability on R+ .

32

0 Introduction

Example 0.25. (see [RVY,VI] or Chap. 2) (Xt , t ≥ 0) denotes here d-dimensional Brownian motion, starting from x = 0. It admits the following skew-product decomposition : Xt = Rt · ΘHt

(0.125)

where

• (Rt , t ≥ 0) = |Xt |, t ≥ 0 is a d-dimensional Bessel process ; • (Θu , u ≥ 0) is a Brownian motion on the unit sphere Sd−1 , independent from (Rt , t ≥ 0) ; t ds • Ht := · 2 0 Rs When d = 2, the decomposition (0.125) writes : Xt = Rt exp(iβHt ) where (βu , u ≥ 0) is a one-dimensional Brownian motion, independent from (Rt , t ≥ 0). i) (d = 2). Let 0 < r < R and : θt := βHt , Stθ := sup θs = sup βu θt−,r

s≤t

u≤Ht

t

:=

θt+,R

1Rs
Ht−,r :=

Let h : R+ × R+ → R+

t

:=

0

1Rs >R dθs 0

t

t ds ds +,R , H := 1Rs >R 2 t 2 R R 0 0 s s such that h(x, y) dx dy = 1 and let n, m two

1Rs
R2+

reals, with 0 < m < 1. −,r +,R We penalise by ((Ht−,r )m (Ht+,R )n h(Stθ , Stθ ), t ≥ 0) : −,r

−,r

+,R

−,r

+,R

Ms(h) = h(Ssθ , Ssθ ) · (Ssθ − θtθ )(Ssθ − θs+,R ) ∞ ∞ θ −,r −,r + dx dy h(x, y) + (Ss − θs ) −,r Ssθ

+(Ssθ

+,R

+,R Ssθ

− θs+,R )

∞ −,r Ssθ

h(x, Ssθ

+,R

)dx

∞ +,R Ssθ

h(Ssθ

−,r

, y)dy (0.126)

Observe that this martingale does not depend on n and m and we may have r = 0 and R = ∞ in this formula. ii) (d ≥ 2 arbitrary). Let C denote a regular cone with vertex the origin, and / C} basis O, an open set of the unit sphere Sd−1 . Let TC := inf{t ≥ 0 ; Xt ∈ the exit time from the cone. Let ρ ≥ 0 and γ a real number such that :

Bibliography

33

μ2 ≥ γ − λ21 ,

with μ =

d − 1, 2

where λ21 (resp. ϕ1 ) is the smallest positive eigenvalue, (resp. eigenfunction and the Dirichlet problem : associated with λ21 ) with respect to −Δ = −λ2 ϕ Δϕ and ϕ|∂O = 0. We penalise Brownian ϕ of class C ∞ in O in O, continuous γ Ht + ρRt , t ≥ 0 . Then : motion with 1TC
Bibliography [AY1 ]

[Be] [BL] [CKMV] [CM] [CP]

[D] [D-M,RVY]

[Gal] [Gia] [Gub]

J. Az´ema and M. Yor. Une solution simple au probl`eme de Skorokhod. In S´eminaire de Probabilit´es, XIII (Univ. Strasbourg, Strasbourg, 1977/78), LNM 721, pages 90–115. Springer, Berlin, 1979. V. Betz. Existence of Gibbs measures relative to Brownian motion. In Markov Processes and Related Fields, 9(1):85–102, 2003. V. Betz and J. Lrinzci. Uniqueness of Gibbs measures relative to Brownian motion. Ann. I.H.P. Prob. Stat., 39(5):877–889, 2003. M. Cranston, L. Koralov, S.Molchanov, and B. Vainberg. Continuous model for homopolymers (private communication). Submitted, 2008. M. Cranston and S.Molchanov. Analysis of a homopolymer model (private communication). Submitted, Dec. 2007. P. Courr`ege and P. Priouret. Temps d’arrˆet d’une fonction al´eatoire: Relations d’´equivalence associ´ees et propri´et´es de d´ecomposition. Publ. Inst. Statist. Univ. Paris, 14:245–274, 1965. J.L. Doob. Classical Potential Theory and its Probabilistic Counterpart. Springer, 1984. C. Donati-Martin, B. Roynette, P. Vallois, and M. Yor. On constants related to the choice of the local time at 0 and the corresponding Itˆ o measure for Bessel processes with dimension d = 2(1 − α), 0 < α < 1. Studia Sci. Math. Hungarica, 44(2), 207–221, 2008. A.R. Galmarino. A test for Markov times. Rev. Un. Mat. Argentina, 21:173–178, 1963. G. Giacomin. Random polymer models. Imperial College Press, London, 2007. M. Gubinelli. Gibbs measures for self-interacting Wiener paths. Markov Process and Related Fields, 12(4):747–766, 2008.

34 [K] [Leb]

[M]

[O1 ] [O2 ] [OY1 ]

[OY2 ]

[OS] [RVY,I]

[RVY,II]

[RVY, III]

[RVY,IV]

[RVY,V]

[RVY,VI] [RVY,VII] [RY,VIII]

[RY,IX]

[SZ]

0 Introduction F.B. Knight. Brownian local times and taboo processes. Trans. Amer. Math. Soc., 143:173–185, 1969. N.N. Lebedev. Special functions and their applications. Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication. P.A. Meyer. Probabilit´es et potentiel. Publications de l’Institut de Math´ematique de l’Universit´e de Strasbourg, N◦ XIV. Actualit´es Scientiﬁques et Industrielles, N◦ 1318. Hermann, Paris, 1966. J. Oblo´j. The Skorokhod embedding problem and its oﬀspring. Probab. Surv., 1:321–390 (electronic), 2004. J. Oblo´j. A complete characterization of local martingales which are functions of brownian motion and its supremum. Bernoulli, 12(6):955–969, 2006. J. Oblo´j and M. Yor. An explicit Skorokhod embedding for the age of Brownian excursions and Az´ema martingale. Stochastic Process. Appl., 110(1):83– 110, 2004. J. Oblo´j and M. Yor. On local martingale and its supremum: harmonic functions and beyond. In A.N. Shyriaev’s Festschrift : From stochastic calculus to mathematical ﬁnance, 517–533. Springer, Berlin, 2006. H. Osada and H. Spohn. Gibbs measures relative to Brownian motion. Ann. Prob., 14(3):733–779, 1999. B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I. Studia Sci. Math. Hungar., 43(2):171–246, 2006. B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time. II. Studia Sci. Math. Hungar., 43(3):295–360, 2006. B. Roynette, P. Vallois, and M. Yor. Limiting laws for long Brownian bridges perturbated by their one-sided maximum. III. Periodica Math. Hungar., 50 (1-2):247–280, 2005. B. Roynette, P. Vallois, and M. Yor. Some extensions of Pitman’s and RayKnight theorems for penalized Brownian motion and their local time. IV. Studia Sci. Math. Hungar., 44(4):469–516, 2007. B. Roynette, P. Vallois, and M. Yor. Penalizing a BES (d) process (0 < d < 2) with a function of its local time. V. Studia Sci. Math. Hungar., 45(1):67–124, 2008. B. Roynette, P. Vallois, and M. Yor. Penalisations of multi-dimensional Brownian motion. VI. To appear in ESAIM P.S., 2009. B. Roynette, P. Vallois, and M. Yor. Brownian penalisations related to excursion lengths. VII. To appear in Annales de l’Inst. H. Poincar´e., 2009. B. Roynette and M. Yor. Ten penalisation results of Brownian motion involving its one-sided supremum until ﬁrst passage time VIII. Journal Funct. Anal., 255(9):2606–2640, 2008. B. Roynette and M. Yor. Local limit theorems for Brownian additive functionals and penalisation of Brownian paths. IX. To appear in ESAIM P.S., 2009. L.A. Shepp and O. Zeitouni. A note on conditional exponential moments and Onsager-Machlup functionals. Ann. Probab., 20(2):652–654, 1992.

Chapter 1

Some Penalisations of the Wiener Measure

Abstract A number of limit laws, which are obtained from various penalisations of the Wiener measure on C (R+ , Rd ), are shown to exist, and are described thoroughly, with the help of path decompositions. Keywords Brownian motion · Wiener measure · 3-dimensional Bessel process · path decompositions · supremum process · L´evy and Pitman theorems · age process of excursions · Brownian windings

1.1 Introduction : A Rough Idea about Penalisation 1.1.a) What does the study of a stochastic process (Xt , t ≥ 0), usually meant to idealize the position of a particle, subjected to some random phenomenon, entail? In many cases, one is mainly interested in the regularity properties of the paths of X, or of functionals related to X, or in the asymptotic behavior of (Xt ), as t → ∞, or t → 0, ..., or in the geometric properties of X, e.g : does the average path have double, triple... points ?

1.1.b) In the present paper, where (Xt ) shall be mainly Brownian motion

in Rd , we are interested in “penalising” X, or its law, that is with the help of appropriate weights placed upon the reference probability P , we shall try to alter radically some of the properties of X, e.g : to create Brownian-like processes which have an overall ﬁnite maximum, etc... A large family of such modiﬁcations has been studied for a long time and formalized as Doob’s h-transforms (see, e.g., 0.4.5), or Feynman-Kac transforms. The latter transforms consist in starting with a Markov process {(Xt , Ft )t≥0 ; (Px )x∈E } with inﬁnitesimal generator L, and function ϕ ∈ D(L), and then to construct the new Markovian laws deﬁned by : ϕ Px|F = Dtϕ •Px|Ft t

B. Roynette, M. Yor, Penalising Brownian Paths, Lecture Notes in Mathematics 1969, DOI 10.1007/978-3-540-89699-9 1, c Springer-Verlag Berlin Heidelberg 2009

(1.1) 35

36

1 Some Penalisations of the Wiener Measure

where : Dtϕ

t Lϕ ϕ(Xt ) exp − = (Xs )ds ϕ(X0 ) ϕ 0

is a ((Px ), (Ft )) martingale. It is the martingale property of {Dtϕ } which allows to deﬁne {Pxϕ } coherently from the equality (1.1). On the contrary, throughout this Monograph, which is devoted to penalisation, the weights {Γt }, which will play a role analogous to {Dtϕ } above, do not allow immediately to create a new probability distribution, although the latter may - hopefully - emerge in the limit as t → ∞. When such a limit exists, we call it the penalised probability associated with P and {Γt }. Many questions in Statistical Mechanics resort to this problematic.

1.1.c) In Section 1.2, we deﬁne precisely our (mainly Brownian) set-up, and what we mean exactly by penalisation, as well as propose some metatheorems which entail the existence of penalised probabilities. Section 1.2 ends with a list of ﬁve case studies which will be developed in the remaining sections of this Chapter. Accordingly, we postpone the presentation of the organisation of the Chapter to the end of Section 1.2.

1.2 Some Meta-Theorems Leading to Penalised Probabilities 1.2.a) Throughout this Chapter, for d = 1, 2, ... an integer, and x ∈ Rd , (d)

Wx will denote the Wiener measure on Ω = C(R+ , Rd ), the canonical space of all continuous functions ω : R → Rd , with Xt (ω) = ω(t), the coordinate (d) process, such that : Wx (X0 = x) = 1. Ω is ﬁtted with the σ-ﬁeld : F∞ = σ{Xs , s ≥ 0}, and with the ﬁltration (Ft = σ{Xs , s ≤ t}, t ≥ 0). (1) We shall simply write : W for W0 , and sometimes even simply P for any (d) Wx , either to simplify notation or with the underlying meaning that the Brownian set-up does not play such an important role in that instance.

1.2.b) Consider a process (Γt , t ≥ 0), taking values in R+ , and such that : 0 < EP (Γt ) < ∞, for every t ≥ 0; (Γt , t ≥ 0) is not necessarily (Ft )-adapted, although in almost all the examples we shall discuss below, (Γt ) is indeed (Ft )-adapted. Γt • P . EP (Γt ) Then, under the two conditions :

Theorem 1.1. Deﬁne : QtΓ =

i) for every s ≥ 0,

EP (Γt |Fs ) a.s. −→ Ms EP (Γt ) t→∞

1.2 Some Meta-Theorems Leading to Penalised Probabilities

37

ii) for every s ≥ 0, EP (Ms ) = 1, the following holds : a) (Ms , s ≥ 0) is a (P, (Fs , s ≥ 0)) martingale; b) for every Λs ∈ Fs , QtΓ (Λs ) −→ EP (1Λs Ms ) t→∞

≥ 0} converges weakly, as t → ∞, to the penalised Thus, the family probability QΓ∞ on (Ω, F∞ ), which is characterized by : {QtΓ , t

QΓ∞ (Λs ) = EP (1Λs Ms )

(s ≥ 0, Λs ∈ Fs )

in the sense that : QtΓ (Λs ) −→ QΓ∞ (Λs ) t→∞

(s ≥ 0, Λs ∈ Fs ) .

This theorem is a direct consequence of Scheﬀe’s lemma, (see : Meyer [M], p. 37), which implies that under the conditions i) and ii) the convergence of EP (Γt |Fs ) also holds in L1 (P ). Thus, with the notation Γt = Γt /EP (Γt ), we EP (Γt ) have : |QΓt (Λs ) − EP (1Λs Ms )| ≤ EP (|EP (Γt |Fs ) − Ms |) −→ 0 t→∞

1.2.c) We shall also be interested in a functional form of Theorem 1.1; here are our hypotheses : (Γt , t ≥ 0) is now a process taking values in Rk (for simplicity), and we consider for ϕ : Rk → R+ , ϕ > 0 : Qϕ t :=

ϕ(Γt ) EP [ϕ(Γt )]

•

P .

We look for (certain) conditions on the process (Γt ) which will ensure that for every, say bounded, continuous ϕ : Rk → R+ , ϕ > 0, ϕ with compact support, Qϕ t converges weakly, as in subsection 1.1.b), when t → ∞ towards a probability which we shall denote by Qϕ ∞ . Of course, we may then try to apply Theorem 1.1 to the processes : Γtϕ = ϕ(Γt ), thus having to consider the asymptotics of EP [ϕ(Γt )|Fs ], s ≥ 0. However, the consideration of a generic function ϕ led us to look for cases of applications of the following two metatheorems. In Theorem 1.2 below, we assume the existence of a (weakly) continuous version of the application : γ → EP (•|Γt = γ). Theorem 1.2. Assume that both following conditions hold : j) for every s > 0, and Λs ∈ Fs , EP (1Λs |Γt = γ) converges, as t → ∞, towards : Q(γ) (Λs ),

38

1 Some Penalisations of the Wiener Measure

jj) there exists a function of t, (αt , t ≥ 0) such that : αt E[ϕ(Γt )] −→ dγ g(γ) ϕ(γ) t→∞

(1.2)

for some locally integrable function g; Then, the limiting result holds : dγ g(γ) ϕ(γ)Q(γ) (Λs ) ϕ Qt (Λs ) −→ . t→∞ dγ g(γ) ϕ(γ)

1.2.d) So far, we have not succeeded in ﬁnding suﬃcient conditions on {Γt }, which ensure that j), in the above meta-theorem 1.2 is satisﬁed, but concerning jj), the following condition will be easily applicable in a number of cases where Γt is a Brownian functional. Theorem 1.3. Assume that, for some function (ht ), which tends to +∞, as t → ∞, one has : 1 (1) (k) (law) (1) (k) (Γt , ..., Γt ) −→ (Γ ∞ , ..., Γ∞ ) t→∞ ht

(1.3)

such that, the LHS of (1.3) admits a density gt (γ) with respect to dγ ≡ dγ (1) ... dγ (k) , and that moreover the RHS of (1.3) admits a density g(γ) with respect to dγ; we assume furthermore that : gt (γ)

converges to

g(γ) ,

as t → ∞ ,

uniformly on every compact set (in γ), and that g is continuous at 0. Then, with : αt = (ht )k , one has : (1) (k) αt E [ϕ(Γt , ..., Γt )] −→ g(0) dγ ϕ(γ) (1.4) t→∞

for every bounded, Borel function ϕ : Rk → R, with compact support. Proof of Theorem 1.3 : It is elementary; the LHS of (1.4) is equal to : γ αt αt dγ gt (γ)ϕ(ht γ) = dγ gt ϕ(γ ) . k (ht ) ht Since αt = (ht )k , it now remains to study : γ dγ gt ϕ(γ ) , ht

1.2 Some Meta-Theorems Leading to Penalised Probabilities

but, from our hypotheses, we can replace gt by g(0).

γ ht

39

by g

γ ht

, and, ﬁnally,

In fact, the hypotheses of this Theorem 1.3 are rarely satisﬁed: we refer the reader to [RY,IX], Theorem 1.1 for a better statement. 1.2.e) The remainder of this Chapter shall be devoted to ﬁve case studies of penalisations associated with (sometimes multidimensional) processes (Γt , t≥0), which are, in that order : t = L0 , which are respectively, the one-sided • Case I : Γt = sup Xs , and Γ t

s≤t

supremum and the local time at 0 of our one-dimensional Brownian motion (Xt , t ≥ 0). t • Case II : Γt = dsq(Xs ), where q : R → R+ , or q : R2 → R+ satisﬁes 0

some integrability condition. • Case III : In this case, (Xt , t ≥ 0) denotes a BES process, with dimension 0 < δ < 2, and Γt = L0t is its local time at 0. • Case IV : (Xt ) is again a one-dimensional Brownian motion; gt = sup{s ≤ t : Xs = 0}, and At = t − gt , t ≥ 0, is the so-called age process (of excursions). We then consider : Σt = sup Au , u≤gt

or

Σt = sup Au ,

or again

u≤t

Σt = sup Au , u≤dt

where dt = inf{s ≥ t : Xs = 0}. • Case V : (Xt , t ≥ 0) denotes the d-dimensional Brownian motion, issued from x = 0 (d ≥ 2). For d = 2, we consider Γt = sup θs , where (θs , s ≥ 0) is a contins≤t

uous determination of the argument of X around 0; we also consider Γt = 1(TC >t) (d ≥ 2), where C is a cone with vertex 0, and basis a regular open set O of Sd−1 , the unit sphere in Rd . Let us mention that the case t

Γt = 0

1(Xs ∈ C) ds is not so well understood.

1.2.f ) A somewhat detailed treatment of each of these cases is presented below in Sections 1.3 to 1.7, each of them corresponding to the cases I-V in that order. As already said in the Introduction, in order to help readers who may not be so familiar with Brownian motion and related processes, each of these sections contains one or several Item where the necessary prerequisites for that Section are recalled; the text of an Item will be inserted in a box, to ensure more visibility. Here is the list of these Items :

40

1 Some Penalisations of the Wiener Measure

• in Section 1.3, devoted to Case I, we present : Item A : Existence of Brownian local times; Item B : L´evy’s equivalence Theorem, i.e. : a representation of reﬂecting Brownian motion in terms of its one-sided supremum; Item C : Pitman’s 2S − X Theorem, which complements L´evy’s Theorem; Item D : the balayage formula which gives a general framework to understand the structure of the Brownian martingales, which are deterministic functions of St and Xt (t ≥ 0) only; • in Section 1.4, devoted to Case II, we present : Item E : Limit laws for Brownian additive functionals, in dimension d = 1 and d = 2; • in Section 1.5, devoted to Case III, we present : Item F : Scale and Clock relations between Bessel processes; • in Section 1.6, devoted to Case IV, we present : Item G : Brownian meander; Item H : A brief sketch of Brownian excursion theory; • in Section 1.7, devoted to Case V, we present : Item I : Limit theorems for planar Brownian windings; Item J : Skew-product decomposition for BM(Rd ); Item K : Local absolute continuity between the laws of the BES processes. Our aim in writing this list right before getting into the main stream of the paper is to give some indication for the reader of the kind of Brownian tools which are being used. We also provide references for each Item .

1.3 Case I : Γt = St := sup Xs, or Γt = L0t (X) s≤t

1.3.a) In this section, we shall only consider (Xt ) a 1-dimensional Brownian motion, and we shall take for (Γt ) either the one-sided supremum of X, or its local time at 0. After recalling brieﬂy the notion of local time(s) in Item A , we shall present L´evy’s equivalence Theorem in Item B , which allows to consider L0t (X) as the one-sided supremum of another Brownian motion. Item C presents the celebrated Pitman’s Theorem, which identiﬁes BES(3) as a particular linear combination of St and Xt , or, via L´evy’s equivalence, of L0t (X) and |Xt |.

1.3 Case I

41

Item A : (Existence of Brownian local times) The local times {Lxt (X); x ∈ R, t ≥ 0} may be deﬁned as a jointly continuous process, in (x, t) ∈ R × R+ , which satisﬁes :

t

∀f : R → R+ , Borel,

f (Xs )ds = 0

+∞

−∞

dx f (x)Lxt (X)

(A.1)

In other terms, for every t ≥ 0, and almost all ω, the random measure : t A ∈ B(R) −→ 1A (Xs (ω))ds 0

is absolutely continuous with respect to Lebesgue measure on R, and it admits a jointly continuous Radon-Nikodym density which is precisely (obtained from) the Brownian local times. As a consequence of (A.1), one has the following approximation of Lxt (X) : Lxt (X)

1 = lim ε→0 2ε

t

0

ds 1|Xs −x|≤ ε

a.s.

(A.2)

We also note that each local time Lxt (X) is an additive functional of Brownian motion. See, e.g., [RY∗ ], Chapter VI. Item B : (L´evy’s equivalence Theorem) P. L´evy showed the following identity in law between two 2-dimensional processes : (law)

(St − Xt , St ; t ≥ 0) = (|Xt |, L0t (X); t ≥ 0)

(B.1)

This identity in law may be proven thanks to Skorokhod’s lemma, which states that, given a continuous function (y(t), t ≥ 0), there exists one and only one continuous solution (z, ) to the following reﬂection equation : ⎧ ⎫ ⎨ (α) z(t) = −y(t) + (t) , ⎬ (β) z(t) ≥ 0 , (B.2) ⎩ ⎭ (γ) (t) is increasing, and d(t) is carried by {t : z(t) = 0} . This solution, which we denote as : (z ∗ (t), ∗ (t)) is given by : ∗ (t) = sup y(s) ; s≤t

z ∗ (t) = −y(t) + ∗ (t)

(B.3)

The identity (B.1) now follows from (B.2), thanks to Tanaka’s formula : |Xt | =

t

sgn(Xs )dXs + L0t (X) , 0

and the fact that 0

t

sgn(Xs )dXs , t ≥ 0 is a Brownian motion.

(B.4)

42

1 Some Penalisations of the Wiener Measure

Thus, L0t (X) is the one-sided supremum of the Brownian motion : t = − X

t

sgn(Xs )dXs , t ≥ 0 .

0

For details, see, e.g., [RY∗ ], Chapter VI, Section 2. Item C : (Pitman’s 2S − X Theorem) Pitman’s celebrated Theorem complements deeply L´evy’s equivalence Theorem. It may be stated as : (law)

(2St − Xt , St ; t ≥ 0) = (Rt , Jt ; t ≥ 0) ,

(C.1)

where, on the right-hand side of (C.1), Jt = inf Rs , and R denotes a BES(3) s≥t process. From L´evy’s equivalence, the identity (C.1) may also be stated as : (law)

(|Xt | + Lt , Lt ; t ≥ 0) = (Rt , Jt ; t ≥ 0) ,

(C.2)

An important diﬀerence between L´evy’s Theorem (B.1) and Pitman’s Theorem (C.1) is that the natural ﬁltration of (St − Xt , t ≥ 0) is equal to that of (Xt , t ≥ 0), whereas the natural ﬁltration, call it {Rt }, of the process ρt := 2St − Xt , t ≥ 0, is strictly contained in that of (Xt ). This fact is conﬁrmed, and made more precise by the following projection formula : 1 ρt E[f (St )|Rt ] = dx f (x) (C.3) ρt 0 for every f : R+ → R+ , Borel. In other terms, conditionally on Rt , and (law)

ρt = r, (St , St − Xt ) = (U r, (1 − U )r), where U denotes a uniform variable on [0, 1]. For details, see, e.g., [RY∗ ], Chapter VII, Exercise (4.15).

1.3.b) We are now in a good position to state functional penalisation results associated with either St or L0t . We shall only state the results involving {St }, and leave the companion results for {L0t } to the reader. Theorem 1.4. Let ϕ : R+ → R+ be a probability density on R+ , and deﬁne : Qϕ t :=

ϕ(St ) (1) •P . E (1) (ϕ(St ))

Then : a) for every s > 0, and Λs ∈ Fs , ϕ (1) ϕ Qϕ t (Λs ) −→ Q (Λs ) := E [1Λs Ms ] , t→∞

1.3 Case I

43

where : Msϕ

= ϕ(Ss ) (Ss − Xs ) +

∞

dy ϕ(y)

(1.5)

Ss

b) Under the probability Qϕ : i) ii) iii) iv) v)

S∞ is ﬁnite a.s and is distributed as ϕ(y)dy. Let g = sup{s; Xs = S∞ }. Then, Qϕ (0 < g < ∞) = 1. The two processes (Xt , t ≤ g) and (Xg+t , t ≥ 0) are independent. ((Xg − Xg+t ), t ≥ 0) is distributed as a BES(3) process starting from 0. Conditionally on S∞ = y, (Xt , t ≤ g) is distributed as (Xt , t ≤ Ty ) under P (1) , where Ty := inf{t ≥ 0, Xt = y}.

c) Under the probability Qϕ , (2St − Xt , t ≥ 0) is a BES(3) process starting from 0, independent of S∞ . The proof of point b i) is easy : Qϕ (St > x) = Qϕ (Tx < t) = E (1) (1Tx
and so, letting t → ∞ : Qϕ (S∞ > x) =

∞

ϕ(y) dy x

An explanation of the last point in Theorem 1.4 is that E (1) (Msϕ |Rs ) = 1

(1.6)

which may be simply deduced from (1.5) and is closely connected to the fact that the only functions ψ : R+ → R+ such that (ψ(Rs ), s ≥ 0) is a martingale, with (Rt , t ≥ 0) a BES(3) process, are the constants. Indeed, we have for any positive functional F : EQϕ (F (2Ss − Xs ; s ≤ t)) = E (1) (F (2Ss − Xs ; s ≤ t)Mtϕ ) = E (1) (F (2Ss − Xs ; s ≤ t)E (1) (Mtϕ |Rt )) = E (1) (F (2Ss − Xs ; s ≤ t))

by (1.6)

and we apply Item C . It is proven in Rogers [R] that the “nice” diﬀusions (Yt , t ≥ 0) such that (RtY := 2StY − Yt , t ≥ 0) is, in its own ﬁltration, a diﬀusion process are essentially Brownian motion with drift. The last point of Theorem 1.4 gives an example of a process (Xt , t ≥ 0) such that, under Qϕ , (2StX − Xt , t ≥ 0) is a diﬀusion (: a BES(3) process). Here, the process (Xt , t ≥ 0), under Qϕ ,

44

1 Some Penalisations of the Wiener Measure

is not Markovian (but the couple (Xt , St ) is Markovian). So, the point c) of Theorem 1.4 is an extension of Pitman’s Theorem in the non-Markovian case. Theorem 1.4 admits a “long bridges” companion which we now state Theorem 1.5. a) The probability Qϕ , as described in Theorem 1.4, may be disintegrated as : ∞ dy ϕ(y)Q(y) , Qϕ = 0 (y)

where (Q

, y ≥ 0) satisﬁes : (1)

• Q(y) (Λs ) = lim E0 [1Λs |St = y] t→∞ y2 2 (1) E [1Λs (y − Xs )|Ss = y] + E (1) [1Λs 1Ss
Q(a,y) (Λs ) := lim E0 [1Λs |Xt = a, St = y] t→∞

Moreover, these limiting laws satisfy : Q(a,y) =

y−a 1 Q(y) + (2y − a) (2y − a)

y

dz Q(z) . 0

Theorem 1.4 and Theorem 1.5 are found respectively in [RVY, II] and [RVY,III].

1.3.c) We close this section 1.3 with a remark, or rather a further Item D , commenting upon the martingales (Msϕ , s ≥ 0) in (1.5). These martingales have played a key role in the solution by Az´ema-Yor [AY1 ] of Skorokhod’s embedding problem for Brownian motion. Here, we simply wish to state that they may be considered as particular cases of the following balayage formula : Item D : (Balayage formula) If (Yt , t ≥ 0) is a continuous semi martingale, and gt = sup{s ≤ t, Ys = 0}, then, for every bounded {Ft } predictable process (kt , t ≥ 0), one has : kgt Yt = k0 Y0 +

t

kgs dYs

(D.1)

0

Applying this simple formula to : Yt = St − Xt , and kt = kgt = ϕ(St ) yields (1.5); likewise, applying the balayage formula to Yt = |Xt |, and kgt = kt = ϕ(L0t ), yields that :

1.4 Case II

45

ϕ(L0t )|Xt | +

∞

is a {Ft } martingale.

dy ϕ(y) L0t

For details, see, e.g., [RY∗ ], Chapter VI, Section 4.

t

ds q(Xs), with (Xs) : BM(Rd),

1.4 Case II : Γt = 0

d = 1, 2

1.4.a) We ﬁrst recall well-known limit theorems, which allow to understand the relevance of Theorem 1.3, in connection with the penalisation Theorems 1.6 and 1.9. Item E :

(Limit laws for Brownianadditive functionals) ∞ 1) (d = 1). Let f : R → R, Borel, satisfy : dx|f (x)| < ∞. Then, −∞ ∞ denoting f = dx f (x) : −∞

1 √ t

t

(law)

(d)

ds f (Xs ) −→ f L01 (X) = f |N | t→∞

0

(E.1)

where N is a standard Gaussian variable; Furthermore, if f and g belong to L1 (R, dx), there is the ratio ergodic Theorem : t ds f (Xs ) f a.s. 0 −→ (E.2) t t→∞ g ds g(Xs ) 0

2) (d = 2). Let f : R2 → R be Borel, locally bounded and satisfy : dx|f (x)| < ∞. Then, denoting X = X (1) + iX (2) , and f = 2 R dx1 dx2 f (x), one has : R2

1 log t

(1)

0

t

(law)

ds f (Xs ) −→ f L0T1 (X (1) )

where T1 = inf{s ≥ 0; Xs = 1}.

t→∞

46

1 Some Penalisations of the Wiener Measure

Furthermore, if g is another such function, then the ratio ergodic Theorem writes : t ds f (Xs ) f a.s. −→ 0 t t→∞ g ds g(Xs ) 0

For details, see, e.g., [IMK], [P1 ] and [PY2 ].

1.4.b) We may now state the following penalisation results. Theorem 1.6. (d = 1). Let q(da) be a positive measure on R, such that : (1 + |a|)q(da) < ∞. Then, R 1 (1) a Ex 1Λs exp − q(da)Lt 2 R 1) Qqx (Λs ) := lim t→∞ 1 (1) Ex exp − q(da)Lat 2 R exists and satisﬁes : Qqx (Λs ) = Ex(1) [1Λs Msq ] , where : Msq =

ϕq (Xs ) exp ϕq (x)

−

1 2

q(da)Las

and ϕq is the unique solution of the Sturm-Liouville equation : ϕ (dx) = ϕ(x)q(dx) with boundary conditions : 2 . lim ϕq (x) = − lim ϕq (x) = x→+∞ x→−∞ π 2) Under Qqx , the canonical process (Xt , t ≥ 0) is a diﬀusion process. It is distributed as the law of the solution to the following stochastic diﬀerential equation : t ϕq (Xs ) ds Xt = x + βt + 0 ϕq (Xs ) where (βt , t ≥ 0) is a Brownian motion starting from 0. The key point for the proof of Theorem 1.6 is the following estimation : Proposition 1.7. (d = 1).

Let q be a positive measure on R, such that : Ex(1)

1 exp − 2

R

Lat

R

(1 + |a|)q(da) < ∞. Then,

ϕq (x) q(da) ∼ √ t→∞ t

For example, if q(dy) = 1[a,b] (y) dy ; (a < b) :

1.4 Case II

47

ϕλ q (x) λ t √ Ex(1) exp − 1[a,b] (Xs )ds ∼ t→∞ 2 0 t ⎧ ⎪ 2 1 ⎪ + x − b if x > b ⎪ ⎪ π λ tanh(λ b−a ⎪ 2 ) ⎪ ⎪ ⎪ ⎨ cosh λ(x− a+b 2 2 ) with ϕλ q (x) = + x − b if x ∈ [a, b] π λ sinh(λ b−a ⎪ 2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 1 ⎪ ⎩ π λ tanh λ b−a + a − x if x < a ( 2 ) Proposition 1.7 admits the following consequence : Theorem 1.8. Let q be as in Theorem 1.6. Then : for any x ∈ R, there exists a positive and σ-ﬁnite measure ν x , supported by R+ , such that : √ (1) t Px Lat q(da) ∈ dy −→ ν x (dy) R

t→∞

in the following sense : for any function f continuous with compact support : √ (1) a t Px Lt q(da) −→ ν x (dy)f (y) f R

t→∞

For example, if q(da) = 1]−∞,0] (a)da (although the measure q(da) doesn’t satisfy the hypothesis of Theorem 1.8, this Theorem may be applied in this case), we obtain : t √ (1) t Px 1]−∞,0] (Xs )ds ∈ dy −→ ν x (dy) t→∞

0

with here : ν x (dy) =

2 π

(x− )2 dy 1 (x+ )δ0 (dy) + √ e− 2y 1y>0 √ y 2π

Theorem 1.9. (d = 2). Let q : R2 → R+ denote a measurable, bounded function such that : i) q has compact support, ii) q ≥ γ1B for some constant γ > 0 and a disc B. Then : 1) For any x ∈ R2 , lim (log

t→∞

t) Ex(2)

t 1 q(Xu )du exp − 2 0

exists.

48

1 Some Penalisations of the Wiener Measure

Let ϕq (x) denote this limit; then ϕq is the unique solution of the SturmLiouville equation : Δϕ(x) = q(x) ϕ(x) with boundary conditions r

∂ϕ (x) −→ 2 r→∞ ∂r

(r = |x|).

2) For any s ≥ 0 and any Λs ∈ Fs : 1 t (2) Ex 1Λs exp − q(Xu )du 2 Qqx (Λs ) := lim t0 t→∞ 1 (2) Ex exp − q(Xu )du 2 0 exists and satisﬁes :

ϕ

Qqx (Λs ) = Ex(2) (1Λs Ms q ) with Msq :=

ϕq (Xs ) exp ϕq (x)

1 s − q(Xu )du 2 0

3) Under Qqx , the canonical process (Xt , t ≥ 0) is a diﬀusion process. It is distributed as the law of the solution to the following 2-dimensional stochastic diﬀerential equation : t ∇ϕq (Xs ) ds Xt = x + βt + ϕq (Xs ) 0 where (βt , t ≥ 0) is a 2-dimensional Brownian motion starting from 0. Theorems 1.6 and 1.9 are found respectively in [RVY,I] and [RVY,VI].

1.5 Case III : Γt = L0t (Rt(δ) ), with R(δ) := BES(δ), 0<δ<2 1.5.a) In this section, we wish to demonstrate that penalisation results may as well be obtained for diﬀusions, instead of Brownian motion. We have chosen as a particular class of diﬀusions that of the BES(δ) processes R(δ) , for 0 < δ < 2. We recall that, for any δ > 0, R(δ) may be deﬁned via the solution of the stochastic diﬀerential equation : t 2 2 Rt = r + 2 Rs dβs + δ t 0

1.5 Case III

49

where we assume a priori that Rt ≡ R(t) ≥ 0, and where (βs , s ≥ 0) is a one-dimensional Brownian motion. Sometimes, it may be more convenient to use the parameter ν instead of δ; the two parameters being related by δ = 2(ν + 1). δ is often called the dimension whereas ν is the index. We shall write R(ν) (t) instead of R(δ) (t). The next item is designed in particular to give a natural deﬁnition of the local times of R(ν) , in the diﬀusion sense, rather than in a semi-martingale sense. Item F :

(Scale and Clock relations between Bessel processes)

• Let p, q be conjugate, i.e.

1 1 + = 1. Then, given (R(ν) (t), t ≥ 0) with p q

1 ν > − , there exists another Bessel process (R(νq) (u), u ≥ 0) s.t. : q t 1 1 − p2 (R(ν) (t)) q = R(νq) [R (s)] ds , t≥0 (F.1) (ν) q2 0 In other terms “a Bessel process is a power of another Bessel process time-changed ”, a “slogan” which is made precise in the above formula (F.1). For details, see, e.g., [RY∗ ], Chapter XI. • Throughout this section, we shall consider (R(−α) (t), t ≥ 0) with 0 < α < 1. Then, as a particular case of (F.1) there is the formula : t 4α2 (R(−α) (t))2α = R (R(−α) (s))4α−2 ds (F.2) 0

where (R(u), u ≥ 0) denotes reﬂecting Brownian motion, i.e. a 1 a Bessel process of index − . 2 • Thanks to formula (F.2), and the density of occupation formula, for the there is the following density of occupation reﬂecting Brownian motion R, formula for (R(−α) (t), t ≥ 0) : t 1 ∞ x 1−2α f (R(−α) (s))ds = Lt x f (x)dx (F.3) α 0 0 where, on the RHS of (F.3), the local times (Lxt ; x ≥ 0, t ≥ 0) may be chosen jointly continuous. As usual we denote (Lt , t ≥ 0) for (L0t , t ≥ 0), and this local time at 0 satisﬁes : ((R(−α) (t))2α − Lt , t ≥ 0) is a martingale (F.4) • (Lt , t ≥ 0) inherits from (R(−α) (t), t ≥ 0) the scaling property : law

(Lct , t ≥ 0) = (cα Lt , t ≥ 0)

(F.5)

50

1 Some Penalisations of the Wiener Measure

and its right continuous inverse (τ := inf{t; Lt > }, > 0), is a stable (α) subordinator. For a concise presentation of local times for diﬀusions, we recommend the text part of Borodin-Salminen [BS].

1.5.b) We now state a theorem which, in the context of R = R(−α) is comparable to Theorem 1.4 in the context of Brownian motion, but where we have replaced (St , t ≥ 0) by (Lt , t ≥ 0). In the next theorem, and up to the end of this Section, Ω is the space of continuous functions from R+ to R+ . We denote as (Rt , t ≥ 0) the coordinate process on Ω, by (Ft , t ≥ 0) its natural ﬁltration and F∞ = ∨ Fs . The s≥0

probability P0 on (Ω, F∞ ) is such that, under P0 , (Rt , t ≥ 0) is a Bessel process of index (−α), i.e. of dimension δ = 2(1 − α), started from 0. In this (y) (x,y) set up, the probabilities Qϕ deﬁned below are probabilities on 0 , Q0 , Q0 (Ω, F∞ ). Theorem 1.10. 1) Let ϕ : R+ → R+ a probability density. Then, there exists a probability Qϕ 0 on (Ω, F∞ ) such that, for any s ≥ 0 and Λs ∈ Fs : lim

t→∞

E0 (1Λs ϕ(Lt )) := Qϕ 0 (Λs ) E0 (ϕ(Lt )) ϕ Qϕ 0 (Λs ) = E0 (1Λs Ms ) where Msϕ := ϕ(Ls )(R(−α) (s))2α + 1 − φ(Ls ) x with φ(x) := ϕ(y)dy (x ≥ 0)

(1.7)

0

2) Under Qϕ 0 , the canonical process (Xt , t ≥ 0) satisﬁes : a) The random variable L∞ is ﬁnite a.s. and admits ϕ as probability density b) Let g := sup{t ≥ 0; Rt = 0}. Then : Qϕ 0 (0 < g < ∞) = 1. c) The following path decomposition before and after g holds : i) the two processes (Rt , t ≤ g) and (Rg+t , t ≥ 0) are independent; ii) the process (Rg+t , t ≥ 0) is a BES (4 − δ) process, starting from 0; iii) Conditionally on L∞ = , the process (Rt , t ≤ g) is distributed as a BES (δ) process, considered up to the ﬁrst hitting time of by (Lt ), i.e. : up to τ = inf{t, Lt > }. Remark 1.11. Note that the martingale (1.7) may be obtained as an ap1 plication of the balayage formula (see Item D ). In particular, if α = (and 2 then R(−α) is the absolute value of a Brownian motion X), (1.7) becomes : Msϕ = ϕ(Ls )|Xs | + 1 − φ(Ls ) It is the last formula of Item D .

1.5 Case III

51

1.5.c) Pursuing the parallel with the Brownian case, we now present a theorem which is the analog, in the present context, of the result concerning the “long Brownian bridges” (see Theorem 1.5 above). Theorem 1.12. We keep the notation in Theorem 1.10. 1) The probability measure Qϕ 0 on the canonical space may be disintegrated as : ∞ (y) = Q0 ϕ(y)dy Qϕ 0 0

where

(y) Q0

may be deﬁned as : (y)

Q0 (Λs ) = lim P0 (Λs |Lt = y) t→∞

and satisﬁes : (y)

Q0 (Λs ) = pLs (y)E0 (1Λs Rs2α |Ls = y) + E0 (1Λs 1Ls
Qϕ 0 (Λ|L∞ = y) = Q0 (Λ)

(Λ ∈ F∞ )

and the law of Qϕ 0 given L∞ = y does not depend on ϕ. 2) More generally, for every s ≥ 0 and Λs ∈ Fs , and every x, y ≥ 0 : (x,y)

Q0

(Λs ) := lim P0 (Λs |Rt = x, Lt = y)

exists

t→∞

and satisﬁes : (x,y)

Q0

(Λs ) =

x pL (y)E0 (1Λs Rs2α |Ls = y) x + y x1−2α s 1 + E0 [1Λs 1Ls
In other terms : (x,y)

Q0

(Λs ) =

x x1−2α (y) Q0 + 1−2α x+yx x + y x1−2α

0

y

(z)

Q0 dz .

Theorem 1.12 may be found in [RVY,V]. 1 For example, in the case α = , the last formula of Theorem 1.12 writes : 2 y x 1 (x,y) (y) (z) Q0 + Q0 = Q0 dz . (1.8) x+y x+y 0

52

1 Some Penalisations of the Wiener Measure

This formula (1.8) is precisely the last formula of Theorem 1.5, via L´evy’s equivalence Theorem (see Item B ).

1.6 Case IV : Γt = Σt(),() where Σt := sup (u − gu), u≤gt

or Σt := sup(u − gu), or Σt := sup (u − gu) u≤t

u≤dt

(with gt = sup{u ≤ t; Xu = 0} and dt = inf{u ≥ t; Xu = 0}). Here, the process (Xt , t ≥ 0) is a one dimensional Brownian motion.

1.6.a) In order to have good means of study of this Case IV, which is relative to diﬀerent suprema associated with the age process of excursions Au := u − gu , two Brownian Items are of some importance : the ﬁrst one yields the conditional law of {Xgt +u ; u ≤ t − gt } given Fgt , whereas the second one is a short Introduction to Brownian Excursion Theory. Item G: (Brownian meander) Let T be a {Fgt } stopping time such that : P (XT = 0) = 0. Then, the σ-ﬁeld FgT , the variable sgn(XT ), and the process : 1 |XgT +u(T −gT ) |, u ≤ 1 mu = √ T − gT are independent; the distribution of the process (mu , u ≤ 1) does not depend on T ; (mu , u ≤ 1) is called the Brownian meander. (law) √ m2 In particular : P (m1 ∈ dm) = m e− 2 dm i.e. : m1 = 2e, with e an exponential variable with parameter 1, and for TaA = inf{u : u − gu = a}, FgT A and the process {XgT A +u , u ≤ a} are independent, and sgn(XTaA ) is a a a symmetric Bernoulli variable. Imhof ’s relation between the laws of (mu , u ≤ 1) and of (Ru , u ≤ 1), the 3-dimensional BES process, is quite useful : % & 2 m1 E[F (R)] = E F (m) (G.1) π or equivalently :

π 1 E[F (m)] = E F (R) 2 R1

In order to memorize easily about the constants some “tips” :

(G.2) 2 and π

π , here are 2

1.6 Case IV

53

i) concerning (G.1), one has : √ √ E(m1 ) = E( 2e) = 2Γ which follows from : Γ

1

=

π 3 = 2 2

√ π;

ii) concerning (G.2), again one may hesitate whether E R11 equals π2 or 1/2 !π 1 1 1 ≤ E . In fact, since : E ≡ 1, the “choice” E 2 R1 R1 = R1 2 2 π (< 1) is the good one!! 2

See, e.g. [RY∗ ], Chap. XII, Section 3, and Exercise (4.18). Item H: (A brief sketch of Brownian excursion theory) Let (Lt , t ≥ 0) denote the Brownian local time at 0 (as deﬁned in Item A ), and τ = inf{t : Lt > }, ≥ 0. Then, the process {e : ω −→ {Xτ− +u , u ≤ τ − τ− }, ≥ 0} is a {Fτ ; ≥ 0} Poisson point process taking values in the space of excursions E = {w : R+ → R; w(0) = 0, ∃ ζ(w) ≥ 0 : w(u) > 0, for 0 < u < ζ(w) and w(u) = 0, ∀ u ≥ ζ(w)}. This means simply that, for any set S measurable with respect to the (nat' ural) σ-ﬁeld on E, if the process NS = u≤ 1(eu ∈S) is ﬁnite valued, then it is a {Fτ } Poisson process. As such, it admits a parameter, denoted as n(S) such that : {NS − n(S), ≥ 0} is a {Fτ } martingale. n : S −→ n(S) is a σ-ﬁnite measure, called Itˆo’s excursion measure, which admits several powerful descriptions, mainly due to D. Williams; see e.g., Chapter XII of [RY∗ ] for a detailed discussion of these descriptions.

1.6.b) We now consider penalisation of Brownian motion with the weight (x) process : Γt := 1(Σt ≤x) , for some x > 0 ﬁxed. The following Proposition 1.13, which is crucial to establish the corresponding penalisation result (see, Theorem 1.14 below), may be obtained via the Tauberian Theorem, by using either Item H or Item G , or a combination of them. !x Proposition 1.13. For every x ≥ 0, P (Σt ≤ x) ∼ t t→∞

We simply give a hint about the use of Item G towards the obtention of Proposition 1.13 : with this Item, it is not diﬃcult to show that, if Sβ denotes an exponential time with parameter β > 0, independent from (Xt , t ≥ 0), then, using the notation TxA = inf{t : t − gt = x}, one obtains : √ E[sinh( 2β|XTxA |)] √ P (ΣSβ ≤ x) = E[cosh( 2β|XTxA |)]

54

1 Some Penalisations of the Wiener Measure

∼

!

β→0

2β E[|XTxA |] =

!

2βx E[m1 ] =

!

πβx

which, thanks to Item G , and which, via the Tauberian Theorem, yields Proposition 1.13.

1.6.c) We may now state the corresponding penalisation result. Ω denotes here the canonical space C([0, ∞[→ R). Theorem 1.14. 1) For every s > 0, and Λs ∈ Fs , Qx (Λs ) := lim

t→∞

E[1Λs 1(Σt ≤x) ] P (Σt ≤ x)

exists, and deﬁnes a probability on (Ω, F∞ ). 2) Q satisﬁes : Qx (Λs ) = E[1Λs Ms ],

(s 1(Σ ≤x) and : M (s = |Xs | 2 + Φ with : Ms = M s πx ∞ u2 2 with : Φ(y) := e− 2 du and : As := s − gs π y

|X | √ s x − As

1As ≤x

(Ms , s ≥ 0) is a positive martingale. Moreover, under Qx , the canonical process (Xt , t ≥ 0) satisﬁes : a) Σ∞ is ﬁnite a.s., and, in fact, Σx∞ is uniformly distributed on [0, 1]; b) The process {Au ≡ u − gu , u ≤ TyA }, for y < x, is identically distributed under P and Qx ; moreover, {Au , u ≤ TyA } and XTyA are independent under P and Qx ; c) Under Qx , if g = sup{t : Xt = 0}, then : i) the processes (Xu , u ≤ g) and (Xg+u , u ≥ 0) are independent; ii) (|Xg+u |, u ≥ 0) is a BES(3) process; πx and conditionally on iii) L∞ (= Lg ) is an exponential r.v. with mean 2 L∞ = , the process (Xu , u ≤ g) is a Brownian motion B, stopped at τ = inf{t : Lt (B) > }.

1.6.d) We now consider penalisation of Brownian motion with the weight process Γt = 1(Σt ≤x) for some x > 0, x ﬁxed. The analogue in this situation of Proposition 1.13 is : Proposition 1.15. 1) Let, for λ ≥ 0, −λ

θ(λ) := e

λ 0

1

eλz √ dz = 2e−λ λΦ z

1 3 , ,λ 2 2

1.6 Case IV

55

where Φ 12 , 32 , • is the conﬂuent hypergeometric function of index 12 , 32 (see [L], p. 266, formula 9.11.1). Then, the function θ(λ) = 1 − θ(λ) has a ﬁrst zero λ0 > 0. 2) For any x > 0, there exists a constant C(x) such that : P0 (Σt < x) ∼ C(x)e−

λ0 t x

t→∞

.

We observe that the behavior of P (Σt < x) (see Proposition 1.13) and of P x) as t → ∞ are strongly diﬀerent : the ﬁrst one is “polynomial” (Σt <

∼ √Ct and the second one is exponential. In fact, we do not know how to prove completely point 2) of this Proposition 1.15. A proof of this point 2) would hinge, after using the inverse of the MellinFourier transform, on the fact that the function y → 1 − θ(λ0 + iy) has no other zero than y = 0, which we are unable to show (see, e.g. [RVY,VII], Section 4). Assuming the validity of this point 2) of Proposition 1.15 as a conjecture, we may now establish the following penalisation result: Theorem 1.16. Let x > 0, x ﬁxed and λ0 > 0 deﬁned in Proposition 1.15. There exists a probability Q x on (Ω, F∞ ) such that, for any s ≥ 0 and Λs ∈ Fs : E(1Λs 1Σt ≤x ) = Q x (Λs ) lim t→∞ E(1Σt ≤x ) 2) Q x satisﬁes : Q x (Λs ) = E(1Λs Ms ) with : Ms = e

λ0 s x

1Σs ≤x

x−As

0

λ0 u |Xs |2 |X | √ s e− 2u + 2 du 2πu3

(Ms , s ≥ 0) is a (Fs , P ) positive martingale. 3) Under Q x , the canonical process (Xt , t ≥ 0) is such that : a) b)

Σ∞ = x a.s. TxΣ = inf{t; Σt = x} = ∞ a.s.

Remark 1.17. The penalisation of Brownian motion with the weight process Γt = 1(Σt ≤x) , (x > 0, x ﬁxed) gives the same result as the penalisation by Γt . More precisely, there exists a probability Q x on (Ω, F∞ ) such that, for any s ≥ 0 and Λs ∈ Fs : lim

t→∞

and :

E(1Λs 1Σt ≤x ) = Q x (Λs ) P (Σt ≤ x)

Q x (Λs ) = E(1Λs Ms ),

with

Ms = Ms

and so

Q x = Q x .

Theorems 1.14 and 1.16 may be found in [RVY,VII].

56

1 Some Penalisations of the Wiener Measure

1.7 Case V : Γt = sup θs (d = 2) or Γt = 1TC >t exp ts≤t ds where Ht = 2 0 Rs

γ2 Ht 2

1.7.a) A fundamental property of planar Brownian motion Xt = Xt(1) +iXt(2)

(here R2 is identiﬁed to C) is its conformal invariance, discovered by P. L´evy in 1943 : if f ∈ H(C), i.e. : f is an entire holomorphic function, then ) t f (Xt ) = X |f (Xs )|2 ds ,

(1.9)

0

)u , u ≥ 0) is another planar Brownian motion; this easily yields that where (X ) and that formula (1.9) extends to meromorphic points are polar for X (or X!), functions f such that X0 does not belong to the set of singular points of f . Assuming that X0 = x = 0, we may write : t dXs Xt = x exp , (1.10) 0 Xs which yields the integral formula : θt = θ0 + Im 0

t

dXs Xs

, t ≥ 0,

(1.11)

for (θt , t ≥ 0) a continuous determination of the argument of X around 0. We also call (θt , t ≥ 0) the winding process of X around 0. As a consequence of (1.10) and (1.11), we obtain the following skew-product decomposition of Xt : if x = 0, we may write : Xt = Rt eiβHt ,

with

θt = βHt

and i) (βu , u ≥ 0) and (Rt , t ≥ 0) are two independent processes ii) (Rt = |Xt |, t ≥ 0) is a BES(2) process t ds iii) Ht = 2 0 Rs iv) (βu , u ≥ 0) is a one dimensional Brownian process. The following Item I completes Item E , point 2), d = 2.

(1.12)

1.7 Case V

57

Item I : (Limit Theorems for planar Brownian windings) The following random vector : t 4 2 2 θ H , f (X )ds , t s t (log t)2 (log t) 0 (log t)

t

ds and where f : C → R is a locally bounded and 2 0 |Xs | integrable function, converges in law as t → ∞ towards : f 0 LT1 (β) , γT1 (β) T1 (β) , (I.1) 2π where Ht =

Here, β and γ are two independent one dimensional Brownian motions and T1 (β) = inf{s; βs = 1} and f = f (x)dx1 dx2 . We note, in particular, that the statement entails a fortiori the celebrated result due to Spitzer ([S], 1958) : 2 (law) θt −→ C1 = γT1 (β) (I.2) (log t) t→∞ where, on the RHS, C1 is a standard Cauchy variable. See, e.g., [RY∗ ], Chap. X, Section 4.

1.7.b) Penalisation of Brownian motion (d = 2) with the weight process Γt := ϕ(Stθ ) = ϕ(sup θ(s)) . s≤t

We shall now apply Theorem 1.3, with k = 1, ht =

1 log t ,

(1)

Γt

= Stθ = sup θ(s) s≤t

and, thanks to (I.1), we get : gt (x) =

2 E π

1 1 2

Ht

exp −

x2 2Ht

This yields : Proposition 1.18. For every integrable function ψ : R+ → R+ , one has : 4 ∞ ψ(x)dx lim (log t)E(ψ(Stθ )) = t→∞ π 0 We may now state the following penalisation result : Theorem 1.19. Let ϕ : R+ → R+ be a probability density and φ(x) := x ϕ(y)dy (x ≥ 0). Then : 0

58

1 Some Penalisations of the Wiener Measure

1) Let x = 0. There exists a probability Qϕ x on (Ω, F∞ ) such that, for any s ≥ 0 and Λs ∈ Fs : Ex (1Λs ϕ(Stθ )) := Qϕ x (Λs ) t→∞ Ex (ϕ(Stθ )) lim

and :

ϕ where : Qϕ x (Λs ) := Ex (1Λs Ms ) 1 [ϕ(Ssθ )(Ssθ − θs ) + 1 − φ(Ssθ )] Msϕ := 1 − φ(θ0 )

(1.13)

where θ0 is the starting point of (θt , t ≥ 0). Moreover (Msϕ , s ≥ 0) is a positive martingale. 2) Under Qϕ x , the canonical process is such that : θ a) The random variable S∞ is ﬁnite a.s. and admits ϕ as its probability density θ θ } = sup{s ≥ 0, θs = S∞ }. b) Let g˜ = inf{s ≥ 0, Ssθ = S∞ ϕ Then, Qx (0 < g˜ < ∞) = 1 c) The process (Xt , t ≥ 0) admits the skew product representation

Xt = Rt exp (iθt ) = Rt exp (iβHt ) where : i) Rt = |Xt | is a 2-dimensional Bessel process, independent from the process (βs , s ≥ 0) ii) Let (Au , u ≥ 0) the inverse of (Ht , t ≥ 0), i.e. Au = inf{t; Ht > u} and deﬁne g = Ag˜ . Then : • (βs , s ≤ g) and (βg − βg+s , s ≥ 0) are independent • (βg − βg+s , s ≥ 0) is a 3-dimensional Bessel process θ = y, (βs , s ≤ g) is a Brownian motion consid• Conditionally on S∞ ered up to the ﬁrst time when it reaches y. It is interesting to compare Theorem 1.19 with the above Theorem 1.4. The ﬁrst point of Theorem 1.19 is an easy consequence of the following estimates : 4 (b − a) t→∞ π 4 • lim (log t)Ex (ϕ(a + Stθ )1Stθ >b−a ) = (1 − φ(b)) t→∞ π • lim (log t)Px (Stθ < b − a) =

These estimates are themselves direct consequences of Proposition 1.18 by choosing as function ψ, respectively ψ(x) = 1[0,b−a] (x) and ψ(x) = ϕ(a + x) 1[b−a,∞[ (x).

1.7 Case V

59

1.7.c) Penalisation of Brownian motion (d ≥ 2) with the weight process Γt = 1TC >t exp

γ

2

Ht + ρ R t

(γ ∈ R, ρ ≥ 0)

In this subsection, we consider d ≥ 2 and C is a cone in Rd , with basis O, a connected, regular set of Sd−1 , the unit sphere of Rd , C = {λ • o; λ ≥ 0, o ∈ O} The process (Γt , t ≥ 0) is then taken to be Γt = 1TC >t exp

γ

2 Ht

+ ρ Rt with :

TC = inf{t; Xt ∈ / C} t ds . the exit time of C, and Ht = 2 0 |Xs | A key tool in this case is the following : Item J : (Skew-product decomposition for BM (Rd )) (Xt , t ≥ 0), a d-dimensional Brownian motion, with X0 = x = 0, may be written as : Xt = Rt ΘHt t ≥ 0 (J.1) where : i) (Rt = |Xt |, t ≥ 0) is a Bessel process of dimension d, i.e. with index μ = d2 − 1 ii) (Θu , u ≥ 0) is a standard Brownian motion in Sd−1 , i.e. a diﬀusion S , with Δ S the Laplace-Beltrami operator on Sd−1 associated to 12 Δ d−1 d−1 iii) (Θu , u ≥ 0) and (Rt , t ≥ 0) are two independent processes t t ds ds iv) Ht = = 2 2 0 Rs 0 |Xs | Formula (J.1) may be considered as a probabilistic expression of the change of variables formula from cartesian to polar coordinates. The formula (1.12) is a particular case of (J.1) when d = 2. For details, see, e.g., [IMK]. Let now 0 < λ21 < λ22 ≤ λ23 ... ≤ λ2n < ...; ϕ1 , ϕ2 , ...ϕn , ... a spectral decomposi in O, associated with the Dirichlet problem : tion of Δ ϕn = −λ2 ϕn Δ n ϕn : O → R, ϕn = 0 on ∂ O, ϕn is C ∞ in O (ϕn , n ≥ 1) is an orthonormal basis of L2 (O)(for the Riemannian measure on O) iv) ϕ1 > 0 on O

i) ii) iii)

With the help of Item J and this spectral decomposition, we obtain the density of the process (Θu , u ≥ 0) killed when it reaches the boundary of O :

60

1 Some Penalisations of the Wiener Measure

p˜O (u, a, b) =

∞

e−

λ2n 2

u

ϕn (a)ϕn (b)

(see [BGM])

n=1

and so, because of the identities : (TC > t) = (TOΘH > t) , (TC > t) = (HTC > Ht ) , HTC = TOΘ = inf{u; Θu ∈ / O} , 2 λ1 (d) Pa (TC > t) ∼ Er exp − Ht , with r = |a| t→∞ 2

(1.14)

(d)

and Pr denotes the distribution of the BES(d) process, starting from r > 0. 2 In order to be able to state the penalisation result for Γt = 1TC >t exp γ2 Ht , we have to estimate Pa (TC > t) as t → ∞. We need the following Item K : Item K : (Local absolute continuity between the laws of the BES processes) (ν) If Pr denotes the law of the BES process (Rt , t ≥ 0) starting from r > 0, with dimension δ = 2(1 + ν), and Rt := σ{Rs , s ≤ t}, then : ν 2 t Rt ds ν (ν) (0) Pr|Rt = exp − •Pr|R , t r 2 0 Rs2 (ν)

(when ν < 0, the LHS should be replaced by Pr|Rt ∩(t t) as t → ∞, which is the main tool for our penalisation result (see Theorem 1.21 below). Proposition 1.20. We have, for a = r • θ ∈ C : 2 α r 1 Pa (TC > t) ∼ k ϕ1 (θ) t→∞ 2 tα with : α=

−μ +

Γ α + d2 μ2 + λ21 , k=

ϕ1 (θ)dθ 2 Γ 2α + d2 O

!

(see R. Banuelos and R. Smits, [BSm]). Theorem 1.21. Let x = 0 and γ ∈ R such that μ2 ≥ γ − λ21 , and ρ ≥ 0. ! 2 Deﬁne ν := μ + λ21 − γ. Then : 1) For every s ≥ 0 and Λs ∈ Fs , the limit as t → ∞ of :

Ex 1Λs 1TC >t exp γ2 Ht + ρ Rt

exists. Ex 1TC >t exp γ2 Ht + ρ Rt

1.7 Case V

61

This limit equals :

(Λs ) := Ex (1Λs MsC,γ,ρ ) QC,γ,ρ x

where

2 γ ρ := k exp − s + Hs (1.15) ϕ1 (ΘHs )Rs−μ Iν (ρ Rs ) 2 2 −1 x |x|−μ Iν (ρ |x|) (with k = ϕ1 |x| , and Iν denotes the modiﬁed Bessel MsC,γ,ρ

function with index ν). (MsC,γ,ρ , s ≥ 0) is a positive martingale. the canonical process (Xt , t ≥ 0) satisﬁes : 2) Under QC,γ,ρ x i) QC,γ,ρ (TC = ∞) = 1 x ii) (Xt , t ≥ 0) admits the skew-product decomposition Xt = Rt ΘHt

where :

• The two processes (Rt , t ≥ 0) and (Θu , u ≥ 0) are independent and t ds Ht = 2 R 0 s • (Rt , t ≥ 0) is the “Bessel process with drift” whose generator is given by : 1 + 2ν ρ Iν+1 (ρ r) 1 + LR : f → LR f (r) = f (r) + f (r) 2 r Iν (ρ r) • (Θu , u ≥ 0) is a diﬀusion taking values in O, with generator : LΘ : f → LΘ f (θ) =

∇ϕ1 1 Δf (θ) + (θ)•∇f (θ) 2 ϕ1

where the scalar product and the gradient are taken in the sense of Riemannian metric on Sd−1 . In the particular case γ = ρ = 0, Theorem 1.21 may be applied. The QC,0,0 process is then the well known Brownian motion conditioned to live in the cone! C. In particular, (Rt = |Xt |, t ≥ 0) is then a Bessel process with index ν = μ2 + λ21 . The martingale {MsC,0,0 } derived from (1.15) is : −1 x MsC,0,0 = k ϕ1 (θHs ) Rsν−μ , with k = ϕ1 rν−μ . r It is the product of the two martingales : 2 2 λ1 λ1 ν−μ Hs and Rs exp − Hs ϕ1 (θHs ) exp 2 2 in agreement with the martingale densities found in Item K . Theorems 1.19 and 1.21 may be found in [RVY,VI].

62

1 Some Penalisations of the Wiener Measure

1.8 Concluding Remarks 1.8.a) In this survey, we have presented the most basic, and, we hope, interesting, results to be found in our series of papers [RVY,...] about Brownian penalisations. For a number of developments, other cases, etc..., we refer the reader to the following papers; here is a very brief sketch of their contents: • [RVY,I]: Penalisation of the 1-dimensional Brownian motion by (Γt = exp(−

t

q(Xs )ds), t ≥ 0) with q ≥ 0

0

• [CM]: Penalisation of the d-dimensional Brownian motion by (Γt = exp(β

t

q(Xs )ds), t ≥ 0) with β > 0, q ≥ 0

0

• [RY,IX]: A local limit theorem for Brownian additive functionals; detailed t study of the penalisation by a function of At := 0 1Xs <0 ds; penalisation t of “long Brownian bridges” by (Γt = exp(− 0 q(Xs )ds), t ≥ 0) • [RVY, II]: Penalisation of 1-dimensional Brownian motion by a function of its one-sided supremum, its local time at 0 and so on... • [Deb]: Penalisation of symmetric random walk Z-valued by a function of its one-sided supremum, its local time at 0 and so on... • [RVY,V]: Penalisation of a recurrent Bessel process by a function of its local time at 0 • [N1 ]: Penalisation of Walsh’s Brownian motion on rays (alias: the Brownian spider) by a function of its local time at 0 • [Pro]: Penalisation of a linear diﬀusion by an exponential of its local time at 0 • [RVY,III]: Penalisation of “long Brownian bridges” • [RVY,IV]: Extension of the Pitman and Ray-Knight theorems to processes obtained by penalisation • [RVY,X]: Penalisation of 1-dimensional Brownian motion by a function of its one-sided maximum and its amplitude; application to a Skorokhod type embedding theorem • [RY,VIII]: To (Xt , t ≥ 0), a 1-dimensional Brownian motion, we associate St = sups≤t Xs , gt = sup{s ≤ t, Xs = 0}, dt = inf{s > t; Xs = 0}. Then, we penalise X by a function of Sgt , resp. Sdt • [RVY,VI]: Penalisation of 2-dimensional Brownian motion • [RVY,VII]: Penalisation of 1-dimensional Brownian motion by a function of the length of its longest excursion • [DG]: Penalisation of birth and death processes by a function of the local time at 0

Bibliography

63

1.8.b) We consider this work as a meaningful test before we study more complicated penalisations which may be considered in relation with the Domb-Joyce and Edwards models [E]; see, in particular, the remarkable thesis Monograph of van der Hofstadt [H], as well as Westwater’s papers [W1 ], [W2 ], [W3 ]. As an appetizer for future research, and in view of our penalisation studies, the Edwards-Westwater program may be phrased as follows : consider, for dimensions d = 1, 2, 3, the Wiener measure W (d) penalised with the weight process : t t Γt = exp −β ds du δ(Xs − Xu ) 0

0

where β > 0 is the “strength of self-repellence”, and δ is the Dirac function (β) at 0; the corresponding probabilities Qt are easily deﬁned in terms of Brownian local times for d = 1; for d = 2, their existence, and Radon-Nikodym equivalence with W (2) has been shown thanks to Varadhan’s renormalisation (β) result (see, e.g. [LG1 ] and [LG2 ]); for d = 3, the existence of Qt is due to Westwater (see the three references [W1 ], [W2 ], [W3 ] and [Bo]); these probabilities are singular with respect to W (3) . A question which remains largely open is : (β) what can be said of Qt , as t → ∞ ?

Bibliography [AY1 ] [Bo] [BGM] [BS] [BSm] [CM] [Deb] [DG] [E] [H] [IMK]

J. Az´ema and M. Yor. Une solution simple au probl`eme de Skorokhod. In S´eminaire de Probabilit´es, XIII (Univ. Strasbourg, Strasbourg, 1977/78), LNM 721, p. 90–115. Springer, Berlin, 1979. E. Bolthausen. On the construction of the three-dimensional polymer measure. PTRF, 97 (1-2), 81–101, 1993. M. Berger, P. Gauduchon, and E. Mazet. Le spectre d’une vari´et´e riemannienne. LNM 194, Springer, Berlin 1971. A. Borodin and P. Salminen. Handbook of Brownian motion. Facts and formulae. Birkh¨ auser. Second edition, 2002. R. Banuelos and R.G. Smits. Brownian motion in cones. PTRF, 108, p. 299– 319, 1997. M. Cranston and S. Molchanov. Analysis of a homopolymer model (private communication). Submitted, Dec. 2007. P. Debs. P´enalisations de la marche sym´etrique a` valeurs dans Z. Th`ese de l’Universit´e H. Poincar´e, 2007. P. Debs and M. Gradinaru. Penalisation for birth and death processes. Journal of Theoretical Prob., 21(3), 745–771, 2008. S.F. Edwards. The statistical mechanics of polymers with excluded volume. Proc. Phys. Sci 85, p. 613–614, 1965. R. van der Hofstadt. One-dimensional random polymers. Thesis Monograph, 1997. K. Itˆ o and H.P. McKean. Diﬀusion processes and their sample paths. Springer, 1965.

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1 Some Penalisations of the Wiener Measure

N.N. Lebedev. Special functions and their applications. Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication. [LG1 ] J.F. Le Gall. Sur le temps local d’intersection du mouvement brownien plan, et la m´ethode de renormalisation de Varadhan. S´em. Prob. XIX, LNM 1123, p. 314–331, Springer, 1985. [LG2 ] J.F. Le Gall. Some properties of planar Brownian motion. Ecole d’Et´e de Saint-Flour XX, 1990. LNM 1527, p. 112–234, Springer, 1992. [M] P.A. Meyer. Probabilit´es et potentiel. Publications de l’Institut de Math´ematique de l’Universit´e de Strasbourg, N◦ XIV. Actualit´es Scientiﬁques et Industrielles, N◦ 1318. Hermann, Paris, 1966. [N1 ] J. Najnudel. P´enalisations de l’araign´ee brownienne. Annales Inst. Fourier 57(4):1063–1093, 2007. [N2 ] J. Najnudel. Temps locaux et p´enalisations browniennes. Th`ese de l’Universit´e Paris VI, June 2007. [P] J. Pitman. One-dimensional Brownian motion and the three-dimensional Bessel process. Adv. App. Prob. 7(3):511–526, 1975. [Pro] C. Profeta. P´enalisation d’une diﬀusion lin´eaire par une exponentielle de son temps local en 0. Thesis in preparation, 2008. [PY1 ] J. Pitman and M.Yor. Asymptotic laws of planar Brownian motion. Ann. Probab., 14(3):733–779, 1986. [PY2 ] J. Pitman and M.Yor. Further asymptotic laws of planar Brownian motion. Ann. Probab., 17(3):965–1011, 1989. [R] L. Rogers. Characterizing all diﬀusions with the 2M-X property. Ann. Prob. 9(4):561–672, 1981. [RVY,I] B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I. Studia Sci. Math. Hungar., 43(2):171–246, 2006. [RVY, II] B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time. II. Studia Sci. Math. Hungar., 43(3):295–360, 2006. [RVY,III] B. Roynette, P. Vallois, and M. Yor. Limiting laws for long Brownian bridges perturbated by their one-sided maximum. III. Periodica Math. Hungar., 50 (1-2):247–280, 2005. [RVY,IV] B. Roynette, P. Vallois, and M. Yor. Some extensions of Pitman’s and RayKnight theorems for penalized Brownian motion and their local time. IV. Studia Sci. Math. Hungar., 44(4):469–516, 2007. [RVY,V] B. Roynette, P. Vallois, and M. Yor. Penalizing a BES (d) process 0 < d < 2) with a function of its local time. V. Studia Sci. Math. Hungar., 45(1):67–124, 2008. [RVY,VI] B. Roynette, P. Vallois, and M. Yor. Penalisations of multi-dimensional Brownian motion. VI. To appear in ESAIM P.S., 2009. [RVY,VII] B. Roynette, P. Vallois, and M. Yor. Brownian penalisations related to excursion lengths. VII. To appear in Annales de l’Inst. H. Poincar´e., 2009. [RY,VIII] B. Roynette and M. Yor. Ten penalisation results of Brownian motion involving its one-sided supremum until ﬁrst passage time VIII. Journal Funct. Anal., 255(9):2606–2640, 2008. [RY,IX] B. Roynette and M. Yor. Local limit theorems for Brownian additive functionals and penalisation of Brownian paths. IX. To appear in ESAIM P.S., 2009. [RVY,X] B. Roynette, P. Vallois, and M. Yor. Penalisation of Brownian motion with its maximum and minimum processes: some relation with Skorokhod embedding applications. To appear in Theory of Stoch. Proc., 2009.

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[W4 ]

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D. Revuz and M.Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999. F. Spitzer. Some theorems about 2-dimensional Brownian motion. Trans. Amer. Math. Soc., vol. 87, p. 187–197, 1958. J. Westwater. On Edwards model for long polymer chains. Comm. Math. Phys. 72, p. 131–174, 1980. J. Westwater. On Edwards model for long polymer chains, II. The selfconsistent potential. Comm. Math. Phys. 79, p. 53–73, 1981. J. Westwater. On Edwards model for long polymer chains. In Trends and developments in the Eighties (S. Albeverio and P. Blanchard, eds.). Bielefeld Encounters in Math. Phys. 4/5, World Scientiﬁc, Singapore, 1984. J. Westwater. On Edwards model for long polymer chains, III. Borel Summability. Comm. Math. Phys. 84(4):459–470, 1982.

Chapter 2

Feynman-Kac Penalisations for Brownian Motion

Abstract Among the various examples of penalisations of Wiener measure discussed in this Monograph, the ones which are obtained by putting a Feynman-Kac type weight with respect to Wiener measure, up to time t, are undoubtedly quite natural, and such transforms of Wiener measure have a long history. In this Chapter, we show that the asymptotic behavior of all these penalised measures may be expressed in terms of ≥0, σ-ﬁnite measures Wx on C(R+ → R+ ). If Λx denotes the image of Wx by the local times application, then this positive, σ-ﬁnite measure Λx on the space C(R → R+ ) characterizes the asymptotic behavior, as t → ∞, of the law of the Brownian local times (Lyt , y ∈ R). These measures Λx , x ∈ R, are described in detail. Keywords Feynman-Kac weights · σ-ﬁnite asymptotic measures

2.1 Introduction 1) Let (Ω, (Xt , Ft )t≥0 , F∞ , Wx (x ∈ R)) denote the canonical Brownian motion. Ω = C(R+ → R) is the space of continuous functions from R+ to R, (Xt , t ≥ 0) denotes the coordinate process on this space, (Ft , t ≥ 0) its natural ﬁltration and F∞ = ∨ Ft . (Wx ; x ∈ R) denotes the set of Wiener t≥0

measures on F∞ such that Wx (X0 = x) = 1. When x = 0, we simply note W instead of W0 . (Lxt ; t ≥ 0, x ∈ R) denotes the jointly continuous family of the local times of Brownian motion. 2) The present work is closely related to our paper [RVY,I], as shall be explained in 3) below. The main novelty in this Chapter is the existence, for every x ∈ R, of a = C(R → R+ ), which is closely related to the σ-ﬁnite ≥ 0 measure Λx , on Ω “spatial asymptotic behavior, as t → ∞, of the Brownian local time L•t ”. This B. Roynette, M. Yor, Penalising Brownian Paths, Lecture Notes in Mathematics 1969, DOI 10.1007/978-3-540-89699-9 2, c Springer-Verlag Berlin Heidelberg 2009

67

68

2 Feynman-Kac Penalisations for Brownian Motion

family (Λx ; x ∈ R) of σ-ﬁnite measures allows to give a global description of the set of Feynman-Kac type penalised Wiener measures, as we shall show in Section 2.4 of this Chapter. 3) A summary of the results of [RVY,I] Let I (resp. I+ ) denote the set of Borel functions q : R → R+ (resp. : R+ → R) such that : 0< R

(1 + |x|)q(x)dx < ∞

(2.1)

Consequently, from the density of occupation formula :

t

q(Xs )ds = 0

Lxt q(x)dx

R

a.s.

(2.2)

and we shall use indiﬀerently either one or the other of the sides of (2.2). To any function q : R → R, we associate q+ its restriction to R+ , and q− the image by the application : x → −x of the restriction of q to R− , i.e. : (x ≥ 0)

q− (x) = q(−x)

(2.3)

For every function Y : R → R+ and q ∈ I, we denote :

∞

< Y, q >= −∞

Yy q(y)dy

(2.4)

and, for every function Y : R+ → R+ and q ∈ I+ , we denote : ∞ Yy q(y)dy < Y, q >= 0

The law of the square of the δ-dimensional Bessel process, started from a (δ) (a ≥ 0, δ ≥ 0) is denoted as Qa and this symbol is used either to denote (δ) − the probability or related expectations. Thus, ] : Qa [e the notation ∞

denotes the expectation of the r.v. exp −

Yy q(y)dy , where (Yy , y ≥ 0)

0

is the canonical δ dimensional Bessel process started from a, on C(R+ → R+ ). We now recall the main result of [RVY,I] : Theorem 2.1. Let q ∈ I. Consider the Wiener measures (Wx , x ∈ R) penalised with the help of the multiplicative functional : 1 t (q) Et := exp − q(Xs )ds , t ≥ 0 (2.5) 2 0

2.1 Introduction

69 (q)

that is :

Et

(q)

Wx,t :=

Wx

(2.6)

•

(q)

EWx (Et )

Then : 1) For every s ≥ 0 and Fs ∈ b(Fs ), the space of bounded Fs measurable r.v’s : ϕq (Xs ) (q) (q) Es Wx,t (Fs ) −→ Wx(q) (Fs ) := EWx Fs (2.7) t→∞ ϕq (x) where the function ϕq may be deﬁned in either of the following manners : i) ϕq is the unique solution of : ϕ = q •ϕ

2 , lim ϕ (x) = π x→+∞

lim ϕ (x) = − √ (q) ii) ϕq (x) = lim t Wx (Et )

such that :

x→−∞

2 π

(2.8) (2.9)

t→∞

2) One has : 1 ϕq (0) = √ 2π

∞

− − {Q(0) ]Q(2) ] a [e a [e

0 − − ]Q(0) ]}da +Q(2) a [e a [e

(and ϕq (x) is then obtained by replacing q by q(x + •)). ϕq (Xs ) (q) Es , s ≥ 0 is a (Wx , (Fs , s ≥ 0)) positive 3) Ms(q) := ϕq (x) martingale which converges a.s. to 0, as s → ∞.

(2.10)

(2.11)

(q)

4) Formula (2.7) induces a probability Wx on (Ω, F∞ ), with respect to which, the canonical process (Xt , t ≥ 0) solves the SDE : Xt = x + Bt + 0

t

ϕq (Xs )ds ϕq

(2.12) (q)

where, in (2.12), (Bt , t ≥ 0) denotes a ((Ft , t ≥ 0), Wx ) Brownian motion (q) starting from 0. In particular, under (Wx , x ∈ R), the process (Xt , t ≥ 0) is a transient diﬀusion. Remark. We note that the results of Theorem 2.1 are valid in fact for all positive Radon measures q(dx) such that 0 < (1 + |x|)q(dx) < ∞ (2.13) R

70

2 Feynman-Kac Penalisations for Brownian Motion

4) Here are two examples of functions q which we considered in [RVY,I] : i) q(x) =

λ2 2

1[a,b] (x)

(a < b, λ > 0)

⎧ ⎪ 2 1 ⎪ + x − b if x > b ⎪ ⎪ π λ tanh(λ b−a ⎪ 2 ) ⎪ ⎨ cosh(λ(x− a+b 2 2 )) ϕq (x) = if x ∈ [a, b] π λ sinh(λ b−a ⎪ 2 ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ π2 λ tanh 1λ b−a + a − x if x < a ( 2 ) ii) q(dx) =

λ2 2

(2.14)

δa (dx) (where δa denotes the Dirac measure at point a) 2 2 ϕq (x) = + |x − a| (2.15) π λ2

5) Although the statement of Theorem 2.1 is satisfactory, we wish here to consider it from a diﬀerent view point, for the following reasons : i) In [RVY,I] we have given a proof of points 1) and 2) of Theorem 2.1 which hinges on an elementary version of the Tauberian theorem by studying the ∞

asymptotic behavior, as λ → 0, of

e−λt Wx (Et )dt. Here, we shall give a (q)

0

direct proof of points 1) and 2) of Theorem 2.1, without using the Tauberian theorem and we shall obtain thus a more general version of Theorem 2.1. ii) In some sense, in [RVY,I], our study is “individual”, in that we ﬁxed a function q, without having much interest in the “dependence in q” of our results. Here, on the contrary, our study is more global and we exhibit a = C(R → R+ ) (x ∈ R). We family of positive σ-ﬁnite measures Λx on Ω will then draw from the existence of this family of measures (Λx , x ∈ R) several interesting consequences, among which : asymptotic behavior, when (q)

t → ∞ of the law of At

t

:=

q(Xs )ds, Ray-Knight type theorem for the 0

local times of the canonical process (Xt , t ≥ 0) under the probability family (q) (Wx , x ∈ R), “meta theorem of penalisation by a functional of the form Γ(L•t )”, where L•t = {Lyt , y ∈ R} and where Γ is an appropriate application = C(R → R+ ) to R+ , as t → ∞. from Ω 6) The plan of this Chapter 2 is the following : In Section 2.2, we recall some useful results about the solutions of the SturmLiouville equation : (q ∈ I+ ) (2.16) ϕ = qϕ and about the associated Brownian martingales. In Section 2.3, we give a direct proof (i.e. without using the Tauberian Theorem) of points 1) and 2) of Theorem 2.1.

2.1 Introduction

71

In Section 2.4, we obtain some absolute continuity relations (on the ﬁnal (q) σ-algebra F∞ ) between the diﬀerent probabilities Wx (q ∈ I) when q varies. = C(R → R+ ) some positive σ-ﬁnite meaThis allows us to construct on Ω sures Λx (x ∈ R). In Chapter 3, we shall obtain, in relation with another family of Brownian penalisations, some similar ≥0 and σ-ﬁnite measures; this is also reminiscent of Itˆo’s measure of Brownian excursions, or again of the L´evy-Khintchine measures M and N which allow to represent the prob(δ) abilities Qa (a ≥ 0, δ ≥ 0) (see [D-M,Y1 ], [P1 ], [PY1 ] or [PY2 ]) : ∞ (δ) ν(dy)Yy Qa exp − 0 = exp −(xM + δN) 1 − exp − ν(dy)Yy for every positive Radon measure ν. In Section 2.4 again, we deduce three important Corollaries from the existence of the measures Λx (x ∈ R) (q)

• the existence, for every q ∈ I, of σ-ﬁnite measures ν x , carried by R+ , which characterize the asymptotic behavior as t → ∞ of the distribution t (q) of the additive functional At := 0 q(Xs )ds, t ≥ 0 (see [RY,IX]) • a Ray-Knight type theorem for the local times (Ly∞ (X), y ∈ R) with (q) respect to the probabilities (Wx , x ∈ R) • a “meta-theorem” of penalisation by a functional of the form (Γt := taking values in R+ , Γ(L•t )), where Γ is an adequate application from Ω y • and Lt is the local time function i.e. : ((y) ≡ Lt , y ∈ R) ∈ Ω. In Section 2.5 we generalize Theorem 2.1 by replacing the penalisation func(q) (q ,q ,q ) tional Et by the more general functional Et 1 2 3 where q1 , q2 , q3 are three elements of I+ , and : gt t dt 1 (q1 ,q2 ,q3 ) ( q) := Et = exp − q1 (Xs )ds + q2 (Xs )ds + q3 (Xs )ds Et 2 gt 0 t (Xt , t ≥ 0) denotes here a reﬂecting Brownian motion and : gt = sup{s ≤ t, Xs = 0} , dt = inf{s ≥ t, Xs = 0} In Section 2.6, we establish a penalisation theorem with a functional of the form h(Lat 1 , Lat 2 ), with a1 = a2 . In order to facilitate the reading of this chapter, in conjunction with [RVY,I], we indicate some changes of notation from [RVY,I] : i) the potential function which we now denote by q, was denoted as V in [RVY,I]. This change is justiﬁed by the fact that in Chapter 3, the letter V indicates, in an essential manner, the lengths of various excursions;

72

2 Feynman-Kac Penalisations for Brownian Motion (q)

ii) the limiting measures are now denoted as Wx - instead of Qx as in [RVY,I] - this is justiﬁed by our desire to see clearly the dependency in q ∈ I, and also by the fact that {Q(δ) a } denotes the laws of squares of Bessel processes.

2.2 On the Solutions of Sturm-Liouville Equations and Associated Brownian Martingales 2.2.1 Notation They are, for the essentials, borrowed from [JY], ([RY∗ ], Chap. XI), and [D-M,Y1 ]. Let q : R+ → R+ (q ∈ I+ ), a Borel function such that : ∞ 0< (1 + x)q(x)dx < ∞ (2.17) 0

We denote by Φq and Ψq the two fundamental solutions deﬁned on R+ , of the Sturm-Liouville equation : ϕ = qϕ

(q ∈ I)

(2.16)

such that : i) Φq (0) = 1 , Φq is decreasing on R+ ii) Ψq (0) = 0 , Ψq (0) = 1 It is well known that :

j) Φq (∞) > 0 , since x jj) Ψq (x) = Φq (x) 0

Thus :

Ψq (x) x

∞

(2.18)

(1 + x)q(x)dx < ∞

0

dy Φ2q (y)

(2.19)

is a bounded function on R+ and : Ψq (x) 1 −→ x x→∞ Φq (∞)

(2.20)

We also note that Φq (0+) < 0. For x > 0, Doob’s optional stopping Theorem applied to the bounded mar(q) tingale Φq (Xt∧T0 )Et∧T0 , where T0 := inf{t; Xt = 0}, implies : Wx

1 exp − 2

T0

q(Xs )ds 0

= Φq (x)

(2.21)

2.2 On the Solutions of Sturm-Liouville Equations

(q)

2.2.2 The process Φq (Xt+ )Et exp −

Φq (0+) 2

73

L0t

,t ≥ 0

is a positive martingale with moments of all orders on [0, T ], for every T > 0. Consequently, Doob’s optional Theorem applied at time τ := inf{s ≥ 0; L0s > } (which bounds the above martingale) implies : Φ ) = exp (0+) (2.22) W0 (Eτ(q) 2 q or equivalently : 1 y Φ (0+) W0 exp − L q(y)dy = exp 2 R+ τ 2 q For q : R → R+ , a function which belongs to I, the independence of the two processes (Lyτ , y ≥ 0) and (Lyτ , y ≤ 0) implies, from (2.22) : (Φ ) = exp (0+) + Φ (0+)) (2.23) W0 (Eτ(q) q− 2 q+ 2.2.3 We come back to the situation where q is deﬁned on R+ . Denoting (3) of thelaws of the 3-dimensional by (Px , x ≥ 0) the family Bessel process Ψq (Rt ) 1 t ((Rt , Rt ), t ≥ 0), then exp − 2 0 q(Rs )ds , t ≥ 0 is a ((Rt , t ≥ 0), Rt (3)

Px ) bounded martingale (from

(2.20)) ∞ which,therefore, converges a.s. as t → ∞ towards Φq 1(∞) exp − 12 0 q(Rs )ds . Thus, from (2.20), we deduce : Ψq (x) 1 1 ∞ = Ex(3) exp − q(Rs )ds (x ≥ 0) (2.24) x Φq (∞) 2 0 (Note : From Jeulin’s Lemma (see [JY]), there is the closely related result : ∞ (3) 0 q(Rs )ds < ∞ P0 a.s. ⇔ q ∈ I+ .) We also recall the absolute continuity property : Xt∧T0 (3) •Wx (2.25) Px|F = |Ft t x which expresses the BES(3) process as Doob’s h-transform of Brownian mo(q) tion (see 0.4.5). From this property, we deduce that (Ψq (Xt∧T0 )Et , t ≥ 0) is, for every x ≥ 0, a Wx -martingale. 2.2.4 It is of interest to compare formulae (2.22) and (2.23) with the following ones (of L´evy-Khintchine type) :

74

2 Feynman-Kac Penalisations for Brownian Motion

Q(δ) x

1 exp − 2

∞

Yy q(y)dy

= (Φq (∞))δ/2 exp

0

x 2

Φq (0+)

(2.26)

(see [RY∗ ], Chap. XI, Th. 1.7, p. 444). 2.2.4.i) Taking δ = 0 and x = in (2.26), we recover the second Ray-Knight theorem : (0) under W0 , (Lyτ , y ≥ 0) is distributed as Q . 2.2.4ii) Likewise, taking δ = 2 and x = 0 in (2.26), we recover a third Ray-Knight theorem (see [D-M,Y1 ] or [W]) : (3)

(2)

under P0 , (Ly∞ , y ≥ 0) is distributed as Q0 . 2.2.5 Our aim is now to express, for every function q which is deﬁned on R and belongs to I, the function ϕq deﬁned by (2.8) in terms of the functions Φq± and Ψq± which were deﬁned in 2.2.1 above. More generally, for q ∈ I deﬁned on R, and α, β any two reals, we denote by the function which solves : ϕα,β q ϕ = qϕ

(q ∈ I)

lim ϕ (x) = α , lim ϕ (x) = β

x→−∞

x→+∞

(2.27)

Proposition 2.2. Denote by q+ the restriction of q to R+ and by q− the image by the application : x → −x of the restriction of q to R− (i.e. : q− (x) = q(−x) for x ≥ 0). Then : i) ϕα,β q (x)

=

AΦq− (−x) − αΦq− (∞)Ψq− (−x) if x < 0 , if x > 0 AΦq+ (x) + βΦq+ (∞)Ψq+ (x)

with A=

αΦq− (∞) − βΦq+ (∞) (Φq+ (0+) + Φq− (0+))

(2.28)

(2.29)

ii) In particular, the function ϕq deﬁned by (2.8), which corresponds to −α = β=

2 π

equals :

⎧ ⎨ Cq Φq (−x) + 2 Φq (∞)Ψq (−x) if x < 0 , − − π − ϕq (x) = ⎩ Cq Φq (x) + 2 Φq (∞)Ψq (x) if x > 0 + + + π with :

ϕq (0) = A := Cq = −

2 (Φq− (∞) + Φq+ (∞)) π (Φq+ (0+) + Φq− (0+))

(2.30)

2.3 A Direct Proof of Point 1) of Theorem 2.1

75

Proof of Proposition 2.2 Of course, it suﬃces to prove point 1) of this Proposition. From the deﬁnition of ϕα,β q , this function is a linear combination of the functions Φq± and Ψq± , satisﬁes ϕ = qϕ. Furthermore, and ϕα,β q α,β lim (ϕα,β q ) (x) = α , lim (ϕq ) (x) = β

x→−∞

x→+∞

since : Φq+ (+∞) = Φq− (+∞) = 0, from (2.18), and Ψq+ (+∞) = 1 Ψq− (+∞) = − Φq (+∞) , from (2.19). − Furthermore, it is clear again from (2.18), that :

1 Φq+ (+∞) ,

α,β ϕα,β q (0+ ) = ϕq (0− ) = A

and that, with the choice of A given by (2.29) : α,β (ϕα,β q ) (0+ ) = (ϕq ) (0− )

This proves Proposition 2.2. Remark 2.3. We note that, in Theorem 2.1, point 2) (i.e. : formula (2.10)) is a consequence of point 1) of the same Theorem 2.1 and of formula (2.26). Indeed, we have : ∞ 1 − 12 − 12 √ da Q(0) ]Q(2) ] a [e a [e 2π 0 (2) − 12 (0) − 12 ]Qa [e ] +Qa [e ∞ Φq− (0+) + Φq+ (0+) 1 = √ a (Φq+ (∞) + Φq− (∞)) da exp 2 2π 0 by application of (2.26) with δ = 0 and x = a, then with δ = 2 and x = a Φq+ (∞) + Φq− (∞) 2 = −√ = Cq = ϕq (0) (2.31) (Φ 2π q+ (0+) + Φq− (0+)) from (2.30)

2.3 A Direct Proof of Point 1) of Theorem 2.1 2.3.1 We recall that, from the above Remark 2.3, point 1) of Theorem 2.1 implies point 2) of the same Theorem. Furthermore, our aim here is to prove point 1) of Theorem 2.1 without using the Tauberian Theorem.

76

2 Feynman-Kac Penalisations for Brownian Motion

We shall prove that, if q ∈ I, then for every s ≥ 0 and every bounded functional F : & % (q) Et ϕq (Xs ) (q) E = W lim Wx F (Xu , u ≤ s) , u ≤ s) F (X (2.32) x u (q) t→∞ ϕq (x) s Wx (Et ) where the function ϕq which will appear in this manner from our estimates is precisely the function ϕq which has been deﬁned in point 2) of Proposition 2.2. On the other hand, since Brownian motion is invariant under space translation, it suﬃces to prove (2.32) for x = 0. t (q) (q) 2.3.2 We write, for s < t : Et = Es exp − 12 s q(Xh )dh , and we take the conditional expectation with respect to Fs . Then, the result (2.32) (with x = 0) will be established once we show that, as t → ∞ : √ 1 t q(Xh )dh → ϕq (0) = Cq (2.33) Dt := tW0 exp − 2 0 and √ 1 t Ns,t := tW0 exp − q(Xh )dh Fs → ϕq (Xs ) (2.34) 2 s 2.3.3 We begin by showing (2.33). We decompose the expectation which deﬁnes Dt with respect to the law of gt = sup{s ≤ t : Xs = 0} which admits as density : fgt (u) = Thus, we obtain : Dt =

√ 1 t π

0

t

1 1 ! 1[0,t] (u) π u(t − u)

!

du u(t − u)

I (q) (u)J (q) (t − u)

(2.35)

(2.36)

with

1 u I (u) := W0 exp − q(Xh )dh |Xu = 0 (2.37) 2 0 s s 1 (s) 1 1 (q) q+ (Xh )dh + exp − q− (Xh )dh exp − J (s) := M 2 2 0 2 0 (2.38) (q)

where M (s) denotes the law of the Brownian meander with duration s (see Chapter 1). We note that the half-sum which appears in (2.38) comes from the occurrence with probability 1/2 after gt of a “positive meander” and a “negative meander”. We examine (2.38). From Imhof’s relation (see Chapter 1) :

2.3 A Direct Proof of Point 1) of Theorem 2.1

77

%

J

(q)

√ s 1 (3) 1 s (s) := E0 exp − q+ (Xh )dh 2 2 0 CXs

1 + exp − 2 (3)

where C = E0

1 X1

=

0

√ & s q− (Xh )dh CXs

(2.39)

s

√ q± (Xh )dh and Xs / s are asymptotically in-

0

dependent; consequently : J

s

2 π.

As s → ∞, the variables

(q)

% 1 (3) 1 ∞ (s) −→ E0 exp − q+ (Xh )dh s→∞ 2 2 0

1 + exp − 2

&

∞

q− (Xh )dh

(2.40)

0

1 (Φq+ (∞) + Φq− (∞)) 2

=

from the relation (2.24) applied with x = 0, using also Ψq (0) = 1. Thus, we deduce from (2.36) and (2.40) that : √ 1 t t 1 1 u √ √ W0 exp − Dt ∼ du q(Xh )dh Xu = 0 × . . . t→∞ π 0 2 0 t−u u 1 (1) (2) . . . (Φq+ (∞) + Φq− (∞)) := Dt + Dt 2 with (1)

Dt = (2) Dt

and ψ(u) := W0

1 exp − 2

0

u

1 π

1 = π

t/2

0

t

t/2

√ t du √ √ ψ(u) t−u u

√ t du √ √ ψ(u) t−u u

1 (Φq+ (∞) + Φq− (∞)) q(Xh )dh Xu = 0 • 2

78

2 Feynman-Kac Penalisations for Brownian Motion

Let us assume for a moment that:

∞ 0

du √ ψ(u) u

< ∞. Then for u ≤ 2t :

* √ √ t t ψ(u) 2 ψ(u) 1 1 1 √ √ ψ(u) ≤ √ = √ π t−u u π t/2 u π u Hence, from the dominated convergence theorem, since (1)

√ √ t −→ t−u t→∞

1:

1 (Φq+ (∞) + Φq− (∞)) . . . 2π ∞ du 1 u √ W0 exp − ... q(Xh )dh Xu = 0 2 0 u 0

Dt −→

t→∞

On the other hand, from Lemma 3.2 of [RY,IX]: C(1 + |x|)(1 + |y|) (x−y)2 1 t Wx exp − e 2t q(Xh )dh Xt = y ≤ 2 0 1+t Hence, since here: x = y = 0, we have: ψ(u) ≤ (2) Dt

C 1+u

and:

√ t du 1 t √ √ ψ(u) = π t/2 t − u u √ 1 1 t dv √ √ ψ(tv) = π 1/2 1 − v v √ 1 √ 1 C t t dv dv C ! ! ≤ ≤ π 1/2 v(1 − v) 1 + tv π(1 + t/2) 1/2 v(1 − v) −→ 0

t→∞

We now prove that: 1 (Φq+ (∞) + Φq− (∞)) 2 Indeed:

∞ 0

du √ ψ(u)du = ϕq (0) u

∞ 1 1 √ ... (Φq+ (∞) + Φq− (∞)) 2 π u 0 u 1 . . . W0 exp − q(Xh )dh Xu = 0]du 2 0 ∞ 1 1 u = √ (Φq+ (∞) + Φq− (∞))W0 exp − q(Xh )dh dL0u 2 0 2π 0 du (since E(dL0u ) = √ ) 2πu

2.3 A Direct Proof of Point 1) of Theorem 2.1

79

∞ 1 1 τ = √ (Φq+ (∞) + Φq− (∞))W0 d exp − q(Xh )dh 2 0 2π 0 (after making the change of variable = L0u ) ∞ Φq+ (0+) + Φq− (0+) 1 exp = √ (Φq+ (∞) + Φq− (∞)) d 2 2π 0 (from (2.23)) 2 Φq+ (∞) + Φq− (∞) = Cq = ϕq (0) (from (2.30)) (2.41) =− π (Φq+ (0+) + Φq− (0+)) (we recall that Φq+ (0+) and Φq− (0+) are negative.) This ﬁnishes the proof of (2.33). 2.3.4 We now prove (2.34). We write :

(1)

(2)

Ns,t := Ns,t + Ns,t with : (1) Ns,t

√ 1 t := tW0 1(ds >t) exp − q(Xh )dh Fs 2 s

(2.42)

(2.43)

(where ds = inf{u ≥ s; Xu = 0} = s + T0 ◦ θs , where θs denotes the usual time translation operator.) √ 1 t = tW0 1(T0 ◦θs >t−s) exp − q(Xh )dh Fs 2 s √ 1 t−s = tWXs 1(T0 >t−s) exp − q(Xh )dh (2.44) 2 0 from the Markov property and likewise : √ 1 t−s (2) q(Xh )dh Ns,t := tWXs 1(T0
(2.45)

As we already observed, all we need to show is that, Ns,t converges as t → ∞, towards ϕq (Xs ). Thus, replacing t − s by t, and using the explicit formula (2.30) which gives ϕq we will have proven (2.34) once we have established that, if x > 0 : √ 2 1 t Φq (∞)Ψq+ (x) tWx 1(T0 >t) exp − q(Xu )du −→ t→∞ 2 0 π + (2.46) and

√ 1 t tWx 1(T0
(2.47)

80

2 Feynman-Kac Penalisations for Brownian Motion

as well as symmetrical formulae for x < 0, the proof of which is quite similar to the one we shall now develop for x > 0. 2.3.4i) We now prove (2.46). We have, for x > 0 : √ 1 t tWx 1(T0 >t) exp − q(Xu )du 2 0 √ (3) x 1 t = tEx exp − q+ (Xu )du 2 0 Xt (from the absolute continuity formula (2.25) and because Xu is positive for every u between 0 and T0 ). 2 1 ∞ x Ex(3) exp − −→ q+ (Xu )du t→∞ π 2 0 (from the asymptotic independence, as t → ∞, of

t

q+ (Xu )du and 0

=

2 Ψq+ (x) x Φq+ (∞) = π x

√ t Xt )

2 Ψq (x)Φq+ (∞) π +

from (2.24). 2.3.4ii) We now show (2.47). √ 1 t tWx 1(T0
2.4 Absolute Continuity Relationships

81

The computations for x < 0 are quite similar to the ones we have just developed for x > 0. In Section 2.5, we shall generalize what we have just done (q) now to more complicated functionals than Et .

2.4 Absolute Continuity Relationships between the Probabilities Wx(q) (q ∈ I ) and Deﬁnition of the σ-ﬁnite Measures Wx (x ∈ R) 2.4.1 In order to be able to state the main result of this section, we need the following proposition. Proposition 2.4. For every q ∈ I and every x ∈ R, 1)

If λ < 1, If λ ≥ 1,

∞ λ Wx(q) exp q(Xs )ds <∞ 2 0 ∞ λ q(Xs )ds =∞ Wx(q) exp 2 0

(2.48) (2.49)

2) For every positive r.v. F , deﬁned on Ω, F∞ -measurable, the quantity : ∞ 1 Wx (F ) := q(Xs )ds F exp 2 0 ∞ 1 (q) y q(y)L∞ dy F exp = ϕq (x)Wx 2 0 ϕq (x)Wx(q)

(2.50) (2.51)

does not depend on q ∈ I, and allows to deﬁne a positive σ-ﬁnite measure on F∞ . In particular, formula : Wx

1 exp − 2

∞

q(Xs )ds

= ϕq (x)

(2.52)

0

holds. 2.4.2 Before proving Proposition 2.4, we make a few comments about its contents. 2.4.2i) We recalled, in Theorem 2.1 that the process (Xt , t ≥ 0) under (q) (Wx , x ∈ R) is a transient diﬀusion. In a way, formula (2.48) which im∞ plies in particular that 0 q(Xs )ds is ﬁnite W (q) a.s. states that the process (q) (Xt , t ≥ 0) is very transient under Wx , although this “very transient” character is moderated by the property (2.49).

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2 Feynman-Kac Penalisations for Brownian Motion

2.4.2ii) The fact that Wx has inﬁnite total mass follows from (2.49) since : ∞ 1 Wx (1) = ϕq (x)Wx(q) exp q(Xs )ds = ∞ from (2.49) 2 0 whilst the σ-ﬁniteness of Wx is a consequence of the next remark : let An = ∞ 0 q(Xs )ds < n . Then, An ↑ Ω (from (2.48)) and : n→∞

∞ n 1 (q) ϕq (x) Wx (An ) = ϕq (x)Wx q(Xs )ds 1An ≤ exp exp 2 0 2 2.4.2iii) We now replace in (2.50) the functional F by a r.v. Fu , assumed to be Fu measurable. The Markov property then implies : Wx (Fu ) ∞ 1 = Wx(q) Fu ϕq (x) exp q(Xs )ds 2 0 u ∞ 1 + (q) exp 1 )s )ds = Wx(q) Fu ϕq (x) exp q(Xs )ds W q( X Xu 2 0 2 0 (2.53) Thus, from (2.49), the quantity (2.53) is either 0 or inﬁnite depending upon (q) (q) the fact that Wx (Fu ) is 0 or strictly positive; since Wx and Wx are equivalent on Fu , for every u > 0 this fact amounts to the same property with q = 0. In other terms, the measure Wx , although it is σ-ﬁnite on F∞ , is not σ-ﬁnite on any Fu for any u > 0. Thus, Wx cannot be described from the σ-ﬁniteness its restriction on Fu and then letting u → ∞; to describe ∞ character of Wx it is necessary to use sets such as { 0 q(Xs )ds ≤ α}, which belong to F∞ and not to Fu for any u < ∞. 2.4.3 Proof of Proposition 2.4 We begin with the proof of (2.48). For this purpose, we write from (2.7), for λ ∈]0, 1[ : t λ (q) q(Xs )ds exp Wx 2 0 ϕq (Xt ) 1−λ t exp − = Wx q(Xs )ds ϕq (x) 2 0 % & ϕ(1−λ)q (x) 1−λ t ϕq (Xt ) ϕ(1−λ)q (Xt ) = Wx exp − q(Xs )ds ϕq (x) ϕ(1−λ)q (Xt ) ϕ(1−λ)q (x) 2 0 (2.54) It is licit to write (2.54) since the functions ϕq and ϕ(1−λ)q are strictly positive. Together with the fact that these functions have the same behavior at ±∞,

2.4 Absolute Continuity Relationships

83

where they are equivalent to π2 |x|, one ﬁnds that there exist two constants 0 < C1 (λ, q) ≤ C2 (λ, q) < ∞ such that : C1 (λ, q) ≤ inf

y∈R

ϕq (y) ϕq (y) ≤ sup ≤ C2 (λ, q) ϕ(1−λ)q (y) ϕ y∈R (1−λ)q (y)

(2.55)

Thus, from (2.54), one obtains : Wx(q)

t ϕ(1−λ)q (x) λ ϕq (y) sup W ((1−λ)q) (1) q(Xs )ds ≤ exp 2 0 ϕq (x) y∈R ϕ(1−λ)q (y) x ≤

C2 (λ, q) C1 (λ, q)

(2.56)

To obtain (2.48), it now remains to let t → ∞ in (2.56) and to use the Monotone Convergence Theorem. Concerning (2.49), this relation results from (2.54) written for λ = 1 since : t √ 1 ϕq (Xt ) Wx(q) exp q(Xs )ds = Wx (2.57) ∼ k(x) t 2 0 ϕq (x) t→∞ 2 as ϕq (y) ∼ π |y|; therefore, letting t → ∞ in this latter relation, we |y|→∞

obtain, using again the Monotone Convergence Theorem : ∞ 1 q(Xs )ds =∞ Wx(q) exp 2 0 We now prove the second point of Proposition 2.4 : Let u ≤ t and consider Fu a positive, bounded, Fu measurable r.v. Then, (q) from the deﬁnition of Wx , we have : Wx(q)

Fu ϕq (x) exp

t 1 q(Xs )ds 2 0

= Wx [Fu ϕq (Xt )] ϕq (Xt ) ϕq (Xt ) = Wx F u ϕq (Xt ) (where q is another element of I) t 1 ϕq = Wx(q ) Fu ϕq (x) (Xt ) exp q (Xs )ds ϕq 2 0

(2.58)

Since the relation (2.58) holds for every Fu Ft measurable and since u ≤ t, t we may replace Fu by F = Fu exp −ε 0 (q + q )(Xs )ds (ε > 0).

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2 Feynman-Kac Penalisations for Brownian Motion

We obtain : t t 1 −ε Wx(q) Fu ϕq (x) exp q(Xs )ds exp −ε q (Xs )ds 2 0 0 t t 1 ϕq (q ) = Wx −ε (Xt ) exp q (Xs )ds exp −ε q(Xs )ds Fu ϕq (x) ϕq 2 0 0 (2.59) (q)

However, under (Wx , x ∈ R), |Xt | → +∞ a.s., from point 4) of Theorem ϕ 2.1; on the other hand, the function ϕ q is bounded. Using (2.48) and the q dominated convergence Theorem, we obtain, letting t → ∞ in (2.59) : ∞ ∞ 1 −ε q(Xs )ds exp −ε q (Xs )ds Wx(q) Fu ϕq (x) exp 2 0 0 (2.60) ∞ ∞ 1 (q ) = Wx −ε q (Xs )ds exp −ε q(Xs )ds Fu ϕq (x) exp 2 0 0 (q ) ϕ 2 since ϕ q (Xt ) −→ 1 , Wx a.s., since ϕq (y) ∼ π |y| for every q ∈ I. q

t→∞

|y|→∞

Now, as a consequence of the monotone class Theorem, (2.60) holds for every bounded positive F∞ measurable r.v. F . Thus : ∞ ∞ 1 −ε q(Xs )ds exp −ε q (Xs )ds Wx(q) F ϕq (x) exp 2 0 0 (2.61) ∞ ∞ 1 (q ) = Wx −ε q (Xs )ds exp −ε q(Xs )ds F ϕq (x) exp 2 0 0 It now remains to let ε → 0 in (2.61) and to use the monotone convergence Theorem to obtain point 2) of Proposition 2.4. t Of course, we have introduced - artiﬁcially - the factor exp(−ε 0 (q + q ) (Xs )ds) in this proof uniquely to take care of the diﬃculty mentioned in point 2.4.2iii) above, that is : Wx is not σ-ﬁnite on the σ-algebra Fu , for every u > 0. We also note, for further use, that the function from R+ to R+ : λ → ϕλq (x) (for q ∈ I and x ∈ R ﬁxed) is the Laplace transform of the law of the r.v. 1 ∞ 2 0 q(Xs )ds considered under Wx since, from (2.52) : λ ∞ ϕλq (x) = Wx exp − q(Xs )ds (2.62) 2 0 When we take q(dx) = δ0 (dx), the Dirac measure in 0, the relation (2.50) becomes, for x = 0 :

2.4 Absolute Continuity Relationships

W0 = 2

85

2 exp π

1 L∞ 2

•

(δ0 )

W0

(2.63)

2 2 since : ϕδ0 (x) = (2 + |x|), hence : ϕδ0 (0) = 2 . π π (δ ) We now recall the description of the canonical process (Xt , t ≥ 0) under W0 0 which we gave in [RVY,II], Theorem 8, p.339, where : x h+ (x) = h− (x) = exp − , x≥0 2 (δ )

The following results hold under W0 0 : • The variable g := sup{u ≥ 0 : Xu = 0} is ﬁnite a.s. and L∞ (= Lg ) admits the density : (δ ) 1 W 0 fL∞0 () = exp − (2.64) 1( ≥ 0) 2 2 • i) the processes (Xu , u ≤ g) and (Xg+u , u ≥ 0) are independent; ii) (Xg+u , u ≥ 0) is distributed with P03,sym , where : P03,sym =

1 3 P0 + P03,− 2

(2.65)

with P03 , resp. P03,− , the law of a 3-dimensional Bessel process, starting from 0, resp. : its opposite. iii) Conditionally on L∞ = , (Xu , u ≤ g) is distributed with W0τ , the law of Brownian motion starting from 0, considered up to τ = inf{t : Lt > }. These results may be summarized via the formula : ∞ 1 (δ0 ) (2.66) exp − W0τ ◦ P03,sym W0 = d 2 2 0 where we denote by W0τ ◦ P03,sym the concatenation of the laws W0τ and P03,sym , that is the image of the product W0τ ⊗ P03,sym by the concatenation application ◦ from Ω × Ω to Ω, deﬁned by : Xt (ω1 ) , if t ≤ τ (ω1 ) (2.67) Xt (ω1 ◦ ω2 ) = Xt−τ(ω1) (ω2 ) , if t ≥ τ (ω1 ) The previous description (2.66) now yields to the following Theorem 2.4.* The measure W0 deﬁned by (2.50) satisﬁes : 2 ∞ τ 1) W0 = W0 ◦ P03,sym d π 0 2) Let g = sup{t ≥ 0 : Xt = 0}

(2.68)

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2 Feynman-Kac Penalisations for Brownian Motion

Then : dt i) W0 (g ∈ dt) = √ (t ≥ 0) π t ii) Conditionally on g = t, the process (Xu , u ≤ g) is, under W0 , a Brownian bridge with length t : for every positive functional F , W0 (F (Xs , s ≤ t)|g = t) = E[F (b(t) u , u ≤ t)] (t)

where (bu , 0 ≤ u ≤ t) denotes a Brownian bridge of length t, whose law we (t) shall denote by Π0 . Consequently : W0 = 0

∞

dt (t) √ Π0 ◦ P03,sym π t

(2.69)

3) For every Ft ∈ b(Ft ), W0 (Ft 1(g≤t) ) = W0 (Ft |Xt |) Note: A full discussion and proof of Theorem 2.4.* is not provided here, as it would bring us too far; this is the aim of [NRY]. For now, we only mention a few important points : a) Formula (2.68) is an easy consequence of (2.66), (2.64) and (2.65); b) The passage from (2.68) to (2.69) is proven - up to the presence of the “factor” P03,sym - in Biane ([B], Theorem 11), or Biane-Yor [BY], or [RY∗ ], p.506, Exercise (4.18); c) Formula (2.67) allows to “represent the sub-martingale (|Xt |, t ≥ 0), for X a Brownian motion, in terms of W0 and the increasing process (1(g≤t) , t ≥ 0)”. This terminology is explained in [NRY], where formula (2.67) appears there as the prototype of more general “representations of sub-martingales”. d) It follows easily from (2.68) or (2.69) that : 2 exp − 2u W0 (g ∈ du, L∞ ∈ d) = ddu πu3/2 2.4.4 We shall now prove the existence of some σ-ﬁnite and positive measures Λx , x ∈ R. = C(R → R+ ) and we let S (S stands We ﬁrst set some notation. We denote Ω → R+ measurable for sub-exponential) denote the set of applications Γ : Ω such that there exist q ∈ I and two strictly positive constants C1 and C2 for which : , ∀λ ∈ Ω with < λ, q >:=

∞

−∞

0 ≤ Γ(λ) ≤ C1 exp(−C2 < λ, q >)

λ(y)q(y)dy.

(2.70)

2.4 Absolute Continuity Relationships

87

Theorem 2.5. For every x ∈ R, there exists a positive, σ-ﬁnite measure Λx such that : on Ω √ 1) ∀Γ ∈ S tWx (Γ(L•t )) −→ Γ(λ)Λx (dλ) (2.71) Ω

t→∞

deﬁned by where, on the LHS of (2.71) L•t denotes the element of Ω λ(y) = Lyt

(y ∈ R)

(2.72)

2) The measure Λx is characterized by : for every q ∈ I :

1 ∞ Λx exp − λ(y)q(y)dy = ϕq (x) 2 −∞

or equivalently

Λx

1 = ϕq (x) exp − < λ, q > 2

(2.73)

(2.74)

3) Λx is the image of Wx , deﬁned in Proposition 2.4 by the application “total local time” : Θ : Ω→Ω Θ(X) = (Ly∞ (X), y ∈ R)

(2.75)

where X denotes here the canonical process (Xt , t ≥ 0) deﬁned on Ω. 4) The measure Λ0 may be represented as 1 Λ0 = √ 2π

0

∞

− (Λ+ + Λ )d

(2.76)

− where in (2.76), Λ+ , resp. Λ , is the law of the process (Yy , y ∈ R) indexed by R such that :

• (Yy , y ≥ 0) is a 2-dimensional squared Bessel process, resp. 0dimensional starting from • (Y−y , y ≥ 0) is a 0-dimensional squared Bessel process, resp. 2dimensional starting from • conditionally on Y0 = , these two processes are independent. In Lemma 2.15 below, we give another description of Λ0 . In his thesis [N2 ] and publication [N3 ], J. Najnudel gives more precise condi → R+ satisﬁes point 1) of Theorem 2.5 tions under which a functional Γ : Ω (see [N2 ] for more details). Before proving Theorem 2.5 we start by giving a simple description of the − measures Λ+ 0 and Λ0 : ∞ 1 ± Λ± Λ0 = √ d 2π 0

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2 Feynman-Kac Penalisations for Brownian Motion

The two following ﬁgures illustrate Λ± Description of Λ+

+ σ+ =∞

−σ+ −

0-dimensional squared Bessel process starting from

←→

2-dimensional squared Bessel process starting from

Description of Λ−

− −σ− =−∞

− σ+

2-dimensional squared Bessel process starting from

←→

Λ0 =

√1 2π

0

∞

0-dimensional squared Bessel process starting from

− (Λ+ + Λ )d

A second description of the measure Λ+ 0 shall now be given from Williams’ Bessel processes illustrations of the time reversal Theorem (of Nagasawa) : the time reversal of a 4-dimensional Bessel process, from its last passage time γ at level is a Bessel process with dimension 0 starting from , considered up to its ﬁrst hitting time of level 0, which is absorbing for the Bessel process with dimension 0. We have : ∞ 1 √ = Λ+ (a) Λ+ 0 d 2π 0 (and the same description holds, mutatis mutandis, for Λ− 0 ). We consider the following processes, and/or random variables : • a positive r.v. D, with “distribution” 2dt • the square of a 4-dimensional Bessel process starting from 0 : (4) (Z0 (t), t ≥ 0)

2.4 Absolute Continuity Relationships

89

• a family of 2-dimensional squared Bessel processes, starting from : (2) (Z (t), t ≥ 0, ≥ 0) (4)

(2)

independent. We assume that D, Z0 and Z are √ Then, the canonical process under 2πΛ+ 0 is distributed as the process (Yu , u ∈ R) deﬁned by : ⎧ if u ≤ −D ⎪ ⎨ Yu = 0 (4) Yu = Z0 (D + u) if − D ≤ u ≤ 0 (b) ⎪ ⎩ Yu = Z (2)(4) (u) if u ≥ 0 Z0 (D)

Description of

√ 2πΛ+ 0

Y0

0 −D →

→

Y , 4-dimensional squared Bessel process starting from 0

2-dimensional squared Bessel process starting from Y0

Description of

√

2πΛ− 0

Y0

0 ←

← D

2-dimensional squared Bessel process starting from YD

Y , 4-dimensional squared Bessel process starting from 0

+ Λ0 = Λ− 0 + Λ0

Λ± 0 =

√1 2π

∞

0

Λ± d

90

2 Feynman-Kac Penalisations for Brownian Motion

Proof of the Preceding Description of

√

2πΛ+ 0

We compute, for F, G positive functionals and h : R+ → R+ , Borel : ⎫ √ + + + ; −σ 2πΛ+ − ≤ u ≤ 0)G(Yv , v ≥ 0)] ⎪ 0 [h(−σ− )F (Yu+σ− ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + ⎪ ⎪ with − σ− := inf{v; Yv > 0} ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ ⎪ (4) (2) = Q0 (F (Yu , u ≤ γ )h(γ ))Q (G(Yv , v ≥ 0))d ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ∞ ⎪ ⎪ (4) (4) ⎪ = d Q0 (γ ∈ dt)h(t)Q0 (F (Yu , u ≤ t)|γ = t) ⎪ ⎪ ⎪ ⎪ 0 0 ⎪ ⎪ (2) ⎪ Q (G(Yv , v ≥ 0)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ after conditioning upon γ = t. (from (a) and Williams’ time reversal Theorem)

(c)

But, we know that (see [PY3 ]) : (4)

(4)

Q0 (F (Yu , u ≤ γ )|γ = t) = Q0 (F (Yu , u ≤ t)|Yt = ) (4)

(d)

, where e denotes a standard exponential variable, and, under Q0 , γ = 2e

(4) dt thus : Q0 (γ ∈ dt) = 2t2 exp − 2t (t ≥ 0). Plugging this relation in (c), we obtain : ∞ ∞ d (4) (2) h(t)dt exp − Q0 (F (Yu , u ≤ t)|Yt = )Q (G(Yv , v ≥ 0)) 2t2 2t 0 0

(4) But, since Q0 (Yt ∈ d) = 4t12 exp − 2t d, this expression equals : ∞ ∞ (4) (2) (4) 2h(t)dt Q0 (F (Yu , u ≤ t)|Yt = )Q (G(Yv , v ≥ 0))Q0 (Yt ∈ d) 0 0 ∞ (4) (2) = 2h(t)dt Q0 [F (Yu , u ≤ t)QYt (G(Yv , v ≥ 0))] 0

which is the announced result √ (b). Thus, under the measure 2πΛ+ 0 , the process (Yu , u ∈ R) may be described as follows : • we consider a r.v. D, , with “distribution” 2 1[0,∞[ dt • we make a 4-dimensional squared Bessel process start at time -D and we continue like this until time 0 • at time 0, this Bessel process has reached, say, point

2.4 Absolute Continuity Relationships

91

• then we make a 2-dimensional squared Bessel process start from , with D and the two squared Bessel processes used in this computation being independent. Proof of Theorem 2.5 i) We deﬁne the measure Λx as the image of Wx by the application Θ, where Wx is deﬁned in Proposition 2.4 and Θ is deﬁned in point 3) of the statement of Theorem 2.5. Thus, for every q ∈ I : Λx

1 exp − 2

∞

λ(y)q(y)dy −∞

1 ∞ = Wx exp − q(Xs )ds 2 0 = ϕq (x) (from (2.52)) (2.77)

ii) Now, in order to prove point 1) of Theorem 2.5, using the deﬁnition of S, it suﬃces with the help of

the Laplace transform to do it for functionals Γ of the form : Γ(λ) = exp − 12 < λ, q > , with q ∈ I. Thus, we need to prove that : √ 1 ∞ 1 y (2.78) tWx exp − q(y)Lt dy −→ Λx exp − < λ, q > t→∞ 2 −∞ 2 However, from (2.77), the RHS of (2.78) is equal to ϕq (x), whereas, from point 1) of Theorem 2.1 (or Section 2.3 of this chapter) : √ 1 ∞ tWx exp − q(y)Lyt dy 2 −∞ √ 1 t = tWx exp − q(Xs )ds 2 0 −→ ϕq (x) (2.79) t→∞

iii) Finally, point 4) of Theorem 2.5 is exactly point 2) of the above Theorem 2.1; it also coincides with Remark 2.3. Remark 2.5.* a) The description of the measure Λ0 we gave in point 4) of Theorem 2.5 may also be obtained from the representation formula (2.68) for W0 found in point 1) of Theorem 2.4.*, with the help of the following : i) Λ0 is the image of W0 by the “total local time application” Θ : Θ(X) = (Ly∞ (X), y ∈ R) (3)

ii) The process (Ly∞ (R0 ), y ≥ 0) of the total local times of a 3-dimensional Bessel process starting from 0 is distributed as Q20 , the law of the square of a 2-dimensional Bessel process starting from 0. This is the “third” RayKnight theorem, as remarked by D. Williams (see, e.g., [RY∗ ], Chapter XI, Exercise 2.5).

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2 Feynman-Kac Penalisations for Brownian Motion

iii) The processes (Lyτ , y ≥ 0) and (L−y τ , y ≥ 0) of the local times up to τ of a Brownian motion starting from 0, are independent and are distributed as Q0 , the law of a squared Bessel process with dimension 0, starting from . This is the “second” Ray-Knight theorem (see, e.g., [RY∗ ], Chapter XI). iv) For every δ, δ ≥ 0, and x, x ≥ 0,

Qδx ∗ Qδx = Qδ+δ x+x where Qδx denotes the law of a squared Bessel process with dimension δ, starting from x (see, e.g., [RY∗ ], Chapter XI), and P ∗ Q denotes the convolution of P and Q, two probabilities on C(R+ → R+ ). b) Finally, we remark that, if we note Λ+ , resp. Λ− , the image of W+ , resp. W− by the application Θ deﬁned by (2.68) (recall that: W+ = 1Γ+ •W, and W− = 1Γ− •W with Γ+ = {ω; Xt (ω) −→ +∞}, Γ− = {ω; Xt (ω) −→ −∞}), t→∞ t→∞ we have: Λ = Λ+ + Λ− and: 1 Λ+ = √ 2π

0

∞

1 Λ− = √ 2π

Λ+ d ,

0

∞

Λ− d

In the end, we ﬁnd: Λ− = Λ− 0

Λ+ = Λ+ 0 ,

We shall now deduce three interesting Corollaries from Theorem 2.5. (q)

2.4.5 A ﬁrst corollary of Theorem 2.5 : the existence of ν x . (q) 2.4.5i) To any element q ∈ I we associate the additive functional (At , t ≥ 0) deﬁned by : t ∞ (q) At := q(Xs )ds = Lyt q(y)dy (t ≥ 0) (2.80) −∞

0 (q)

Corollary 2.6. Let q ∈ I and (At , t ≥ 0) deﬁned by (2.80). For every (q) x ∈ R there exists a positive, σ-ﬁnite measure ν x carried by R+ such that : √ (q) tWx (At ∈ da) −→ ν (q) (2.81) x (da) t→∞

In (2.81), the convergence is understood in the following sense : for every function h : R+ → R+ , continuous with compact support (or with subexponential growth in +∞), we have : √ (q) tWx (h(At )) −→ h(a)ν (q) (2.82) x (da) t→∞

R+

2.4 Absolute Continuity Relationships

93

(q)

The measure ν x is characterized by : ∞ λa exp − ν (q) x (da) = ϕλq (x) 2 0

(2.83)

Proof of Corollary 2.6 (q)

i) It suﬃces to take for ν x the image of the measure Λx by the application → R+ deﬁned by : γq : Ω ∞ (y)q(y)dy (2.84) γq () =< , q >= −∞

(q)

ii) It is also possible to show the existence of the measure ν x without using Theorem 2.5 in the following manner : from Theorem 2.1 or Section 2.3 of this chapter we have : √ λ t tWx exp − q(Xs )ds −→ ϕλq (x) t→∞ 2 0 However, from (2.62), the ∞function : λ →ϕ∞λq (x) is the Laplace transform of the distribution of 12 0 q(Xs )ds = 12 −∞ Ly∞ q(y)dy under Wx . This ∞ (q) shows the existence of ν x which is thus the distribution of 12 0 q(Xs )ds under Wx , or equivalently from the deﬁnition of Λx , as the image of Wx by Θ, the distribution of 12 < •, q > under Λx . (q) νx .

2.4.5ii) Here are now some examples of measures These examples are lifted from [RY,IX] and are presented without the computations which led to the formulae below (the reader may refer to [RY,IX] for the detailed computations). It is noteworthy that in these examples, the function q does not always belong to I, either because it is a Radon measure ∞ (instead of a function), or because it does not satisfy the hypothesis −∞ (1 + |x|)q(x)dx < ∞. However, in all the examples below, the conclusion of Corollary 2.6 still holds. (q) In the following list of examples, we simply write ν x instead of ν x . (the Dirac measure in 0) q = δ0 2 2 ν x (dy) = 1[0,∞[ (y)dy + |x|δ0 (dy) π π

Example 2.4.5.a)

Example 2.4.5.b) q = δa + δ b (a < b) 2 1 1[0,∞[ (y)dy + (x − b)1x>b δ0 (dy) + (a − x)1x
94

2 Feynman-Kac Penalisations for Brownian Motion

(x ∈ R) e2x 1 dy ν x (dy) = √ exp − 1[0,∞[ (y) 2y y 2π

Example 2.4.5.c)

q(x) = e2x

Example 2.4.5.d)

q(x) = 1]−∞,0] (x) (x− )2 2 1 dy ν x (dy) = x+ δ0 (dy) + exp − 1[0,∞[ (y) √ π π 2y y

Example 2.4.5.e)

q(x) = xα 1[0,∞[ (x) ν 0 (dy) =

Example 2.4.5.f )

Cα

(x > 0)

1+α

1[0,∞[ (y)dy

q(x) = 1[a,b] (x)

(a < b)

y 2+α

ν x (dy) = ⎧ 2 ' n (b−a)2 2 ⎪ dy (x − b)δ0 (dy) + π√1 y 1[0,∞[ (y) 1 + 2 ∞ ⎪ n=1 exp − π 2y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (if x > b) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ '∞ ⎪ n (b−a)2 2 1 ⎪ √ ⎨ dy (a − x)δ (dy) + 1 (y) 1 + 2 exp − 0 [0,∞[ n=1 π π y 2y ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(if x < a) 1 √ π y

'∞ n=0

2 2 exp − (n(b−a)+(b−x)) + exp − (n(b−a)+(x−a)) 1[0,∞[ (y)dy 2y 2y (if a < x < b)

2.4.6 Second corollary of Theorem 2.5 : the Ray-Knight Theorem under (q) (Wx , x ∈ R) (q) Let q ∈ I. We recalled (point 4) of Theorem 2.1) that under (Wx , x ∈ R), the process (Xt , t ≥ 0) solves the equation : Xt = x + Bt + 0

t

ϕq (Xs )ds ϕq

(2.85)

(q)

where (Bt , t ≥ 0) is a ((Ft , t ≥ 0), Wx ) Brownian motion starting from 0. (q) Corollary 2.7. i) Let Λ x denote the law on Ω = C(R → R+ ) of the local y (q) times (L∞ , y ∈ R) of the transient diﬀusion which solves (2.85). Λ x is absolutely continuous with respect to Λx and its density fΛ (q) is given by : x

2.4 Absolute Continuity Relationships

95

1 1 exp − < , q > fΛ (q) () = x ϕq (x) 2 In other terms : (q) (d) = Λ x

1 1 exp − < , q > •Λx (d) ϕq (x) 2

(2.86)

(2.87)

∞ (q) ii) In particular, the law of 0 q(Xs )ds under Wx is absolutely continu(q) ous with respect to ν x (deﬁned in Corollary 2.6) and its density equals exp(− 12 a) ϕq (x) . In other terms : Wx(q)

∞

q(Xs )ds

h 0

= R+

a h(a) exp − ν (q) x (da) ϕq (x) 2

(2.88)

for every h : R+ → R+ Borel. Proof of Corollary 2.7 i) We already show that point 1) of Corollary 2.7 implies point 2) of this corollary. We have, with h(y) = exp λy (0 ≤ λ < 1, y ≥ 0): 2 ∞ λ λ (q) (q) • Wx < L∞ , q > q(Xs )ds = Wx exp exp 2 0 2 λ (q) = Λx exp < , q > 2 (q) ) (by deﬁnition of Λ x λ−1 1 = Λx exp < , q > ϕq (x) 2 ϕ(1−λ)q (x) (from (2.86)) = ϕq (x) ∞ 1−λ 1 a ν (q) exp − (from (2.73)) = x (da) ϕq (x) 0 2 (2.89) from (2.83), which proves (2.88). ii) We now prove point 1) of Corollary 2.7.

∞ For every g ∈ I, using (2.50) with F = ϕq1(x) exp − 21 0 (g + q)(Xs )ds we obtain, on one hand : 1 ∞ g(Xs )ds Wx(q) exp − 2 0 1 1 ∞ = Wx exp − (g + q)(Xs )ds ϕq (x) 2 0

96

2 Feynman-Kac Penalisations for Brownian Motion

=

ϕ(g+q) (x) 1 1 = Λx exp − < g + q, > ϕq (x) ϕq (x) 2

(2.90)

(from (2.52) and (2.74)) (q) and, on the other hand, from the deﬁnition of Λ x : ∞ 1 1 g(Xs )ds = Wx(q) exp − Ly∞ g(y)dy Wx(q) exp − 2 0 2 R 1 (q) =Λ < g, > (2.91) exp − x 2 Thus, from (2.90) and (2.91) : 1 1 1 (q) Λ < g, > = Λ < g + q, > exp − exp − x x 2 ϕq (x) 2 which implies (2.86) or (2.87). (q) Λ x

Remark 2.8. The formula (2.87) which gives the density of with respect to Λx is interesting but has an “abstract” character. Of course, it would be (q) preferable to give a “more concrete” form to the law Λ x e.g. : to express this law in terms of the functions Φq± and Ψq± deﬁned in Section 2.2. This is what we shall now achieve. 2.4.7 A concrete form of Corollary 2.7 In order to present this new formulation of Corollary 2.7, we need to introduce some notations. Let γ : R+ → R+ an element of I+ . We denote : (2),γ

i) Q the law of the diﬀusion (Yb , b ≥ 0) issued from , solution of the SDE : b! b Φγ Yb = + 2 Yc dβc + 2b + 2 (2.92) (c)Yc dc Φγ 0 0 where, in (2.92), (βc , c ≥ 0) is a Brownian motion issued from 0 and where (2),0 Φγ is the function deﬁned by (2.18). Let us note that if γ ≡ 0, then Q is the law of a squared Bessel process of dimension 2 issued from . (0),γ ii) Q the law of the diﬀusion (Yb , b ≥ 0) issued from solution of the SDE : b! b Φγ (2.93) Yc dβc + 2 (c)Yc dc Yb = + 2 Φγ 0 0 where, in (2.93), (βc , c ≥ 0) is a Brownian motion issued from 0 and (0),0 where Φγ is the function deﬁned by (2.18). If γ ≡ 0, then Q is the law of a squared Bessel process of dimension 0 issued from .

2.4 Absolute Continuity Relationships

97

iii) If P− and P+ are two probabilities deﬁned on C(R+ → R+ ) such that P− (Y0 = ) = P+ (Y0 = ) = 1, we denote as P− P+ the law of the process (Yy , y ∈ R) where : (Yy , y ≥ 0) follows the law P+ (Y−y , y ≥ 0) follows the law P− We may now present the following “concrete form” of Corollary 2.7. (q) of local times (Ly (X), y ∈ R) under W (q) Corollary 2.9. The law Λ ∞ 0 0 equals : ∞ (q) = √1 Λ (Λ+,q + Λ−,q (2.94) 0 )d 2π 0 with : Λ+,q

Λ−,q

(0),q (2),q (Φ (0+) + Φq− (0+)) Q − Q + 2 q+ (2.95) Φq− (∞) (2),q (0),q exp (Φ (0+) + Φq− (0+)) Q + Q − = ϕq (0) 2 q+ Φq+ (∞) exp = ϕq (0)

We may already note that formulae (2.94) and (2.95) imply : ∞ 1 (q) (1) = 1 = √1 Λ (Φ exp (0+) + Φ (0+)) d × . . . q− 0 2 q+ 2π ϕq (0) 0 . . . (Φq+ (∞) + Φq− (∞)) Hence :

1=− i.e.

ϕq (0) = −

2 1 π ϕq (0) 2 π

Φq+ (∞) + Φq− (∞) Φq− (0+) + Φq+ (0+)

Φq+ (∞) + Φq− (∞) Φq+ (0+) + Φq− (0+)

This formula agrees with (2.30). Proof of Corollary 2.9 (q) , Λ+ , Λ− , ...) Most of the probabilities we are going to study hereafter (e.g. Λ 0 = C(R → R+ ). We shall note (Yy , y ∈ R) the process are probabilities on Ω of coordinates on this space. According to Theorem 2.5 and formulae (2.76) and (2.87) we have : ∞ 1 1 1 ∞ (q) + − √ Λ0 = Yy q(y)dy • (Λ (2.96) exp − + Λ )d 2 −∞ 2π ϕq (0) 0

98

2 Feynman-Kac Penalisations for Brownian Motion

and it then remains to understand the probability : 1 ∞ +,q Λ := exp − Yy q(y)dy • Λ+ 2 −∞

(2.97)

(and in the same manner by replacing the index + by −). We shall only deal −,q is analogous. with Λ+,q , as the computation for Λ Let a and b be positive and denote by Fa,b the σ-algebra (of measurable equal to : subsets of Ω) Fa,b := σ(Yy ; −a ≤ y ≤ b) Then, the expression of Λ+,q over this σ-algebra may be given as follows: for every pair of positive functionals F and G, respectively deﬁned on C([0, a] → R) and C([0, b] → R), one has, by (2.97): (0) (k) Λ+,q (F (Y−y ; 0 ≤ y ≤ a)G(Yy ; 0 ≤ y ≤ b)) = Σ Σ

where :

1 a q(−z)Yz dz • F (Yy ; 0 ≤ y ≤ a) exp − 2 0 ∞ 1 (0) QYa exp − q(−(a + u))Yu du 2 0

(2.98)

(0)

Σ(0) := Q

(2.99)

and (2)

Σ

1 b := q(z)Yz dz • G(Yy ; 0 ≤ y ≤ b) exp − 2 0 1 ∞ (2) QYb exp − q(b + v)Yv dv 2 0 (2) Q

(2.100)

following the deﬁnition of Λ+ given in point 4) of Theorem 2.5 and the Markov property. But, in addition, formula (2.26), applied for δ = 0 (and the function q− translated by a) and for δ = 2 (and the function q+ translated by b) yields : 1 ∞ q(−(a + u))Y du exp − ha (y) := Q(0) u y 2 0 ∞ 1 = Q(0) q (a + u)Y du exp − − u y 2 0 y Φq− (a) = exp (2.101) 2 Φq− (a)

2.4 Absolute Continuity Relationships

and

99

1 ∞ kb (y) := Q(2) q(b + v)Y dv exp − v y 2 0 ∞ 1 = Q(2) q (b + v)Y dv exp − + v y 2 0 Φq+ (∞) y Φq+ (b) = exp Φq+ (b) 2 Φq+ (b)

(2.102)

Plugging back (2.101) in (2.99) and (2.102) in (2.100), we obtain : Ya Φq− (a) 1 a (0) (0) − q− (z)Yz dz Σ = Q F (Yy ; 0 ≤ y ≤ a) exp 2 Φq− (a) 2 0 Σ(2)

(2.103) b Φ (b) Φq (∞) Yb q+ 1 (2) exp − = Q G(Yy ; 0 ≤ y ≤ b) + q(z)Yz dz Φq+ (b) 2 Φq+ (b) 2 0 (2.104)

Hence, according to Girsanov : (0),q Σ(0) = exp Φq− (0+) Q − (F (Yz ; 0 ≤ z ≤ a)) 2 (0),q−

where Q

Za0,q−

(2.105)

(0)

is the law Q transformed by the exponential martingale : , 2 1 a Φq− (b) 1 a Φq− (b) dYb − := exp d b (2.106) 2 0 Φq− (b) 8 0 Φq− (b)

(with d b = 4Yb db). (0),q In other terms, Q − is the law of the diﬀusion (Ya , a ≥ 0) issued from and (0) deﬁned by (2.93), since, under Q the canonical process (Ya , a ≥ 0) satisﬁes : a! Ya = + 2 Yc dβc 0

Likewise, according to Girsanov once again : (2),q (2) Φ (0) Q + (G(Yy ; 0 ≤ y ≤ b)) Σ = Φq+ (∞) exp 2 q+ (2),q+

where Q (2),q+

Zb

(2)

(2.107)

is the law Q transformed by the exponential martingale : 1 Φq+ (b) 1 b Yb − Φq+ (0) − 2 log Φq+ (b) − := exp q(z)Yz dz 2 Φq+ (b) 2 0 (2.108)

100

2 Feynman-Kac Penalisations for Brownian Motion (2),q

In other terms, Q + is the law of the diﬀusion (Yb , b ≥ 0) issued from and (2) deﬁned by (2.92), since, under Q the canonical process (Yb , b ≥ 0) satisﬁes :

b

Yb = + 2

!

Yc dβc + 2b

0

Corollary 2.9 is therefore proven by plugging (2.105) and (2.107) respectively in (2.98) and (2.96), and by completing the same computation we just made with the index − instead of the index +. (0),q (2),q It is interesting to observe that the martingales (Za − , a ≥ 0) and (Zb + , b ≥ 0) which appear in an essential manner in the preceding proof, are precisely the ones that are used to prove formula (2.26), i.e. Theorem 1.7 in ([RY∗ ], Chap. XI, p. 444). 2.4.8 Third Corollary of Theorem 2.5 (: A penalisation meta-theorem) We now consider the Wiener measures (Wx , x ∈ R) penalised by the functional (Γt := Γ(L•t ), t ≥ 0) where Γ ∈ S, i.e. : Γ := Wx,t

Γ(L•t ) • Wx Wx (Γ(L•t ))

(2.109)

We state here, in this general framework, a penalisation result although we do not know how to prove it completely in such a general situation. In his Thesis, (see [N2 ]), and in [N3 ], J. Najnudel gives suﬃcient conditions on the functional Γ which allow to obtain a rigorous penalisation Theorem. Corollary 2.10. (: A penalisation meta-theorem) For every s ≥ 0 and Fs ∈ b(Fs ) Γ(L•t ) Wx F s −→ Wx [Fs MsΓ ] EWx (Γ(L•t )) t→∞

(2.110)

where (MsΓ , s ≥ 0) is a positive martingale deﬁned by : MsΓ = = Γ(λ + λ) where : Γλ (λ)

1 < ΛXs , ΓL•s > < Λx , Γ >

(2.111)

∈ Ω). (λ, λ

“Proof ” of Corollary 2.10 We know (see Chapter 0, section 0.3.3) that, in order to prove Corollary 2.10, it suﬃces to show the two following assertions : Γ(L•t ) Fs −→ MsΓ a.s. (2.112) i) Wx t→∞ Wx (Γ(L•t ))

2.4 Absolute Continuity Relationships

ii) for every s,

101

Wx (MsΓ ) = 1

(2.113)

(where MsΓ is deﬁned by (2.111)). i) Among these two points, the ﬁrst one (2.112) is clear. Indeed, according to Theorem 2.5 : √ t Wx [Γ(L•t )] −→ Γ(λ)Λx (dλ) (2.114) t→∞

Ω

and √ √ tWx [Γ(L•t )|Fs ] = tW [Γ(L•s + (Lt−s ◦ θs )• )|Fs ] √ x •−x = tW0 [Γ(λ + Lt−s )] λ=L• s x=Xs

(from the Markov property) X (dλ) −→ Γ(L•s + λ)Λ s

t→∞

Ω

(2.115)

→ Γ(λ+ λ) belongs to S. Thus, (2.114) still following Theorem 2.5, since λ and (2.115) imply (2.112). ii) However, we are not able to prove (2.113) in a general manner. We shall study in detail, in Section 2.6, a situation - where Γt is a function of Lat 1 and Lat 2 , with a1 = a2 - where (2.113) is true and where Corollary 2.10 applies. Remark 2.11. Here is now an example of the functional Γ where the penalisation meta-theorem applies in a rigorous manner. Besides, this example is interesting as it allows to make a link and recover - in a very diﬀerent way - a penalisation result by the functional f (St ) (where f : R+ → R+ is a ∞ Borel function such that 0 f (x)dx = 1 and where St := sup Xs ), a result s≤t

we obtained in [RVY,II] and which is related to the Az´ema-Yor martingales. Let us be more precise. Let σ+ (λ) denote the right extremity of the support : of λ, for λ ∈ Ω σ+ (λ) = sup{x; λ(x) > 0}

(possibly equal to +∞) (2.116) ∞ and let f : R+ → R+ , Borel, such that 0 f (x)dx = 1 and f with compact support. → R+ be deﬁned by : Let Γ : Ω Γ(λ) = f (σ+ (λ))

(2.117)

Γ(L•t ) = f (St )

(2.118)

It is clear that :

102

2 Feynman-Kac Penalisations for Brownian Motion

The application of Corollary 2.10 yields therefore : Γ(L•t ) f (St ) F s = W0 Fs W0 W0 (Γ(L•t )) W0 (f (St )) −→ MsΓ

t→∞

with : MsΓ =

1 < ΛXs , ΓL•s > < Λ0 , Γ >

(2.119)

(2.120)

Using then the form (2.76) of the measure Λ0 : < Λ0 , Γ > ≡ Γ(λ)Λ0 (dλ) Ω ∞ 1 √ = d Λ− (dλ)Γ(λ) 2π 0 Ω (because, under Λ+ , σ+ = ∞ a.s., and Γ(λ) = 0 a.s.) ∞ 1 (0) Q (f (T0 ))d = √ 2π 0 (from the deﬁnition of Λ− given in point 4) of Theorem 2.5) ∞ 1 = √ d E f 2e 2π 0 where e is a standard exponential r.v., since (law) (0) (see [RY∗ ], Chap.XI, Ex.1.27) under Q , T0 = 2e ∞ ∞ 2 2 1 (2.121) f (x)dx = dx f (x) = = √ E(2e) π 0 π 2π 0 Likewise :

< ΛXs , ΓL•s > ≡ X (dλ) = Γ(Ls + λ)Λ s

Ω

•

Ω

Ω

X (dλ) Γ(L•s + λ)Λ s f (σ+ (λ) ∨ (x + σ+ (λ)))Λ 0 (dλ)

(with λ = L•s and x = Xs ) ∞ 2 = f (σ+ (λ) ∨ (x + t))dt π 0 (from (2.121) applied to the function f(y) := f (σ+ (λ) ∨ (x + y))) ∞ 2 = f (y)dy f (σ+ (λ))(σ+ (λ) − x) + π σ+ (λ)

2.5 An Extension of Theorem 2.1

103

(with λ = L•s and x = Xs ) (from (2.118)) ∞ 2 = f (y)dy (2.122) f (Ss )(Ss − Xs ) + π Ss Finally, taking the ratio of (2.122) by (2.121), we obtain : ∞ Γ(L•t ) Fs −→ f (Ss )(Ss − Xs ) + f (y)dy (2.123) W0 t→∞ W0 (Γ(L•t )) Ss ∞ It is clear that (MsΓ := f (Ss )(Ss −Xs )+ Ss f (y)dy, s ≥ 0) is a martingale (see [RVY,II]) : it is an Az´ema-Yor martingale. We have in particular E(MsΓ ) = 1 for every s ≥ 0. In conclusion, Corollary 2.10 (the penalisation meta-theorem) is a “real theorem” in the case: Γ(λ) := f (σ+ (λ)) , i.e. Γ(L•t ) = f (St ) In his thesis, and in [N3 ], J. Najnudel also treats this example and, more generally, provides some suﬃcient conditions bearing on the functional Γ which allow to obtain a rigorous penalisation Theorem (see [N2 ] and [N3 ]).

2.5 An Extension of Theorem 2.1 2.5.1 We have studied, in Theorem 2.1, the penalisation of the Wiener mea(q) sures (Wx , x ∈ R) by the exponential functional (Et , t ≥ 0), with q ∈ I. We shall do the same here but in a more general framework. Let us start by ﬁxing some notations. For every t ≥ 0, let : gt := sup{s ≤ t; Xs = 0} , dt := inf{s ≥ t; Xs = 0}

(2.124)

gt is the last zero of X before t and dt is the ﬁrst zero after t, so that dt − gt is the excursion length which straddles t, and : dt = t + T 0 ◦ θ t

(2.125)

where T0 is the hitting time of zero and θt is the usual operator of timetranslation. Now, let q (1) , q (2) , q (3) be three elements of I. We note as q the triplet (q (1) , q (2) , q (3) ) and : ( q)

Et

1 gt (1) 1 t (2) 1 dt (3) := exp − q (Xs )ds − q (Xs )ds − q (Xs )ds 2 0 2 gt 2 t (2.126)

104

2 Feynman-Kac Penalisations for Brownian Motion ( q)

and we shall study the Wiener measures (Wx , x ∈ R) penalised by Et , i.e. : ( q)

Et

( q)

Wx,t :=

( q)

•

Wx (Et )

Wx

(2.127) ( q)

In other terms, we take up again the study of the penalisation by Et by “distinguishing what happens before gt , between gt and t, and between t and dt ”. 2.5.2 In order to study this penalisation, the role of the “denominator” : ( q)

Dt := Wx (Et )

(2.128)

is important, and we shall start by studying its asymptotic behavior as t→∞. To simplify, we assume that x = 0 and we replace the Brownian motion (Xt , t ≥ 0) by the reﬂected Brownian motion (|Xt |, t ≥ 0) (or, which is equivalent, we assume the functions q (i) , i = 1, 2, 3, to be symmetric with respect to the origin) and the functions q (i) , i = 1, 2, 3 are then elements of I+ (deﬁned on R+ ). Theorem 2.12. In this case, (Xt , t ≥ 0) is the reﬂected Brownian motion issued from 0. Then : √ √ 2 Φq(2) (∞)Φq(3) (∞) ( q) (2.129) tDt = tW0 (Et ) −→ − t→∞ π Φq(1) (0+) Let us remark that, in this formula, q (i) , i = 1, 2, 3 are deﬁned on R+ and are elements of I+ . Besides, if we take q (2) = q (1) with q (3) = 0 in (2.129), we obtain : √ 2 Φq(2) (∞) (2.130) tDt −→ − t→∞ π Φq(2) (0+) since Φq (+∞) = 1 if q ≡ 0. Besides, this formula is coherent with (2.33) since : ( q)

(q)

i) If q (2) = q (1) = q and q (3) = 0, Et = Et ii) Thus we have, by plugging (2.33) with the function q obtained by extending q (2) (or q (3) ) by symmetry with respect to 0 (to take into account the fact that in Theorem 2.12, it is the reﬂected Brownian motion which intervenes, whereas in Theorem 2.1, the studied process is non reﬂected Brownian motion), we have : √ 2 Φq+ (∞) + Φq− (∞) 2 Φq(2) (∞) =− tDt −→ ϕq (0) = Cq = − t→∞ π Φq+ (0+) + Φq− (0+) π Φq(2) (0+) since Φq+ (∞) = Φq− (∞) = Φq(2) (∞), Φq+ (0+) = Φq− (0+) = Φq(2) (0+).

2.5 An Extension of Theorem 2.1

105

Proof of Theorem 2.12 To simplify the writing, we note : 1 gt (1) (1) q (Xs )ds μt = exp − 2 0 t 1 (2) (2) μt = exp − q (Xs )ds 2 gt 1 dt (3) (3) μt = exp − q (Xs )ds 2 t ( q)

so that Et

(1)

(2)

(3)

= μt •μt •μt . (3)

i) The ﬁrst step in the proof consists to “get rid of ” the term μt . For this purpose, we write : √ √ 1 t+T0 ◦θt (3) (1) (2) tDt := tW μt μt exp − q (Xs )ds 2 t √ 1 T0 (3) ) (1) (2) + = tW μt μt W q ( X )ds exp − Xt s 2 0 (from the Markov property) √ (1) (2) = tW [μt μt Φq(3) (Xt )] (from (2.21)) (2.131) But, from Williams’ time reversal result : 1 γa (3) Φq(3) (a) = E0 q3 (Ru )du exp − 2 0

(2.132)

where γa denotes the last passage time at level a of the 3-dimensional Bessel process (Ru , u ≥ 0) starting from 0; hence : Φq(3) (a) −→ Φq(3) (∞) > 0 a→∞

(2.133)

since q (3) ∈ I+ and with the help of Jeulin’s lemma recalled in Section 2.2. ii) Working as we did in subsection 2.3.3 (Section 2.3), we condition now with respect to gt = u : √ t √ t du ! I (q1 ) (u)J (q2 ,q3 ) (t − u) tDt = (2.134) π 0 u(t − u) with :

I

(q1 )

(u) = W0

1 u (1) q (Xh )dh |Xu = 0 exp − 2 0

(2.135)

106

2 Feynman-Kac Penalisations for Brownian Motion

s 1 (q2 ,q3 ) (s) (2) J (s) = M q (Xh )dh Φq(3) (Xs ) exp − 2 0 (2.136) where M (s) denotes the law of Brownian meander. Thanks to Imhof’s absolute continuity formula, we get : √ s 1 s (2) (3) (q2 ,q3 ) (s) = E0 q (Xh )dh Φ (3) (Xs ) (2.137) exp − J 2 0 CXs q √ (3) s 1 2 = . From the asymptotic independence of with C = E0 X1 π Xs

and (Xv , v ≤ A) as s → ∞, from the scaling property and (2.132), (3) (2.133), we deduce, since Xs −→ ∞ a.s. under P0 : s→∞

1 ∞ (2) q (Xh )dh exp − t→∞ 2 0 = Φq(2) (∞)•Φq(3) (∞) (2.138) (3)

J (q2 ,q3 ) (t − u) −→ Φq(3) (∞)•E0

Hence : √ 1 tDt ∼ t→∞ π

0

t

√ t du (q1 ) √ √ I (u)du Φq(2) (∞)Φq(3) (∞) t−u u (2.139)

1 −→ √ Φq(2) (∞)Φq(3) (∞) . . . t→∞ 2π ∞ 1 τ (1) q (Xh )dh d ...E exp − 2 0 0 using the same arguments as those at the end of subsection 2.3.3 (Section 2.3). Then, since from (2.22) : Φq(1) (0+) 1 τ (1) W0 exp − , q (Xh )dh = exp 2 0 2 √ 2 1 tDt −→ − √ (0+) Φq (2) (∞)Φq (3) (∞) t→∞ Φ 2π q(1) 2 Φq(2) (∞)Φq(3) (∞) = − π Φq(1) (0+) Let us remark that, if we take as the underlying process the Brownian motion itself instead of the reﬂected Brownian motion (and then the functions q (i) , i = 1, 2, 3 are deﬁned on R and belong to I), we ﬁnd :

2.5 An Extension of Theorem 2.1

107

√ √ 1 (q) tDt = t W0 exp − Et 2 Φ (2) (∞)Φ (3) (∞) + Φ (2) (∞)Φ (3) (∞) 2 q+ q+ q− q− −→ − t→∞ π Φ (1) (0+) + Φ (1) (0+) q+

q−

2.5.3 Here is now the main Theorem of this Section Theorem 2.13. In this case, (Xt , t ≥ 0) is the reﬂected Brownian motion issued from 0. Let q = (q (1) , q (2) , q (3) ) three elements of I+ . For every s ≥ 0 and Fs ∈ b(Fs ) : lim W0

t→∞

Fs

( q)

Et

( q)

= W0 (Fs Ms(q) ) := W0 (Fs )

( q) W0 (Et )

(2.140)

( q)

where (Ms , s ≥ 0) is a positive martingale deﬁned by : (2) (q ) (2.141) Ms(q) = −Φq(1) (0+)Ψq(2) (Xs )μ(1) s μs + Φq (1) (Xs )Es gs s 1 1 = −Φq(1) (0+)Ψq(2) (Xs ) exp − q (1) (Xu )du − q (2) (Xu )du 2 0 2 gs (2.142) 1 s (1) +Φq(1) (Xs ) exp − q (Xu )du 2 0 (1)

We easily verify : ( q)

• since Φq(1) (0) = 1 and Ψq(2) (0) = 0, that M0 • since

−Φq(1) (0+)

> 0 , that

( q) Ms

= 1;

≥ 0.

Proof of Theorem 2.13 As we have recalled in subsection 2.4.8 (see the proof of Corollary 2.10), Theorem 2.13 will be proven if we demonstrate that : i) W0

( q)

Et

( q)

W0 (Et )

ii) W0 (Ms(q) ) = 1

|Fs

−→ Ms(q)

t→∞

a.s.

for every s

(2.143) (2.144)

Let us begin with the proof of (2.143). As we have already established (see Theorem 2.12) that : ( q) W0 (Et )

k 2 Φq(2) (∞)Φq(3) (∞) ∼ √ , with k = − t→∞ π Φq(1) (0+) t

(2.145)

108

2 Feynman-Kac Penalisations for Brownian Motion

it now remains to study the asymptotic behavior of : ( q)

E0 (Et |Fs )

as t → ∞

Thus, with s < t : √ √ ( q) (1) (2) (3) tW0 (Et |Fs ) = tW0 (μt μt μt |Fs ) √ (1) (2) ∼ tW0 (μt μt |Fs )Φq(3) (∞)

Ns,t :=

t→∞

(2.146) (2.147)

from point i) of the preceding proof. As in (2.143), the quantity which remains to study, will be multiplied in the numerator and denominator by the same quantity Φq(3) (∞). Thus, it remains to study : )s,t := N

√ (1) (2) tW0 (μt μt |Fs )

(2.148)

Let us ﬁrst remark that, when we shall have obtained our estimates, it will re(1) (2) (3) (1) (2) sult from (2.147) that W (q ,q ,q ) = W (q ,q ,0) ; in other terms, the function (3) q does not play any role in the penalisation. Now, working as in subsection 2.3.4, let us write : (1) (2) )s,t = N )s,t )s,t N +N (2.149) with (1) )s,t N :=

√

(1) (2)

tW0 [μt μt 1ds >t |Fs ] √ 1 t (2) (1) q (Xu )du 1ds >t |Fs = tW0 μt exp − 2 gs

(2.150)

because, on the set (ds > t), we have : gs = gt , and : (2) )s,t N :=

√

(1) (2)

tW0 [μt μt 1ds
(1) (2) )s,t )s,t We shall study N and N successively. (1) ) i) Asymptotic behavior of Ns,t Since (ds > t) = (T0 ◦ θs > t − s), we have, from (2.150) : √ (1) 1 s (2) (1) ) Ns,t = q (Xu )du × . . . tμs exp − 2 gs 1 t (2) . . . E 1T0 ◦θs >t−s exp − q (Xu )du |Fs 2 s s 1 (2) ∼ μ(1) exp − q (X )du F (t, t − s, Xs ) u s t→∞ 2 gs

(2.151)

2.5 An Extension of Theorem 2.1

109

with F (t, t − s, x) =

√

1 t−s (2) tWx 1T0 >t−s exp − q (Xu )du 2 0

from the Markov property. But, from (2.46) F (t, t − s, x) −→

t→∞

Hence : (1) )s,t N

−→

t→∞

2 Φ (2) (∞)Ψq(2) (x) π q

2 (2) Φ (2) (∞) μ(1) s μs Ψq (2) (Xs ) π q

(2.152)

(2.153)

)s,t ii) Asymptotic behavior of N Since, on (ds < t), it holds that : gt > ds , we have : (2)

√ 1 ds (1) 1 gt (1) tW0 1ds
(2) )s,t N =

with G(t, t − s, x) =

1 T0 (1) tWx 1(T0
√

after having exploited the Markov property (at time s). By applying the Markov property this time in T0 , we obtain : √ 1 T0 (1) G(t, t − s, x) = tWx 1T0
110

2 Feynman-Kac Penalisations for Brownian Motion

from Theorem 2.12 applied with q (3) = 0. Hence, plugging this estimate (2.156) in (2.155) and then in (2.154), we obtain : Φ (∞) 1 T0 (1) (1) (2) + )s,t −→ − 2 q(2) • WX q (X )du E)sq exp − N u s t→∞ π Φq(1) (0+) 2 0 2 Φq(2) (∞) q(1) = − •E •Φq (1) (Xs ) (2.157) π Φq(1) (0+) s from (2.21). Then, gathering the estimates (2.157), (2.153) and (2.145) we obtain : ( q) Et |Fs −→ Ms(q) a.s. W0 ( q) t→∞ W0 (Et ) with (2) q Ms(q) = −Φq(1) (0+)μ(1) s μs Ψq (2) (Xs ) + Es •Φq (1) (Xs ) 1 gs (1) 1 s (2) q (Xu )du − q (Xu )du = −Φq(1) (0+)Ψq(2) (Xs ) exp − 2 0 2 gs s 1 +Φq(1) (Xs ) exp − q (1) (Xu )du 2 0 (1)

( q)

It remains to prove that W0 (Ms ) = 1 for every s ≥ 0 (1) (2) Let us ﬁrst note that, although s → μs μs is not continuous - this function has a jump at the passage of a zero of X - however, the function (1) (2) s → Ψq(2) (Xs )μs μs is continuous because Ψq(2) (Xs ) = 0 if Xs = 0. Thus, ( q)

the martingale (Ms , s ≥ 0), as any Brownian martingale, is continuous. Ap( q) plying Itˆ o’s formula and taking into account the fact that M0 = 1 and the preceding remark, we obtain : s 1 ( q) (1) (2) 0 −Φq(1) (0+)μu μu Ψq(2) (Xu )(dβu + dLu ) + Ψq(2) (Xu )du Ms = 1 + 2 0 s 1 (1) + Eu(q ) Φq(1) (Xu )(dβu + dL0u ) + Φq(1) (Xu )du 2 0 s 1 (2) (1) (2) + −Φq(1) (0+)Ψq(2) (Xu )μu μu − q (Xu ) du 2 0 s 1 (1) + Φq(1) (Xu )Eu(q ) − q (1) (Xu ) du (2.158) 2 0

2.5 An Extension of Theorem 2.1

111

where we have used the expression of the reﬂected Brownian motion Xt under the (Tanaka) form βt + L0t (where (βt , t ≥ 0) is a Brownian motion) and there (1) (2) is no contribution, in Itˆ o’s formula, coming from the jumps of s → μs μs due to the preceding remark. Besides, one has:

Ψq(2) = q (2) Ψq(2)

and

Φq(1) = q (1) Φq(1) ; ( q)

then, since the measure dL0u only charges the zeros of X, Ms writes: s (2) (q (1) ) Ms(q) = 1 + [−Φq(1) (0+)μ(1) Φq(1) (Xu )]dβu u μu Ψq (2) (Xu ) + Eu ∞0 (2) (q (1) ) 0 + [−Φq(1) (0+)μ(1) (2.159) gu μgu Ψq (2) (0) + Egu Φq (1) (0+)]dLu 0

(q (1) )

Then, it remains to observe that : μgu = Egu , μgu = 1, Ψq(2) (0) = 1, to deduce that the term in dL0u in (2.159) is null and so : s (2) (q (1) ) Ms(q) = 1 + [−Φq(1) (0+)μ(1) Φq(1) (Xu )]dβu (2.160) u μu Ψq (2) (Xu ) + Eu (1)

(2)

0

(q (1) )

and we easily deduce from (2.160) (since Ψq(2) , Φq(1) , μu , μu and Eu (1)

bounded functions) that

( q) (Ms , s

≥ 0) is a martingale and

(2)

( q) E(Ms )

are

= 1.

Remark 2.14. i) As we saw in the proof of Theorem 2.13, W (q not depend on q (3) . In particular : (q (1) ,q (2) ,q (3) )

W0

(1)

,q (2) ,q (3) )

does

(q (1) ,q (2) ,0)

= W0

ii) The application of Girsanov’s Theorem allows to show the existence of a W (q) Brownian motion (Bt , t ≥ 0) such that, under W (q) , the canonical process (Xt , t ≥ 0) satisﬁes : Xt = Bt + L0t (X) t −Φ (0+)μ(1) μ(2) Ψ (X ) + E (q(1) ) Φ (X ) u u u u u q (1) q (2) q (1) du + ( q) 0 Mu (2.161) In particular, Theorem 2.13 proves the existence of a (weak) solution of this equation. Let us note the very particular aspect of this equation : indeed, (1) (2) (q (1) ) in the second term of (2.161), the quantities μu , μu , Eu are, of course, (1) 1 gu (X) (1) the functionals μu (X) = exp − 2 0 q (Xs )ds (and it is also true for

112 (2)

2 Feynman-Kac Penalisations for Brownian Motion (q (1) )

μu , Eu ). We are therefore dealing with a SDE with coeﬃcients depending on (gu (X), u ≥ 0). iii) Working as for Theorem 2.5, we deduce from Theorem 2.12 (with q (2) = (1) q (3) ≡ 0) the existence, for any x, of a positive and σ-ﬁnite measure Λx on such that, for every Γ ∈ S : Ω, √ • tWx [Γ(Lgt )] −→ Γ(λ)Λ(1) (2.162) x (dλ) Ω

t→∞

In the same spirit, we can prove the existence of a measure Λ(1,2,3) describing the asymptotic behavior, as t → ∞, of local times until gt , then between gt and t, and ﬁnally between t and dt . (1) iv) Relation between the measure Λ0 (see formula (2.162)) and the measure Π, deﬁned and studied in Chapter 3. Once again here, (Xt , t ≥ 0) is the reﬂected Brownian motion issued from 0. Let q ∈ I+ . We have, from (2.162) : gt √ (1) tE0 exp − q(Xs )ds −→ exp(− < q, λ >)Λ0 (dλ) (2.163) t→∞

0

Ω

But, on the other hand : gt q(Xs )ds = [βn (gt ) − αn (gt )] 0

1

q(Xαn (gt )+u(βn (gt )−αn (gt )) )du 0

n≥1

where ([αn (gt ), βn (gt )], n ≥ 1) are the excursion intervals away from 0 of X (1) (2) before gt and whose lengths are equal to, in decreasing order Vgt ≥ Vgt ≥ (n) ... ≥ Vgt ≥ ... (see Chapter 3, Section 3.8). By scaling, we then have : gt √ tW0 exp − q(Xs )ds 0 1 ! [βn (gt ) − αn (gt )] q( βn (gt ) − αn (gt )rn (u))du = W exp − 0

n≥0

where, in this case, the (rn (•), n ≥ 1) are Bessel bridges of dimension 3, which are independent, and are independent of excursion lengths. Thus, denoting : 1 √ √ (q) q( λru )du (2.164) O ( λ) := E exp −λ 0

we have :

√ tW0 exp −

gt

q(Xs )ds 0

√ (n) (q) = tE( Π O ( Vgt )) n≥1

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

113

Let us deﬁne ) (q) (x1 , x2 , ..., xn ...) := Π O(q) (xn ) O

(x1 ≥ x2 ... ≥ xn ...)

n≥1

(2.165)

we then have, from Theorem 3.16, Chapter 3, Section 3.8 of this Monograph : √ (n) (q) ) (q) (s)Π(ds) O tE( Π O ( Vgt )) −→ (2.166) n≥1

t→∞

S↓

Then, comparing (2.166) and (2.163) : (1) exp(− < q, λ >)Λ0 (dλ) = Ω

) (q) (s)Π(ds) O

(2.167)

S↓

) (q) is deﬁned by (2.165) and (2.164). where O (1) The formula (2.167) is the announced relation between Λ0 and Π. Let us note that (2.167) is true for any q ∈ I+ . A minor comment about our notation The reader may have noticed, with the above identities, the appearance of Π for inﬁnite products, and Π for our σ-ﬁnite measure on S ↓ ; this should

n≥1

not induce any confusion; in fact, this notation highlights the presence of the sequence (V (n) ) in the discussion.

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2 2.6.1 Our aim in this section 2.6 is to describe results for the penalisation of the Wiener measures Wx (x ∈ R) with a functional Γ(L•t ), t ≥ 0, diﬀerent from the one presented in Remark 2.11, and for which Corollary 2.10 may be applied; i.e. : we obtain a penalisation Theorem. The penalisation functional we choose here is : {Γt := f (Lat 1 )g(Lat 2 ), t ≥ 0}

(a1 < a2 )

(2.168)

where f and g are two functions from R+ to R+ , of class C 2 say, with compact support (we might consider in a more general manner Γt in the form Γt = h(Lat 1 , Lat 2 ), with h : R+ × R+ → R+ , but this would not change much apart from having heavier notations). We shall therefore study the asymptotic behavior, as t → ∞, of the family of measures : f,g := Wx,t

f (Lat 1 )g(Lat 2 ) • Wx Wx (f (Lat 1 )g(Lat 2 ))

(2.169)

114

2 Feynman-Kac Penalisations for Brownian Motion

and show, in this framework, a penalisation Theorem (Theorem 2.18), below. We think of this theorem as the prototype of a truly more general theorem which would describe the penalisation with {Γt := h(Lat 1 , Lat 2 , ..., Lat n ), t ≥ 0} (a1 < a2 < ... < an ) (see [N3 ]) and, hopefully, we would then be able to let n → ∞. But, we have not reached this goal yet, and, for now, we shall study more modestly the penalisation with the functional (Γt , t ≥ 0) deﬁned by (2.168). What do we know ? • Corollary 2.6, once extended in an obvious way to a couple of additive functionals, Aat 1 = Lat 1 and Aat 2 = Lat 2 yields the existence of a family of positive, and σ-ﬁnite measures (ν ax1 ,a2 ) carried by R+ × R+ , such that : √ tWx (f (Lat 1 )g(Lat 2 )) −→ f (1 )g(2 )ν ax1 ,a2 (d1 , d2 ) (2.170) t→∞

[0,∞]2

• The (proven) part of Corollary 2.10 (the penalisation “meta-theorem”) allows to assert that : f (Lat 1 )g(Lat 2 ) Fs −→ Msf,g Wx a.s. (2.171) t→∞ Wx (f (Lat 1 )g(Lat 2 )) with : Msf,g

:=

[0,∞]2

f (Las 1 + 1 )g(Las 2 + 2 )ν aX1s,a2 (d1 , d2 ) a1 ,a2 (d1 , d2 ) [0,∞]2 f (1 )g(2 )ν x

(2.172)

Thus, our program is the following : i) First, we give the explicit form of the measure ν ax1 ,a2 ; ii) Then, we use this explicit form in order to also compute explicitly Msf,g for every s ≥ 0; iii) We then show that (Msf,g , s ≥ 0) is a (true) martingale, from which we deduce that Ex (Msf,g ) = 1 for every s > 0. This allows us ﬁnally - see Corollary 2.10 - to obtain Theorem 2.18, which we only state at the end of this program, because for this statement we need quite a few notations which we will introduce little by little. 2.6.2 Computation of the measure ν ax1 ,a2 2.6.2.i) From (2.170), we have, for λ1 , λ2 > 0 : √ 1 a1 a2 tWx exp − (λ1 Lt + λ2 Lt ) 2 1 −→ exp − (λ1 1 + λ2 2 ) ν ax1 ,a2 (d1 , d2 ) t→∞ 2 [0,∞]2

(2.173)

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

115

and, according to Theorem 2.1, point 1), applied with q = λ1 δa1 + λ2 δa2 : √ 1 tWx exp − (λ1 Lat 1 + λ2 Lat 2 ) −→ ϕ(λ1 , λ2 , x) (2.174) t→∞ 2 where ϕ (as a function of x) satisﬁes (in the sense of distributions) : ϕ = ϕ(λ1 δa1 + λ2 δa2 ) ϕ (−∞) = − π2 ϕ (+∞) = π2 ,

(2.175)

Formula (2.175) implies that ϕ(λ1 , λ2 , •) is aﬃne on the intervals, ] − ∞, a1 ], [a1 , a2 ], [a2 , ∞[. Denoting Ai = ϕ(λ1 , λ2 , ai ) i = 1, 2 (A1 and A2 are unknown), (2.175) implies : ⎧ 2 ⎪ (A + a1 − x) (x < a1 ) ⎪ ⎪ ⎨π 1 A2 −A1 a2 A1 −a1 A2 2 (2.176) ϕ(λ1 , λ2 , x) (x ∈ [a1 , a2 ]) π a2 −a1 x + a2 −a1 ⎪ ⎪ ⎪ 2 ⎩ (x > a2 ) π (A2 + x − a2 ) Letting γ =

1 a2 −a1 ,

formula (2.175) then yields : γ(A2 − A1 ) + 1 = λ1 A1 1 − γ(A2 − A1 ) = λ2 A2

A1 =

i.e.

2γ + λ2 2γ + λ1 1 , A2 = , γ= γ(λ1 + λ2 ) + λ1 λ2 γ(λ1 + λ2 ) + λ1 λ2 a2 − a1

2.6.2.ii) From (2.173) and (2.174), we then have : 1 exp − (λ1 1 + λ2 2 ) ν ax1 ,a2 (d1 , d2 ) = ϕ(λ1 , λ2 , x) 2 [0,∞]2

(2.177)

(2.178)

where ϕ(λ1 , λ2 , x) is given by (2.176) and (2.177). It remains then, to compute ν ax1 ,a2 , to invert the Laplace transform (2.178). This is the following step. We denote by Iν the modiﬁed Bessel function of index ν (see [Leb], p. 108). For ν=0: 1 z 2k (k!)2 2 k=0 γ √ θ(x, y) := exp − (x + y) I0 (γ xy) 2 I0 (x) :=

and let

∞

(2.179) (2.180)

A simple computation then proves that : ∞ ∞ 1 γ1 1 exp − (λ1 1 + λ2 2 ) exp − A1 (λ1 , λ2 ) = d1 ⊗ δ0 (d2 ) 2 2 2 0 0 1 ∂θ γ θ ( + , )d d + 1 2 1 2 2 ∂2 2

116

2 Feynman-Kac Penalisations for Brownian Motion

and

1 1 γ2 δ0 (d1 ) ⊗ exp − exp − (λ1 1 + λ2 2 ) d2 2 2 2 0 1 ∂θ γ + + θ (1 , 2 )d1 d2 2 ∂1 2 (2.181)

∞ ∞

A2 (λ1 , λ2 ) = 0

2.6.2.iii) Of course, according to point 4) of Theorem 2.5 and the description of the measure Λ0 , the function ϕ(λ1 , λ2 , •) and the measure ν ax1 ,a2 must be related to the laws of Bessel processes of dimension 2 and 0. Besides, we will see this in detail by giving another proof of the explicit form of the measure ν ax1 ,a2 in the following points 2.6.v and 2.6.vi. But, at this point, let us recall the following formulae (see [RY∗ ], Chap. IX), ) (δ) where (Q a , a > 0) denotes the transition semigroup of the Bessel process of (δ) dimension δ (and where qa (1 , 2 ) denotes its density for δ > 0). ⎧ ∞ ⎪ (0) ⎪ ) ⎨ Qa h() = exp − qa(0) (, 1 )h(1 )d1 h(0) + 2a 0 ∞ (2.182) ⎪ ) (2) ⎪ qa(2) (, 1 )h(1 )d1 ⎩Q a h() = 0

where : ⎧ − 12 √ 1 1 1 ( + 1 ) ⎪ (0) (4) ⎪ ⎨ qa (, 1 ) = qa (1 , ) = exp − I1 2a 2a a √ (2.183) ⎪ ( + 1 ) 1 1 (2) (2) ⎪ ⎩ qa (, 1 ) = qa (1 , ) = exp − I0 2a 2a a We then have : ⎧ ∂θ (2) (0) ⎪ ⎪ ( , ) + γθ(1 , 2 ) = q 1 (1 , 2 ) + q 1 (1 , 2 ) ⎪ ⎨ ∂ 1 2 γ γ 2

⎪ ⎪ ∂θ (2) (0) ⎪ ⎩ (1 , 2 ) + γθ(1 , 2 ) = q 1 (1 , 2 ) + q 1 (2 , 1 ) γ γ ∂1 Let us show e.g. the ﬁrst assertion of (2.184). Since γ ! θ(1 , 2 ) := exp − (1 + 2 ) I0 (γ 1 2 ) 2 we obtain: γ ! ∂θ γ = − exp − (1 + 2 ) I0 (γ 1 2 ) ∂2 2 2 γ γ √ ! 1 + exp − (1 + 2 ) √ I0 (γ 1 2 ) 2 2 2

(2.184)

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

117

But, since I0 = I1 (see [Leb], p.108), we get: γ ! ∂θ γ + γθ = exp − (1 + 2 ) I0 (γ 1 2 ) ∂2 2 2 √ γ ! 1 + exp − (1 + 2 ) √ I1 (γ 1 2 ) 2 2 (2)

(0)

γ

γ

= q 1 (1 , 2 ) + q 1 (2 , 1 ) and the proof of the second assertion of (2.184) is analogous. 2.6.2.iv) Let us return to formula (2.181) which we write, thanks to (2.184) : ∞ ∞ 1 γ1 1 exp − (λ1 1 + λ2 2 ) exp − A1 (λ1 , λ2 ) = d1 ⊗ δ0 (d2 ) 2 2 2 0 0 1 (2) (0) + q 1 (1 , 2 ) + q 1 (1 , 2 ) d1 d2 γ γ 2 ∞ ∞ 1 1 γ2 A2 (λ1 , λ2 ) = δ0 (d1 ) ⊗ exp − exp − (λ1 1 + λ2 2 ) d2 2 2 2 0 0 1 (2) (0) + (2.185) q 1 (1 , 2 ) + q 1 (2 , 1 ) d1 d2 γ γ 2 2.6.2.v) The computation of ν ax1 ,a2 is now ﬁnished : we use (2.178), and ϕ(λ1 , λ2 , x) is the Laplace transform, in λ1 , λ2 of ν ax1 ,a2 , and ϕ is given by 1 ): (2.176), (2.177) and (2.185). We ﬁnd (let us recall that γ = a2 −a 1

2 a1 ,a2 ν (d1 , d2 ) = π x

⎧ ⎪ if x < a1 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if x > a2 , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

(a1 − x)δ0,0 (d1 , d2 ) + 12 exp − γ2 1 d1 ⊗ δ0 (d2 ) (2) (0) 1 + 2 q 1 (1 , 2 ) + q 1 (1 , 2 ) d1 d2 γ γ (x − a2 )δ0,0 (d1 , d2 ) + 12 δ0 (d1 ) ⊗ exp − γ2 2 d2 (2) (0) 1 + 2 q 1 (1 , 2 ) + q 1 (2 , 1 ) d1 d2 γ γ (2.186) γ γ1 d1 ⊗ δ0 (d2 ) if x ∈ [a , a ] , (a − x) exp − ⎪ 1 2 2 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (2) (0) ⎪ + q 1 (1 , 2 ) + q 1 (1 , 2 ) d1 d2 ⎪ ⎪ ⎪ γ γ ⎪ ⎪ ⎪ ⎪ γ γ2 ⎪ d2 + (x − a ) δ (d ) ⊗ exp − ⎪ 1 0 1 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (2) (0) ⎪ + q 1 (1 , 2 ) + q 1 (2 , 1 ) d1 d2 ⎩ γ

γ

118

2 Feynman-Kac Penalisations for Brownian Motion

2.6.2.vi) We now present a second proof of formula (2.186), based upon the decomposition of the measure 1 Λ0 = √ 2π

0

∞

− (Λ+ + Λ )d

(see (2.76))

We assume, for conciseness, that x = 0, and 0 < a1 < a2 . By deﬁnition of Λ0 and ν a0 1 ,a2 , we have :

f (1 )g(2 )ν a0 1 ,a2 (dλ1 , d2 ) =

with Λ0 =

∞

√1 2π

1 Λ+ 0 = √ 2π

0

0

∞

Ω

f (λ(a1 ))g(λ(a2 ))Λ0 (d)

(2.187)

− (Λ+ + Λ )d. Let us denote :

1 − Λ+ d , Λ0 = √ 2π

0

∞

+ − Λ− (2.188) d , so : Λ0 = Λ0 + Λ0

− (let us recall that Λ+ 0 and Λ0 are σ-ﬁnite measures on Ω = C(R → R+ ) and that we denote by (Yy , y ∈ R) the process of coordinates on this space). − Lemma 2.15. Under Λ+ 0 , resp. Λ0 , the law of Ya (a > 0) is given by :

1 dx Λ+ 0 (Ya ∈ dx) = √ 2π 1 Λ− (2aδ0 (dx) + dx) 0 (Ya ∈ dx) = √ 2π

(2.189) (2.190)

Proof of Lemma 2.15 We begin with (2.189). From the description given in point 4) of Theorem − 2.5 for Λ+ 0 and Λ0 , we have : ∞ √ (2) + 2πΛ0 (f (Ya )) = Q (f (Ya ))d 0 ∞ ∞ = d qa(2) (, )f ( )d 0 0 ∞ ∞ = f ( )d qa(2) ( , )d 0 0 ∞ ) (2) 1( ) = f ( )d Q a 0 ∞ = f ( )d 0

(from (2.182)) (from (2.183) and Fubini)

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

119

We now prove (2.190). As previously, we have : ∞ √ (0) 2πΛ− (f (Y )) = Q (f (Ya ))d a 0 0 ∞ ∞ (0) d exp− f (0)+ qa (, )f ( )d (from (2.182)) = 2a 0 ∞ 0∞ = 2af (0) + f ( )d qa(4) ( , )d 0

0

(from (2.183) and Fubini) ∞ ) (4) = 2af (0) + f ( )d Q a 1( ) 0 ∞ = 2af (0) + f ( )d 0

We now plug (2.190) and (2.189) successively in (2.187) and (2.188); we obtain, from the Markov property : 1 √ f (λ(a1 ))g(λ(a2 ))Λ− 0 (dλ) 2π Ω ∞ 1 (0) (0) ) ) f (1 )Qa2 −a1 g(1 )d1 = √ 2a1 f (0)Qa2 −a1 g(0) + 2π 0 ∞ 1 γ1 = √ f (1 )d1 exp − g(0) 2a1 f (0)g(0) + 2 2π 0 ∞ (0) q 1 (1 , 2 )g(2 )d2 + 0

γ

(0)

after having used once more (2.183), and the fact that q 1 (0, ) = 0 γ ∞ 1 γ1 = √ exp − f (1 )d1 2a1 f (0)g(0) + g(0) 2 2π 0 ∞ ∞ (0) + f (1 )g(2 )q 1 (1 , 2 )d1 d2 (2.191) 0

0

γ

and likewise : 1 √ f (λ(a1 ))g(λ(a2 ))Λ+ 0 (dλ) 2π Ω ∞ 1 ) (2) f (1 )Q =√ a2 −a1 g(1 )d1 2π 0 ∞ ∞ 1 (2) =√ f (1 )g(2 )q 1 (1 , 2 )d1 d2 γ 2π 0 0 (2)

(2)

γ

γ

(by using q 1 (1 , 2 ) = q 1 (2 , 1 )).

(2.192)

120

2 Feynman-Kac Penalisations for Brownian Motion

The addition of (2.191) and (2.192) now leads to : ∞ ∞ f (1 )g(2 )ν a0 1 ,a2 (d1 , d2 ) 0 0 ∞ γ1 1 √ 2a1 f (0)g(0) + g(0) exp − = f (1 )d1 2 2π 0 ∞ ∞ (0) (2) + f (1 )g(2 )(q 1 (1 , 2 ) + q 1 (1 , 2 ))d1 d2 0

γ

0

(2.193)

γ

It is now clear that (2.193) is exactly the ﬁrst line of (2.186), in the case x = 0 and 0 < a1 < a2 . But, since : ν ax1 ,a2 = ν a0 1 −x,a2 −x , we have obtained the ﬁrst line of (2.186) in the general case. Of course, the second line of (2.186) follows from the ﬁrst by symmetry. 2.6.2.vii) The arguments presented in 2.6.2.vi) to compute ν ax1 ,a2 when x < a1 < a2 (or a1 < a2 < x) allow, mutatis mutandis, to compute ν ax1 ,a2 when a1 < x < a2 . We do not give these computations. Here is the value a0 1 ,a2 which is found for a1 < 0 < a2 : ν ∞ 1 2 (2) a0 1 ,a2 (d1 , d2 ) = ν exp − q|a1 | (2 , 1 )d2 d1 ⊗ δ0 (d2 ) 2 2a2 ∞ 0 1 1 exp − + d2 δ0 (d1 ) ⊗ qa(2)2 (1 , 2 )d1 2 2|a1 | 0 1 (4,2) (4,2) q|a1 |,a2 (1 , 2 ) + qa2 ,|a1 | (1 , 2 ) d1 d2 + (2.194) 2 where we noted :

(4,2)

qs,t (1 , 2 ) :=

0

=

∞

∞

(2)

qs(4) (1 , )qt (, 2 )d

(2.195)

(2)

qs(0) (, 1 )qt (, 2 )d

0

Of course, it would be preferable to verify that, for a1 < 0 < a2 , we really a0 1 ,a2 = ν a0 1 ,a2 , which is not immediate when we compare (2.194) to have : ν the last line of (2.186), but which hinges on : Lemma 2.16. For a1 < x < a2 (with γ = i) 0

exp −

∞

1 2(x − a1 )

1 a2 −a1 ),

one has :

(2) qa2 −x (1 , 2 )d1

γ2 = γ(x − a1 ) exp − 2

(2 > 0) (2.196)

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

ii)

(4,2)

121

(4,2)

qx−a1 ,a2 −x (1 , 2 ) + qa2 −x,x−a1 (1 , 2 ) (2.189’) (2)

(0)

= qa2 −a1 (1 , 2 ) + qa2 −a1 (2 , 1 )

(1 , 2 > 0)

Proof of Lemma 2.16 i) We now prove (2.196) by computing the Laplace transform of each of the two sides of this relation and by using the classical formula (see [RY∗ ], Chap. XI) : λy −δ/2 [exp(−λ Y )] = (1 + 2λ t) exp − Q(δ) (λ, y ≥ 0) (2.197) t y 1 + 2λ t (2)

Applying (2.197) and the symmetry property of q 1 , we obtain : γ

∞

1 (2) exp − qa2 −x (1 , 2 )d1 2(x − a ) 1 0 ∞ 1 (2) = exp − (2 , 1 )d1 q 2(x − a1 ) a2 −x 0 ⎞ ⎛ 2 x − a1 1 2 2(x−a1 ) ⎠ ⎝ = = exp − 1 exp − 2 −x a2 − a1 2(a2 − a1 ) 1 + ax−a a2 −x 1 1+ x−a 1 γ2 = γ(x − a1 ) exp − 2 ii) We now prove (2.189’) by computing once again the Laplace transform of each of the two sides of this relation and by using (2.197). We successively obtain : ∞ ∞ (2) exp (−λ1 1 − λ2 2 ) qa2 −a1 (1 , 2 )d1 d2 0 ∞ 0 1 λ2 2 exp −λ1 1 − = d1 1 + 2λ2 (a2 − a1 ) 1 + 2λ2 (a2 − a1 ) 0 1 (2.198) = λ1 + λ2 + 2λ1 λ2 (a2 − a1 ) Likewise :

(0)

exp (−λ1 1 − λ2 2 ) qa2 −a1 (2 , 1 )d1 d2 ∞ λ1 2 = exp −λ2 2 − d2 1 + 2λ1 (a2 − a1 ) 0 1 + 2λ1 (a2 − a1 ) = λ1 + λ2 + 2λ1 λ2 (a2 − a1 )

(2.199)

122

And : = =

2 Feynman-Kac Penalisations for Brownian Motion

(4,2)

qx−a1 ,a2 −x (1 , 2 ) exp(−λ1 1 − λ2 2 )d1 d2 (4)

(2)

qx−a1 (1 , y)qa2 −x (y, 2 ) exp(−λ1 1 − λ2 2 )d1 d2 dy (0)

(2)

qx−a1 (y, 1 ) exp(−λ1 1 )qa2 −x (y, 2 ) exp(−λ2 2 )d1 d2 dy (from (2.183)) λ1 y 1 λ2 y = dy exp − exp − 1 + 2λ1 (x − a1 ) 1 + 2λ2 (a2 − x) 1 + 2λ2 (a2 − x) (1 + 2λ2 (a2 − x))(1 + 2λ1 (x − a1 )) 1 = 1 + 2λ2 (a2 − x) λ1 + λ2 + 2λ1 λ2 (a2 − a1 ) 1 + 2λ1 (x − a1 ) (2.200) = λ1 + λ2 + 2λ1 λ2 (a2 − a1 ) And likewise, by symmetry : (4,2) qa2 −x,x−a1 (1 , 2 ) exp(−λ1 1 − λ2 2 )d1 d2 =

1 + 2λ1 (a2 − x) λ1 + λ2 + 2λ1 λ2 (a2 − a1 ) (2.201)

Formula (2.189’) is then proven, as we can notice that the sum of (2.198) and 1 (a2 −a1 ) (2.199) is equal to that of (2.200) and (2.201), i.e. to 2 λ1 +λ1+λ . 2 +2λ1 λ2 (a2 −a1 ) As a conclusion : we have proven, by two diﬀerent methods, that : √ tWx [f (Lat 1 )g(Lat 2 )] −→ f (1 )g(2 )ν ax1 ,a2 (d1 , d2 ) t→∞

R+

R+

where ν ax1 ,a2 is given by (2.186). Let us note that formulae (2.184) have not been obtained in the way we have just presented them : we have, in fact, computed ν ax1 ,a2 following the two methods developed above, and then deduced that these formulae (2.184) were true. 2.6.3 Computation of (Msf,g , s ≥ 0) We recall that Msf,g is given by (2.172). As ν ax1 ,a2 is now explicitly known, the application of this formula (2.172) yields (up to a multiplicative constant) : 1 f,g Ms = 1Xs
2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

1 2

∞ ∞

123

f (Las 1 + 1 )g(Las 2 + 2 )θ2 (1 , 2 )d1 d2 ∞ 1 γ1 a2 a1 +1a1 <Xs a2 (Xs − a2 )f (Las 1 )g(Las 2 ) + f (Las 1 )• 2 ∞ γ 2 g(Las 2 + 2 ) exp − d2 2 0 ∞ ∞ 1 a1 a2 + f (Ls + 1 )g(Ls + 2 )θ1 (1 , 2 )d1 d2 (2.202) 2 0 0

+

0

0

where we have denoted : θ1 (1 , 2 ) :=

∂θ (2) (0) (1 , 2 ) + γθ(1 , 2 ) = q 1 (1 , 2 ) + q 1 (2 , 1 ) γ γ ∂1

(2.203)

∂θ (2) (0) θ2 (1 , 2 ) := (1 , 2 ) + γθ(1 , 2 ) = q 1 (1 , 2 ) + q 1 (1 , 2 ) γ γ ∂2 It is not diﬃcult to verify, by observing what happens near points Xs = a1 or Xs = a2 , that (Msf,g , s ≥ 0) is a continuous function of s. 2.6.4 Let us show that (Msf,g , s ≥ 0) is a (true) martingale We start by showing that it is a local martingale. We apply Itˆ o-Tanaka’s formula and since Msf,g is a partly linear function of Xs , it remains to see o-Tanaka development of Msf,g are that the terms in dLas 1 and dLas 2 in the Itˆ a1 a2 null. As the terms in dLs and dLs have analogous forms, we will only take an interest in the term in dLas 1 . Let us note this term S(s)dLas 1 . So, it remains to show that S(s) ≡ 0. What is the value of S(s) ? We write S(s) = S1 (s) + S2 (s) where : i) S1 (s) is the contribution due to the presence of Xs in the expression of Msf,g ; it is therefore given by the “slope rupture” at point a1 , ii) S2 (s) is the contribution due to the presence of Las 1 in the expression of Msf,g ; it is given by the derivative related to (the letter) Las 1 .

124

2 Feynman-Kac Penalisations for Brownian Motion

i) Computation of S1 (s) We then have : ∞ 1 γ γ2 S1 (s) = g(Las 1 + 2 ) exp − f (Las 1 ) d2 2 2 2 0 ∞ ∞ + f (Las 1 + 1 )g(Las 2 + 2 )θ1 (1 , 2 )d1 d2 0 0 ∞ γ1 a2 a1 −g(Ls ) f (Ls + 1 ) exp − d1 2 0 ∞ ∞ a1 a2 − f (Ls + 1 )g(Ls + 2 )θ2 (1 , 2 )d1 d2 0 0 a1 a2 +f (Ls )g(Ls ) (0)

(0)

γ

γ

As θ1 (1 , 2 ) − θ2 (1 , 2 ) = q 1 (2 , 1 ) − q 1 (1 , 2 ), we then have : ∞ 1 γ γ2 a1 a2 S1 (s) = g(Ls + 2 ) exp − f (Ls ) d2 2 2 2 ∞ 0 γ1 −g(Las 2 ) f (Las 1 + 1 ) exp − d1 2 0 ∞ ∞ (0) (0) a1 a2 + f (Ls + 1 )g(Ls + 2 )(q 1 (2 , 1 ) − q 1 (1 , 2 ))d1 d2 γ γ 0 0 +f (Las 1 )g(Las 2 ) (2.204) ii) Computation of S2 (s) We have : S2 (s)dLas 1 is the sum of 3 terms (1), (2) and (3) : a1 a2 a1 dLas 1 p.s. (1) := ((a1 − X s )f (L Xs = a1 s ∞)g(Ls ))dLs =0 since ∂ γ1 1 f (a + 1 ) exp − (2) := g(Las 2 ) d1 a=Las 1 dLas 1 2 ∂a 0 2 1 = − g(Las 2 )f (Las 1 )dLas 1 2 ∞ γ γ1 a2 a1 (2.205) + g(Ls ) f (Ls + 1 ) exp − d1 dLas 1 2 0 4 1 ∂ (2) a1 f (a + 1 )g(Las 2 + 2 ) q 1 (1 , 2 ) (3) := s γ 2 ∂a a=L (0) +q 1 (1 , 2 ) d1 d2 dLas 1 γ

The computation of the derivative featuring in (3) induces two other terms : (3) = (31 ) + (32 ), with :

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

125

∞ 1 (2) (0) a1 a1 a2 a2 (31 ):=(dLs ) − f (Ls ) g(2 ) q 1 (0, 2 − Ls ) + q 1 (0, 2 − Ls ) d2 a γ γ 2 Ls 2 ∞ γ γ 2 = (dLas 1 ) − f (Las 1 ) g(Las 2 + 2 ) exp − (2.206) d2 4 2 0 (0)

(2)

γ

γ

since, from (2.183) : q 1 (0, x) = 0 and q 1 (0, x) =

γ 2

exp − γx 2 .

2 ∂ θ f (1 )g(2 ) (1 − a, 2 − b) a1 a2 ∂1 ∂2 Ls Ls ∂θ (1 − a, 2 − b) d1 d2 +γ a ∂1 a=Ls 1

(32 ) := −

∞ ∞

(2.207)

a b=Ls 2

The comparison of (2.204) with the addition of (2.205), (2.206) and (2.207) proves therefore that S(s) = S1 (s) + S2 (s) ≡ 0 is an immediate consequence of : Lemma 2.17. The following identity holds: 2 ∂ θ ∂θ γ (0) (0) +γ (1 , 2 ) = (q 1 (2 , 1 ) − q 1 (1 , 2 )) γ ∂1 ∂2 ∂1 2 γ

(2.208)

Once we will have proven Lemma 2.17, then we will have completed the proof that (Msf,g , s ≥ 0) is a local martingale. Proof of Lemma 2.17

√ Since θ(1 , 2 ) := exp − γ2 (1 + 2 ) I0 (γ 1 2 ), we have : γ γ ! ∂θ γ 2 = − θ + exp − (1 + 2 ) I1 (γ 1 2 ) (since I0 = I1 ) ∂1 2 2 2 1 γ (0) (2.209) = − θ + q 1 (2 , 1 ) γ 2 ∂θ γ (0) = − θ + q 1 (1 , 2 ) γ ∂2 2 γ ! γ 1 γ =− θ+ exp − (1 + 2 ) I1 (γ 1 2 ) (since I0 = I1 ) 2 2 2 2 γ ! ∂2θ γ ∂θ γ =− + exp − (1 + 2 ) I1 (γ 1 2 ) ∂1 ∂2 2 ∂1 4 2 γ ! γ 2 1 − exp − (1 + 2 ) I0 (γ 1 2 ) 4 2 2 γ ! γ 2 1 2 + exp − (1 + 2 ) I1 (γ 1 2 ) 4 2 1 2 Since : I1 (z) + zI1 (z) = zI0 (z), (see [Leb] p. 110)

126

2 Feynman-Kac Penalisations for Brownian Motion

γ ! ∂2θ γ ∂θ γ 2 1 =− − exp − (1 + 2 ) I1 (γ 1 2 ) ∂1 ∂2 2 ∂1 4 2 2 γ ! γ2 + exp − (1 + 2 ) I0 (γ 1 2 ) 4 2 γ ∂θ γ (0) γ2 =− + θ − q 1 (1 , 2 ) 2 ∂1 4 2 γ Hence :

∂2θ ∂θ +γ (1 , 2 ) ∂1 ∂2 ∂1 γ ∂θ γ (0) = (1 , 2 ) − q 1 (1 , 2 ) + 2 ∂1 2 γ 2 γ (0) γ = − θ(1 , 2 ) + q 1 (2 , 1 ) + 4 2 γ (from (2.209)) γ (0) (0) = (q 1 (2 , 1 ) − q 1 (1 , 2 )) γ 2 γ

γ2 θ(1 , 2 ) 4 γ2 γ (0) θ(1 , 2 ) − q 1 (1 , 2 ) 4 2 γ

2.6.5 Itˆo-Tanaka’s formula which we have just written yields : s Msf,g = 1 + α(u)dXu

(2.210)

0

Observing formula (2.202) which gives Msf,g , it is not diﬃcult to see, taking into consideration the hypotheses made on f and g that for every t and every p>0: sup E(|α(u)|p ) < ∞ u
We deduce that ≥ 0) is a true martingale, and therefore that E(Msf,g ) = 1 for every s ≥ 0. Thus, we have obtained : (Msf,g , s

Theorem 2.18. Let f and g denote two functions from R+ to R+ , of class C 2 with compact support. Then : i)

√ tWx [f (Lat 1 )g(Lat 2 )] −→ t→∞

0

∞ ∞

f (1 )g(2 )ν ax1 ,a2 (d1 , d2 )

0

where ν ax1 ,a2 is deﬁned by (2.186). ii) For every s ≥ 0 and every Fs ∈ b(Fs ) : f (Lat 1 )g(Lat 2 ) lim Wx Fs = EWx [Fs Msf,g ] := Wxf,g (Fs ) t→∞ Wx [f (Lat 1 )g(Lat 2 )] where (Msf,g , s ≥ 0) is the positive martingale deﬁned by (2.202).

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

127

Let us note that, in (2.202), (Msf,g , s ≥ 0) is only speciﬁed up to one multiplicative constant, and that this constant is such that M0f,g = 1. Let us also note that we can weaken the hypotheses on f and g and only assume that f and g are integrable. (see e.g. [RVY,II] to justify such a weakening of the hypotheses). Besides, the situation where a1 = a2 , and g ≡ 1, has been studied in [RVY,II]. Remark 2.19. i) Theorem 2.18 and the explicit form, given by (2.202), of the martingale (Msf,g , s ≥ 0) invites to seek all the (local) martingales (Ms , s ≥ 0) of the form Ms = F (Las 1 , ..., Las k , Xs ), where F is a regular function. Thus, we have obtained : ii) Let a1 < a2 < ... < ak and B0 , B1 denote two functions from Rk+ to R, of class C k . Let us deﬁne : C1 (α1 , α2 , ..., αk ) = B0 (α1 , α2 , ..., αk ) + ai B1 (α1 , ..., αk )

i = 1, 2, ..., k (2.211)

and Ai := −2

i ∂Ci + (−1)r 2r (ai − ar−1 )(ar−1 − ar−2 )...(a2 − a1 )• ∂αi r=2 1 <2 <...
r

•

∂ C1 ∂α1 ...∂αr−1 ∂αi

,

i = 1, 2, ..., k

(2.212)

Moreover, let : F (α1 , ..., αk , x) := B0 (α1 , ..., αk )+xB1 (α1 , ..., αk )+

k

Ai (α1 , ..., αk )(x−ai )+

i=1

(2.213) With these notations : (F (Las 1 , ..., Las k , Xs ), s ≥ 0) is a local martingale, and this is the general form of all local martingales which are regular functions of Las 1 , ..., Las k , Xs . iii) Proof Thus, let : F : Rk+ × R → R regular such that (F (Las 1 , ..., Las k , Xs ), s ≥ 0) is a local martingale. Itˆo’s formula then implies that, as a function of the last variable, F is aﬃne on a ﬁnite number of intervals. So, it is written as follows : F (α1 , ..., αk , x) = B0 (α1 , ..., αk ) + xB1 (α1 , ..., αk ) +

k

Ai (α1 , ..., αk )(x − ai )+

i=1

(2.214) this writing being canonical. Then, writing Itˆ o-Tanaka’s formula for (F (Las 1 , ..., Las k , Xs ), s ≥ 0), and with the coeﬃcients of the terms in dLas i being necessarily null, we obtain :

128

∂B0 ∂α1 ∂B0 ∂α2 ... ∂B0 ∂αi ... ∂B0 ∂αk

2 Feynman-Kac Penalisations for Brownian Motion

∂B1 1 + A1 = 0 ∂α1 2 ∂B1 ∂A1 1 + a2 + (a2 −a1 ) + A2 = 0 ∂α2 ∂α2 2 + a1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂B1 ∂A1 ∂Ak−1 1 ⎪ + ak + (ak −a1 ) + ... + (ak −ak−1 ) + Ak = 0⎭ ∂αk ∂αk ∂αk 2 + ai

∂B1 ∂A1 ∂Ai−1 1 + (ai −a1 ) + ... + (ai −ai−1 ) + Ai = 0 ∂αi ∂αi ∂αi 2

(2.215)

The diﬀerential system (2.215) may be very simply integrated as it is triangular. As B0 and B1 are chosen arbitrarily, we obtain : ∂B0 ∂B1 + a1 A1 = −2 ∂α1 ∂α1 ∂B0 ∂B0 ∂B1 ∂B1 ∂ A2 = −2 + a2 + a1 + 4(a2 −a1 ) ∂α2 ∂α2 ∂α2 ∂α1 ∂α1 ∂B0 ∂B0 ∂B1 ∂B1 ∂ A3 = −2 + a3 + a1 + 4(a3 −a1 ) ∂α3 ∂α3 ∂α3 ∂α1 ∂α1 ∂B0 ∂B1 ∂ +4(a2 −a1 ) + a2 ∂α3 ∂α2 ∂α2 2 ∂B0 ∂B1 ∂ + a1 −8(a3 −a2 )(a3 −a1 ) ∂α2 ∂α3 ∂α1 ∂α1 and ﬁnally : Ai = −2

i ∂Ci + (−1)r 2r (ai − ar−1 )(ar−1 − ar−2 )...(a2 − a1 )• ∂αi r=2 1 <2 <...
r

•

∂ C1 ∂α1 ...∂αr−1 ∂αi

,

i = 1, 2, ..., k

iv) If we choose B1 ≡ 0 and B0 , along with its k ﬁrst derivatives, to be bounded functions, then (Mt∧T1 , t ≥ 0), where T1 := inf{t; Xt = 1}, is a bounded process. In particular, if we choose for B0 : k B0 (α1 , ..., αk ) = exp − λi αi (2.216) (λi > 0, i = 1, ..., k) i=1

we may apply Doob’s stopping Theorem at time T1 and so, we easily recover the result in [RY∗ ], Exercise 4.17, p. 267 : k ρ(0, λ) (2.217) E exp − λi LaTi = ρ(1, λ) i=1

2.6 Penalisation by a Function of Lat 1 and Lat 2 , with a1 = a2

129

with : ρ(x, λ) := 1 +

k

2(x − ai )+ λi

i=1

+

k

2r

r=2

(a2 −a1 )...(ar −ar−1 )(x−ar )+ λ1 ...λr

1 <2 <...
(2.218) Let us also note that (2.217) gives a proof of the ﬁrst Ray-Knight Theorem (see [RY∗ ], Chap. XI). The reader will ﬁnd in [P2 ] some formulae which are closely related to those of this paragraph. v) The martingale (Msf,g , s ≥ 0) given by (2.202) is of the form described in point 2) of this Remark 2.19 with : B1 (α1 , α2 ) = f (α1 )g(α2 )

(2.219)

1 B0 (α1 , α2 ) = a1 f (α1 )g(α2 ) + [f(α1 )g(α2 ) + f(g(α1 , α2 )] 2

(2.220)

with :

γ1 1 f (α1 + 1 ) exp − γ= d1 2 a2 − a1 0 ∞ γ2 g(α2 + 2 ) exp − g(α2 ) := d2 2 0 ∞ ∞ f (α1 + 1 )g(α2 + 2 )θ2 (1 , 2 )d1 d2 f(g(α1 , α2 ) := f(α1 ) :=

∞

0

(2.221) (2.222) (2.223)

0

(See (2.203) for the deﬁnition of θ2 .) In this situation, the functions A1 and A2 deﬁned in (2.212) are equal to : γ f (α1 ) A1 (α1 , α2 ) = f (α1 )g(α2 ) + g (α2 ) − f(α1 )g(α2 ) (2.224) 2 ( + f(g(α1 , α2 ) − f(g(α1 , α2 ) γ g (α2 ) − f(α1 )g(α2 ) (2.225) f (α1 ) A2 (α1 , α2 ) = f (α1 )g(α2 ) − 2 ( + f(g(α1 , α2 ) − f(g(α1 , α2 ) where : ( f(g(α1 , α2 ) =

∞ ∞

f (α1 + 1 )g(α2 + 2 )θ1 (1 , 2 )d1 d2 0

(2.226)

0

(See (2.203) for the deﬁnition of θ1 .)

130

2 Feynman-Kac Penalisations for Brownian Motion

Bibliography [B]

[BY] [D-M,Y1 ] [JY]

[Leb]

[N2 ] [N3 ] [NRY]

[P1 ] [P2 ] [PY1 ] [PY2 ] [PY3 ] [RY∗ ] [RVY,I]

[RVY,II]

[RVY,J] [RY,VIII]

[RY,IX]

[W]

P. Biane. Decompositions of Brownian trajectories and some applications. In A. Badrikian, P-A Meyer, and J-A Yan, editors, Probability and Statistics; Rencontres Franco-Chinoises en Probabilit´es et Statistiques; Proceedings of the Wuhan meeting, World Scientiﬁc, 51–76, 1993. P. Biane and M. Yor. Valeurs principales associ´ees aux temps locaux browniens. Bull. Sc. Maths., 2(111):23–101, 1987. C. Donati-Martin and M. Yor. Some Brownian functionals and their laws. Ann. Probab., 25(3):1011–1058, 1997. Th. Jeulin and M. Yor, editors. Grossissements de ﬁltrations: exemples et applications, volume 1118 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1985. Papers from the seminar on stochastic calculus held at Universit´e Paris VI, Paris, 1982/1983. N. N. Lebedev. Special functions and their applications. Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication. J. Najnudel. Temps locaux et p´enalisations browniennes. Th`ese de l’Universit´e Paris VI, June 2007. J. Najnudel. Penalisations of the Brownian motion with a functional of its local time. S.P.A. 118, 1407–1433, 2008. J. Najnudel, B. Roynette and M. Yor. A remarkable σ-ﬁnite measure associated with Brownian path penalisations. C.R. Math. Acad. Sci. Paris, 345(8), 459–466, 2007. J. Pitman. One-dimensional Brownian motion and the three-dimensional Bessel process. Advances in Appl. Probability, 7(3):511–526, 1975. J. Pitman. Cyclically stationary Brownian local time processes. PTRF, 106(3):299–329, 1996. J. Pitman and M. Yor. Asymptotic laws of planar Brownian motion. Ann. Probab., 14(3):733–779, 1986. J. Pitman and M. Yor. Further asymptotic laws of planar Brownian motion. Ann. Probab., 17(3):965–1011, 1989. J. Pitman and M. Yor. Bessel processes and inﬁnitely divisible laws. In Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), LNM 851, pages 285–370. Springer, Berlin, 1981. D. Revuz and M. Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 1999. B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I. Studia Sci. Math. Hungar., 43(2):171–246, 2006. B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time. II. Studia Sci. Math. Hungar., 43(3):295–360, 2006. B. Roynette, P. Vallois, and M. Yor. Some penalisations of the Wiener measure. Japanese Jour. of Math. 1, 263–290, 2006. B. Roynette and M. Yor. Ten penalisation results of Brownian motion involving its one-sided supremum until ﬁrst passage time VIII. Journal Funct. Anal., 255(9):2606–2640, 2008. B. Roynette and M. Yor. Local limit theorems for Brownian additive functionals and penalisation of Brownian paths. IX. To appear in ESAIM P.S., 2009. D. Williams. Path decomposition and continuity of local time for onedimensional diﬀusions, I. Proc. London Math. Soc, Ser. 3, 28, 738–768, 1974.

Chapter 3

Penalisations of a Bessel Process with Dimension d (0 < d < 2) by a Function of the Ranked Lengths of its Excursions

Abstract For any integer n = 1, 2, ..., limiting laws, as t → ∞, for a Bessel process with dimension d (0 < d < 2) penalised by the nth -ranked length of its excursions up to t, or up to the last zero before t, or again up to the ﬁrst zero after t, are shown to exist, and are characterized. Under these limiting laws Q(n) , the canonical process admits a last zero g, and the sequence of the normalized ranked lengths of its excursions up to g, is described in terms of the Poisson-Dirichlet distribution studies, e.g., by Pitman-Yor [PY5 ]. As n → ∞, Q(n) is shown to converge to Q(∞) the distribution of the Bessel process penalised by an adequate function of its local time at 0, and the sequence of the normalized ranked lengths of the excursions up to gunder Q(∞) is then precisely the Poisson-Dirichlet distribution P D 1 − d2 , 0 studied in Pitman-Yor [PY5 ]. Keywords Ranked lengths of excursions · Poisson-Dirichlet distributions · Bessel meanders · Penalisations

Generalities 3.1 Introduction 1) During the last ﬁve years, we have been interested in a number of studies which may all be gathered under the following umbrella : let P be a probability on the canonical space Ω = C(R+ , R), where (Xt , t ≥ 0) denotes the canonical coordinate process, Ft = σ{Xs , s ≤ t}, t ≥ 0, the family of σ-ﬁelds generated by this process, and F = ∨t Ft . To any process (Γt , t ≥ 0), usually but not always assumed to be (Ft ) adapted, and satisfying : 0 < EP (Γt ) < ∞ , B. Roynette, M. Yor, Penalising Brownian Paths, Lecture Notes in Mathematics 1969, DOI 10.1007/978-3-540-89699-9 3, c Springer-Verlag Berlin Heidelberg 2009

131

132

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

we associate the following family of probabilities on (Ω, F) : Qt =

Γt • P , t ≥ 0 . EP (Γt )

We have been interested in showing the existence of a limiting probability : Q∞ = limt→∞ Qt , where this limit is to be understood in the following sense : ∀ s > 0 , ∀ Λs ∈ Fs , lim Qt (Λs ) exists; t→∞

this limit then deﬁnes Q∞ on Fs ; hence as s varies, it characterizes Q∞ . To show, in each of the examples we have considered so far, the existence of Q∞ , we have proven that the properties i) and ii) in the next theorem are satisﬁed. Theorem 3.1. Under the two conditions : EP (Γt |Fs ) a.s. −→ Ms i) for every s ≥ 0, EP (Γt ) t→∞ ii) for every s ≥ 0, EP (Ms ) = 1, the following holds : a) b)

(Ms , s ≥ 0) is a (P, (Fs )) martingale; for every Λs ∈ Fs , Qt (Λs ) −→ EP (1Λs Ms ) t→∞

In our survey [RVYJ ] (or in Chap. 1), we have given a sketch of proofs that these two properties i) and ii) are satisﬁed in ﬁve Cases, numbered I to V, which we now recall brieﬂy : t = L0t , which are respectively, the one-sided • Case I : Γt = sup Xs , and Γ s≤t

supremum and the local time at 0 of our one-dimensional Brownian motion (Xt , t ≥ 0) (see Chap. 1, Case I, or [RVY,II]). t • Case II : Γt = ds q(Xs ), where q : R → R+ , or q : R2 → R+ satisﬁes 0

some integrability condition (see Chap. 1, Case II, or [RVY,I]). • Case III : In this case, (Xt , t ≥ 0) denotes a BES process, with dimension 0 < δ < 2, and Γt = L0t is its local time at 0 (see Chap. 1, Case III, or [RVY,V]). • Case IV : (Xt ) is again a one-dimensional Brownian motion; gt = sup{s ≤ t : Xs = 0}, and At = t − gt , t ≥ 0, is the so-called age process (of excursions). We then consider : Σt = sup Au , u≤gt

or

Σt = sup Au , u≤t

or again

Σt = sup Au , u≤dt

where dt = inf{s ≥ t : Xs = 0} (see Chap. 1, Case IV, or [RVY,VII]). • Case V : (Xt , t ≥ 0) denotes the d-dimensional Brownian motion, issued from x = 0 (d ≥ 2).

3.1 Introduction

133

For d = 2, we consider Γt = sup θs , where (θs , s ≥ 0) is a contins≤t

uous determination of the argument of X around 0; we also consider Γt = 1(TC >t) (d ≥ 2), where C is a cone with vertex 0, and basis a regular d open set O of Sd−1 , the unit sphere in R . Let us mention that the case t

Γt = 0

1(Xs ∈ C) ds is not so well understood (see Chap. 1, Case V, or

[RVY,VI]). 2) In the present Chapter, we take up the study of Case IV (in the Brownian set-up) and we systematically develop and extend this study in two directions, by : - ﬁrst, replacing Brownian motion by a d-dimensional Bessel process, where 0 < d < 2 (or, with d = 2(1 − α), 0 < α < 1); the case 0 < d < 2 corresponds precisely to the Bessel processes for which 0 is regular for itself; - second, considering the full sequence of ranked lengths of excursions away from 0 : (1) (2) (n) (Vt , Vt , ...Vt , ...) for the Bessel process (Ru , u ≤ t), and penalising the law P (α) of the Bessel process with dimension d = 2(1 − α) with the sets : (Vg(1) ≤ x) t (1) (Vt

≤ x) or

(3.1) (1) (Vdt

≤ x)

(3.2)

(Vg(n) ≤ x) , n ≥ 2 t

(3.3)

(n) (Vt

(3.4)

(n) ≤ x) or (Vdt ≤ x) for n ≥ 2 (Vg(1) ≤ x1 ; Vg(2) ≤ x2 ; ... ; Vg(n) ≤ t t t

xn )

(3.5)

where gt = sup{s ≤ t : Rs = 0}, and dt = inf{s ≥ t : Rs = 0}. These respective penalisations are studied in Sections : 3.3↔(3.1); 3.4↔(3.2); 3.5↔(3.3); 3.7↔(3.4); 3.9↔(3.5). In order to be able, in each of these cases, to prove that the properties i) and ii) of Theorem 3.1 are satisﬁed, we need to know and recall a number of relevant properties for the Bessel process of dimension d, such as the law of T0 = inf{t > 0 : Rt = 0}, where R0 = 0; we also need to have some good knowledge of the local time at 0 of (Rt ). Such prerequisites are developed in Section 3.2 of this chapter, where we also recall some of the results about the sequence of lengths of excursions obtained by Pitman-Yor ([PY6 ]). In Section 3.8, we obtain some precise asymptotics for the distribution of (1) (n) lengths of excursions, that is some asymptotics for P (Vgt ≤ x1 ; ... ; Vgt ≤ xn ), as t → ∞. (A summary of these results appeared in [RVYCR ]). With the help of such asymptotics, we are able in Section 3.9 to obtain the existence

134

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2) (1)

(n)

of the penalised law of P (α) by (Vgt ≤ x1 ; ... ; Vgt ≤ xn ) which constitutes the most general penalisation result by lengths of excursions which is being obtained in this paper. Going back to 3.5↔(3.3), we also show, in Section 3.6, that as n → ∞, the (n,x) penalised laws Q(n,x) ≡ Q∞ obtained in Section 3.5, converge weakly, as n → ∞, to the penalised laws of P (α) by (Lt ≤ x), so that, in some sense, “a certain diagram is commutative”; see Section 3.6 for a precise formulation. Notations In this paragraph, (Ω, (Rt , Ft )t≥0 , F∞ = ∨ Ft , Px (x ∈ R+ )) shall denote t≥0

the canonical realisation of the Bessel process (Rt , t ≥ 0) of dimension d = 2(1 − α), 0 < α < 1, on the measurable space Ω = C([0, ∞[→ R+ ) ﬁtted with the σ-algebra F∞ generated by the coordinate process (Rt , t ≥ 0). We denote Ft = σ(Rs , s ≤ t); hence (Ft , t ≥ 0) is the natural ﬁltration of the process (Rt , t ≥ 0). Px denotes the law of R starting from x. Since α, or d = 2(1 − α), shall be ﬁxed throughout this work, we omit it (except if explicitly mentioned otherwise) in our notation such as Px , and so on. Let, for any t ≥ 0, θt : Ω → Ω denote the classical translation operator which satisﬁes : (s, t ≥ 0) (N.1) Rs ◦ θt = Rs+t and for any a ≥ 0, note Ta the hitting time of a : Ta := inf{s ≥ 0; Rs = a}

(N.2)

Let for any time t ≥ 0 (random or deterministic) : gt := sup{s ≤ t, Rs = 0}

and

dt := inf{s ≥ t, Rs = 0}

(N.3)

Hence, gt is the last zero before time t and dt the ﬁrst zero after time t and dt − gt is the length of the excursion which straddles t. Deﬁne, for any t ≥ 0 : At := t − gt , and, for y > 0,

A∗t = sup As

(N.4)

s≤t

TyA := inf{s ≥ 0; As ≥ y}

(N.5)

(At , t ≥ 0) is the so-called age (of excursions) process. For any x > 0 and n ≥ 1, we deﬁne : Hx(1) = inf{s ≥ 0; As = x} Hx(n)

(= TxA )

(N.6)

= inf{s ≥ dH (n−1) ; As = x}

(N.7)

x

We note that : Hx(n) = dH (n−1) + H (1) ◦ θd x

(n−1) Hx

= Hx(n−1) + H (1) ◦ θH (n−1) x

(N.8)

3.2 Some Prerequisites about Bessel Processes

135

where : dH (k) = Hx(k) + T0 ◦ θH (k) , x

k = 1, 2, ...

x

Let, for any t (random or deterministic) (1)

(2)

(n)

(Vt , Vt , ...Vt

, ...)

the sequence of excursions of (Rs , s ≤ t) before t (including t − gt ), ranked in decreasing order. Hence : (n) (1) (2) (n) and Vt = t (N.9) Vt ≥ Vt ≥ ... ≥ Vt ≥ ... n≥1

The following relations are easy consequences of the deﬁnitions Vg(1) ≤ A∗t ≤ Vdt , t (1)

A∗t = Vt

(1)

,

(1)

Vdt = Vg(1) ∨ (dt − gt ) t

(N.10)

and, for any x ≥ 0 : (Vg(n) ≤ x) = (gt ≤ Hx(n) ) = (dH (n) ≥ t) t x

(n)

(Vt

≤ x) = (Hx(n) ≥ t)

(N.11)

(n)

(Vdt ≤ x) = (dt ≤ Hx(n) ) = (t ≤ gHxn )

3.2 Some Prerequisites about Bessel Processes of Dimension d = 2(1 − α), 0 < α < 1 In this Section, we gather a number of results relative to the d-dimensional Bessel process (Rt , t ≥ 0) with dimension d = 2(1 − α), 0 < α < 1. Some of these results are classical, others less so. 1) The law of T0 . Under Pr , the law of T0 is given by : (d)

T0 =

r2 2γα

(3.6)

where γα denotes a gamma (α) variable ([G] and [Y]). Consequently, we have : ∞ βr 2 βr 2 1 e− 2u −u uα−1 du Er (e−βT0 ) = E e− 2γα = Γ(α) 0 √ ! (r 2β)α Kα (r 2β) = α−1 2 Γ(α)

136

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

(where Kα denotes the Bessel-Mac Donald function with index α (see [Wat], for such a formula)), and we deduce from the previous equality : Er (e−βT0 ) = 1 −

Γ(1 − α) 2α α r β + o(β α ) 2α Γ(1 + α)

(β → 0)

(3.7)

Indeed, we have : Er (e−βT0 ) ! ! 1 Kα (r 2β) (r 2β)α = α−1 2 Γ(α) ! πz α [I−α (z) − Iα (z)] = α (with z = r 2β, see [Leb] p. 108) 2 Γ(α) sin(απ) z α z −α 1 1 πz α − + o(z α ) (z → 0) = α 2 Γ(α) sin(απ) 2 Γ(1 − α) 2 Γ(1 + α) (see [Leb] p. 108; Iν denotes the modiﬁed Bessel function of the ﬁrst kind) π z 2α Γ(1 − α) 2α + o(z ) = 1− 2α Γ(α)Γ(1−α) sin(απ) 2 Γ(α)Γ(1 − α) sin(απ)Γ(1 + α) z 2α Γ(1 − α) + o(z 2α ) = 1 − 2α 2 Γ(1 + α) Γ(1 − α) 2α α r β + o(β α ) = 1− α (β → 0) 2 Γ(1 + α) from the formula of complements : Γ(α)Γ(1 − α) = We shall also use the notation : ! Φ(α) (r) := P ( 2γα > r) =

1 Γ(α)

∞

π . sin(απ)

e−u uα−1 du

(r ≥ 0)

r 2 /2

Clearly, Φ(α) (0) = 1 and Φ(α) (∞) = 0 and 2 ! r r r Pr (T0 < u) = P √ = Φ(α) √ 2γα u u

(3.8)

(3.9)

In particular : 1 Pr (T0 > t) = Γ(α)

r2 2t

e−u uα−1 du ∼

0

t→∞

Note that, for d = 1, i.e. α = 1/2, we have : Φ(1/2) (r) = P (|N | > r) =

2 π

r2α 2α Γ(α + 1)tα

where N denotes a standard Gaussian variable.

r

∞

e−

u2 2

du

(3.10)

3.2 Some Prerequisites about Bessel Processes

137

2) The local time at 0 and its inverse. As is well known, the local time at 0, (Lt , t ≥ 0) of the Markov process (Rt , t ≥ 0) is unique, up to a multiplicative constant (see [D-M,RVY] for a description of the diﬀerent choices found in the literature). Here, our choice of (Lt , t ≥ 0) is the continuous increasing process such that : (Rt2α − Lt , t ≥ 0)

is a martingale

(3.11)

which may be written as : Nt := Rt2α − Lt = 2α

t

Rs2α−1 dBs

(3.12)

0

where (Bt , t ≥ 0) denotes the “driving” Brownian motion of R. One has :

t

< N >t = 4α2

Rs2(2α−1) ds 0

and αm E(Lm t )=t

Γ(1 + m) Γ(1 + αm)

2α Γ(1 + α) Γ(1 − α)

m (m ≥ 0)

(3.13)

The distribution of (Lt ) is closely related to the Mittag-Leﬄer distribution with parameter α; more precisely for Mα with that distribution, one has : E[(Mα )m ] =

Γ(1 + m) Γ(1 + αm)

(d)

α

Γ(1+α) . Hence, from (3.13), Lt = tα Cα Mα where Cα = 2 Γ(1−α) We denote by (τ , ≥ 0) the right continuous inverse of (Lt , t ≥ 0)

τ := inf{t > 0; Lt > }

(3.14)

(τ , ≥ 0) is a stable (α) subordinator, whose Laplace transform is given by : Γ(1 − α) λα E(e−λτ ) = exp − α 2 Γ(1 + α) The L´evy measure of (τ , ≥ 0) is given by : να (dt) =

1

dt

2α Γ(α)

tα+1

(see, e.g., [D-M,RVY], as well as [RVY,V]).

(λ ≥ 0)

(3.15)

138

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

3) Properties of the Bessel meander and consequences We now introduce the Bessel meander : 1 √ := R ; 0 ≤ u ≤ 1 m(t) gt +u(t−gt ) u t − gt

(3.16)

which is independent from Fgt , and whose distribution does not depend on t. We also recall that : gt (d) t (d) = = βα,1−α (3.17) t dt where βa,b is a beta variable with parameter (a, b)

(0 < a, b < 1).

Proposition 3.2. Let T be a (Fgt , t ≥ 0) stopping time such that P (RT = 0) = 0. (T ) 1) The process (mu , u ≤ 1) is independent from FgT and its law does not depend on T . 2) i) For any Borel function f : R+ → R+ , one has : E[f (RT )|FgT ] = Kf (AT )

(3.18)

where the Markov kernel K is deﬁned by : 2 ∞ ! z z Kf (y) := exp − f (z)dz = E(f ( 2ye)) y 2y 0

(3.19)

with e an exponential variable with mean 1. It is noteworthy that the kernel K does not depend on α. (n) ii) In particular, if T = Hx , then : RH (n) x

and

and

Fg

(n) Hx

are independent

(3.20)

√ P (RH (n) ∈ dz) = K(x, dz) = P ( 2xe ∈ dz)

(3.21)

x

The sequence (RH (n) , n ≥ 1) is i.i.d and there exists ρ > 0 s.t : x

(1)

E[e ρ H1 ] < ∞ (b)

(3.22)

3) Consider the functions : ψ1,α (z) = z 2α , ψ2,α (z) = Φ(α)

√z b

(z, b ≥ 0)

where Φ(α) has been deﬁned in (3.8). Then : 2α K ψ1,α (y) = 2α Γ(α + 1)y α = E(RH (n) ) y α y (b) K ψ2,α (y) = 1 − b+y

(n ≥ 1)

(3.23) (3.24)

3.2 Some Prerequisites about Bessel Processes

4) Let

139

1

)R θ(x, β) := E E

(n)

ψn (x, β) := E(e−β Hx ) ,

(1) Hx

) (e−β T0 )

(3.25)

(n ≥ 1, x, β ≥ 0). Then : ψn (x, β) = [ψ1 (x, β)]n [θ(x, β)]n−1 5) The sequence of r.v. (LH (1) , LH (2) −Ld x

(1) Hx

x

(3.26)

, ...LH (n) −Ld x

α

(n−1) Hx

, ...) is i.i.d and

α

LH (1) is an exponential variable with mean 2 Γ(1 + α)x . Hence : x

LH (n) (= Lg x

(n) Hx

(d)

= Ld

(n) Hx

) = 2α Γ(1 + α)xα γn

(3.27)

where γn is a gamma (n)-variable, so that : E[LH (n) ] = n 2α Γ(1 + α)xα x

(3.28)

6) For any n ≥ 1 and > 0 : 1

(d)

(n) (Vτ )α

In particular :

E

=

1 (n)

(Vτ )α

2α Γ(α + 1) γn

=

n 2α Γ(α + 1)

(3.29)

(3.30)

Proof of Proposition 3.2 The points 1, 2, 3 are proven in [RVY,VII] (except (3.22), which results from (3.20) and Theorem 2 in [DRVY]). We prove (3.23) and (3.24) : We have : ! K ψ1,α (y) = E[( 2ye)2α ] = 2α y α E(eα ) = 2α Γ(1 + α)y α We have : K

1

(b) ψ2,α (y)

2ye (by (3.19)) b 2ye =P
(α)

Φ

By scaling, the two functions in (3.25) only depend on the product xβ, so that below we shall write ψn (xβ) and θ(xβ).

140

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

1) We prove point 4 of Proposition 3.2 We have : % ψn (xβ) = E[e−β

(n) Hx

(1) (n−1) +Hx ◦θd (n−1) Hx Hx

−β[d

]=E e

& ]

(n−1) (1) −β[Hx +T0 ◦θ (n−1) ] Hx =E e E[e−β Hx ]

(by (3.20))

(n−1)

(by the Markov property) = E[e−β Hx

(n−1) Hx

(n−1)

= E[e−β Hx (n−1)

because Hx

)

)R E

(e(−β T0 ) )] ψ1 (xβ)

] E[ER

(n−1) Hx

(e(−β T0 ) )] ψ1 (xβ)

and RH (n−1) are independent (see (3.20)), and so : x

ψn (xβ) = ψn−1 (xβ)ψ1 (xβ)θ(xβ) because : )R E[E

(n−1) Hx

) )R (e(−β T0 ) )] = E[E

)

(1) Hx

(e(−β T0 ) )] ,

the r.v’s (RH (n) , n ≥ 0) having the same law. x

2) We now prove point 5 of Proposition 3.2 The ﬁrst sentence is obvious, using the Markov property. We now show that LH (1) is an exponential variable. Since (Rt2α − Lt , t ≥ 0) is a martingale then, x by Itˆ o’s formula, for any λ > 0, (λ e−λLt Rt2α + e−λLt , t ≥ 0) is also a martingale. We apply Doob’s optional stopping Theorem to the stopping time (1) Hy (= TyA ) : −λL

1 = λ E(e

(1) Hy

−λL

•

(1) Hy

2α RH (1) ) + E(e

)

y

−λL

= λ E(e

(1) Hy

−λL

2α )E(RH (1) ) + E(e

(1) Hy

)

y

(by independence of RH (1) and LH (1) = Lg y

y

−λL

E(e

(1) Hy

)=

(1) Hy

). So :

1 1+

λ 2α Γ(1

(by (3.23))

+ α)y α

3) We prove point 6) of Proposition 3.2 We have, by (N.11), on one hand : ≤ x) = P (τ ≤ Hx(n) ) = P ( ≤ LH (n) ) , P (Vτ(n) x

3.2 Some Prerequisites about Bessel Processes

141

and, on the other hand, by scaling P (Vτ(n) ≤ x) = P (1/α Vτ(n) ≤ x) . 1 %

Hence : P

α

1 (n)

Vτ1

> α x

&

=P

LH (n) x

xα

>

xα

and, by the preceding point : α LH (n) (d) α 1 (d) x = = 2 Γ(1 + α)γn (n) xα Vτ1 and by scaling

1 (n)

Vτ

α

1 =

1

(d)

α

(n)

(d)

=

Vτ1

2α Γ(1 + α) γn

4) Explicit expressions for ψ1 and θ We denote by Φ(α, γ, •) γ = 0, −1, −2, ... the conﬂuent hypergeometric function of index (α, γ) : Φ(α, γ, z) :=

∞ (α)k z k (γ)k k!

(z ∈ C)

(3.31)

k=0

Γ(λ + k) (see [Leb], p. 260). Γ(λ) Let λ0 (α) denote the ﬁrst positive zero of : with (λ)k =

θα (λ) = 1 − λ e−λ

1 Φ(1 − α, 2 − α; λ) (1 − α)

(3.32)

(we prove below that such a zero exists). Proposition 3.3. Let x > 0. Then, there exists two functions ψ, θ : R+ → R+ such that : i)

1 Φ(1, 1 − α; βx)

(3.33)

1 Φ(1 − α, 2 − α; −βx) 1−α

(3.34)

ψ(xβ) := E(exp(−βHx(1) )) =

and Φ(1, 1 − α; βx) = 1 + βx eβx

The function β → ψ(β) is holomorphic in the strip Reβ > −λ0 (α).

142

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

In particular

βx + o(β) 1−α

ψ(xβ) = 1 − ii)

θ(xβ) := E(ER

(1) Hx

(β → 0)

(3.35)

(exp −βT0 ))

= Φ(1, 1 − α; βx) − Γ(1 − α)(βx)α exp(βx) = 1 − Γ(1 − α)(βx)α + o(β α ) (β → 0)

(3.36) (3.37)

Note that formula (3.33) had already been obtained under a slightly diﬀerent form in ([PY4 ], [PY5 ], [PY6 ] Proposition 11). With the notations of point 4) of Proposition 3.2, we have : ψ1 (x, β) = ψ(xβ) ,

θ(x, β) = θ(xβ)

Proof of Proposition 3.3. By scaling, it is suﬃcient to prove (3.33) and (3.36) for x = 1. 1) We prove (3.33) for x = 1 For β ≥ 0, let ψβ (r) := rα I(−α) (βr) (3.38) where I(−α) denotes the modiﬁed Bessel function with index (−α). Since the generator of the Bessel process (Rt , t ≥ 0) is given by : LR f (r) =

1 1 − 2α f (r) + f (r) 2 2r

a simple computation shows that : (ψβ (Rt )e− equals :

2 β

α

1 Γ(1−α)

β2 2

t

, t ≥ 0) is a martingale which

for t = 0. Thus, Doob’s optional stopping Theorem, (1)

applied with the stopping time H1

implies : α (1) β2 2 1 E[ψβ (RH (1) )e− 2 H1 ] = 1 β Γ(1 − α) (1)

(3.39)

With the help of the independence between H1 and RH (1) (following (3.20) 1 and (3.19)), we get : α 2 (1) 2 1 − β2 H1 (3.40) ]= E[e β Γ(1 − α)Kψβ (1) It remains to compute Kψβ (1) ψβ (r) = rα I(−α) (βr) −α+2k ∞ βr 1 = rα 2 Γ(k + 1)Γ(1 − α + k) k=0 α 2k ∞ 2 βr 1 = β 2 Γ(k + 1)Γ(1 − α + k) k=0

3.2 Some Prerequisites about Bessel Processes

143

and by (3.19)), we have : α 2k ∞ 2 β 1 E((2e)k ) β Γ(k + 1)Γ(1 − α + k) 2 k=0 α ∞ 2 k 2 β 1 = Γ(k + 1) β 2 Γ(k + 1)Γ(1 − α + k) k=0 α 2 k ∞ 2 β 1 = β Γ(1 − α + k) 2 k=0 α 2 β2 1 Φ 1, 1 − α, = (3.41) β Γ(1 − α) 2

Kψβ (1) =

Plugging (3.41) in (3.40), we obtain (3.33). The relation (3.34) follows from the classical formula (see, [Leb], formula 9.11.2, p. 267) : Φ(1, 1 − α, −λ) = 1 − λ e−λ

1 Φ(1 − α, 2 − α, λ) 1−α

(3.42)

which we already used in (3.34). The holomorphy of ψ in the set {β : Reβ > −λ0 (α)} follows from ([DRVY], Theorem 2). 2) We now prove (3.36) for x = 1 We have, by (3.7) : Er (e−βT0 ) =

1 2α−1 Γ(α)

[(r

!

2β)α Kα (r

!

2β)]

where Kα is the Bessel-Mac Donald function of index α, and so, by (3.19) : θ(β) =

1 KΛ(1) , 2α−1 Γ(α)

with Λ(r) := Kα (r

! ! 2β)(r 2β)α

But (see [Leb], p. 108) : ! ! ! π (r 2β)α [I(−α) (r 2β) − Iα (r 2β)] 2 sin(πα) ∞ 2k π2α β 1 = r 2 sin(πα) 2 Γ(k + 1)Γ(k − α + 1) k=0 2α+2k ∞ β 1 − r 2 Γ(k + 1)Γ(k + α + 1)

Λ(r) =

k=0

144

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

by the deﬁnition of the functions I−α and Iα . So, by (3.19) : ∞ βk π E(ek ) θ(β) = Γ(α) sin(πα) Γ(k + 1)Γ(k − α + 1) k=0

−

∞ k=0

∞

β α+k E(eα+k ) Γ(k + 1)Γ(k + α + 1) ∞

β α+k βk − Γ(k − α + 1) Γ(k + 1) k=0 k=0 1 π α β Φ(1, 1 − α, β) − β e = Γ(α) sin(πα) Γ(1 − α) =

π Γ(α) sin(πα)

= Φ(1, 1 − α, β) − Γ(1 − α)β α eβ by the formula of complements. Formula (3.37) is an easy consequence of (3.36). 5) One dimensional asymptotics for the lengths of excursions Throughout the sequel, we shall use the following estimates : Theorem 3.4. 1) For any n ≥ 1 and x > 0 : x α ≤ x) ∼ n P (Vg(n) t t→∞ t

(3.43)

For any n ≥ 2 and x > 0 : (n)

(n)

P (Vdt ≤ x) ∼ P (Vt t→∞

≤ x) ∼ (n − 1)

x α t

t→∞

(3.44)

2) There exist constants C1 , C2 and λ0 (α) > 0 such that : (1)

P (Vt

≤ x) = P (A∗t ≤ x) ∼ C1 e−λ0 (α) x t

(3.45)

t→∞

−λ0 (α) xt

(1)

P (Vdt ≤ x) = P (Vg(1) ∨ (dt − gt ) ≤ x) ∼ C2 e t t→∞

(3.46)

(see below the important Remark following the proof of this Theorem). Proof of Theorem 3.4 In the Brownian case (i.e. : d = 1, or α = 1/2), the proof of (3.45) and (3.46) is found in [RVY,VII], where the constant λ0 is given “explicitly”. Now, we shall prove point 1) in the general case d ∈ (0, 2). 1) We ﬁrst prove that ∞ e−βt P (Vg(n) ≤ x)dt ∼ n xα Γ(1 − α)β α−1 t 0

β→0

3.2 Some Prerequisites about Bessel Processes

145

which is equivalent to (3.43), from the Tauberian Theorem (see [Fel], Chap. XIII). i) Let Sβ be an exponential r.v with mean 1/β, independent of (Rt , t ≥ 0). We have, by (N.11) : P (Vg(n) ≤ x) Sβ

= P (Sβ ≤ dH (n) ) x

= 1 − E(exp(−β dH (n) ))

(by deﬁnition of Sβ )

x

= 1 − E(exp −β[Hx(n) + T0 ◦ θH (n) ]) x

= 1 − E[(exp −β

) )R (e−β T0 )](by Hx(n) ) E (n) H

the strong Markov property)

x

)R = 1 − E(exp −β Hx(n) ) E(E

)

(n) Hx

(e−β T0 ))

(by independence of Hx(n) and RH (n) ) x

= 1 − ψn (xβ)θ(xβ) = 1 − [ψ1 (xβ)]n [θ(xβ)]n

(by (3.26)) (1)

(1)

But : ψ1 (xβ) = 1 − β E(Hx ) + o(β), because Hx admits moments of all orders, by (3.22), and )R θ(xβ) = E[E

(1) Hx

)

(e−β T0 )]

= 1 − Γ(1 − α)β α xα + o(β α ) (β ≥ 0) Hence :

by (3.37)

(n)

P (VSβ ≤ x) = n xα Γ(1 − α)β α + o(β α ) ii) But, by deﬁnition of Sβ again : P (Vg(n) ≤ x) = Sβ

∞

0

β e−β t P (Vg(n) ≤ x)dt t

and with the help of (3.47) : ∞ e−β t P (Vg(n) ≤ x)dt ∼ n xα Γ(1 − α)β α−1 t β→0

0

2) We prove, for n > 1: ∞ (n) e−β t P (Vt ≤ x)dt ∼ (n − 1)Γ(1 − α)β α−1 xα 0

β→0

(3.47)

(3.48)

146

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

which is equivalent to (3.44). But, using (N.11) once more, with Sβ an exponential variable with mean 1/β, assumed to be independent of (Ru , u ≥ 0) : (n)

P (VSβ ≤ x) = P (Sβ ≤ Hx(n) ) = 1 − E(e−βHx ) (n)

(by (3.26)) = 1 − [ψ1 (xβ)]n [θ(xβ)]n−1 (1) n−1 = 1 − [1 − β E(Hx ) + o(β)] [1−Γ(1 − α)xα β α + o(β α )]n−1 = (n − 1)Γ(1 − α)xα β α + o(β α )

Since : (n)

P (VSβ ≤ x) = β we obtain :

∞

e−β t P (Vt

(n)

∞

e−β t P (Vt

(n)

≤ x)dt ,

0

≤ x)dt ∼ (n − 1)Γ(1 − α)xα β α−1 , β→0

0

which is our claim. 3) We prove, for n > 1 : ∞ (n) e−β t P (Vdt ≤ x)dt ∼ (n − 1)Γ(1 − α)xα β α−1 β→0

0

which is equivalent to (3.44). With the same notations as for the previous point, we have, by (N.11) : −β g

(n)

P (VdS ≤ x) = P (Sβ ≤ gH (n) ) = 1 − E(e x

β

(n) Hx

)

(n)

But : Hx − gH (n) = x. So : x

(n)

(1)

P (VdS ≤ x) = 1 − eβ x E(e−β Hx ) = (1 − eβ x ) + eβ x (1 − E(e−β Hx )) (n) β

= (n − 1)Γ(1 − α)xα β α + o(β α ) by the previous point. Hence : ∞ 1 (n) (n) e−β t P (Vdt ≤ x)dt = P (VdS ≤ x) ∼ (n − 1)Γ(1 − α)xα β α−1 . β β→0 β 0 4) We now prove point 2) of Theorem 3.4 We already show (3.45) which by scaling, it suﬃces to prove for x = 1. We have by (3.33) and (3.34) : (1)

E(e−λT1 ) = E(e−λH1 ) = A

1 1 1 + λ eλ 1−α Φ(1 − α, 2 − α, −λ)

3.2 Some Prerequisites about Bessel Processes

147

We also know (see [DRVY], Theorem 2, p. 543) that T1A admits small exponential moments. Thus, for λ > 0, small enough : A

E(eλT1 ) =

1 1 1 − λ e−λ 1−α Φ(1 − α, 2 − α, λ)

(3.49)

Now, let (see (3.32)): 1 Φ(1 − α, 2 − α, λ) 1−α 1 1 λ e−λ α (1 − α)eλ = 1+ +o (1 − α) λ λ λ

θα (λ) := λ e−λ

α +o λ

= 1+

1 λ

(λ → ∞)

(λ → ∞)

(see [Leb], p. 271, formula 9.12.8). Thus, θα (λ) := 1 − θα (λ) admits a ﬁrst positive zero, λ0 (α) (since θα (0) = 1 and θα (λ) ∼ − αλ ). Hence, denoting λ→∞

by fT1A the density of T1A (note that T1A ≥ 1) we get :

∞

1

eλx fT1A (x)dx =

1

=

1 − θα (λ)

1 θα (λ)

hence, from Mellin-Fourier : fT1A (x) ∼ C e−λ0 (α)x x→∞

which implies : (1)

P (Vt

≤ x) = P (t < TxA ) = P

T1A >

t x

∞

= t x

fT1A (u)du

∼ C e−λ0 (α) x t

t→∞

This proves (3.45). The proof of (3.46) is quite similar : E[exp(λ gT1A )] = E[exp(λ(T1A − 1))] =

e−λ θα (λ)

Hence, from Mellin-Fourier again : fgT A (x) ∼ e−λ0 (α) C e−λ0 (α)x 1

x→∞

from (3.49)

148

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

and

(1)

P (Vdt ≤ x) = P (t ≤ gTxA ) = P

gT1A >

t x

∼ e−λ0 (α) C e−λ0 (α)x/t

t→∞

Important Remark In fact, the proof of point 2) of Theorem 3.4 we have just given is not entirely satisfactory. Indeed, we use twice the inverse transform of Mellin-Fourier; in order that this use is licit, we should know that: the function: y → θα (λ0 (α) + iy) has no other zero than y = 0, but the proof of this eludes us. To take care of this gap in our proof, we should make the following conjecture (C), and we shall clearly indicate the statements which rely on (C) P (Vdt ≤ x)(≡ P (A∗t ≤ x)) ∼ C1 e−λ0 (α) x (1)

t

t→∞

P (Vdd ≤ x)(≡ P (Vgt ∨ (dt − gt ) ≤ x)) ∼ C2 e−λ0 (α) x (1)

(1)

t

(C)

t→∞

t

Penalisations of a BES(d) Process, Involving One Ranked Length of its Excursions 3.3 Penalisation by (Vg(1) ≤ x) t In this section, we study the penalisation of the Bessel process (Rt , t ≥ 0) (1) with dimension d = 2(1 − α) (0 < α < 1) by the set (Vgt ≤ x), for x > 0, ﬁxed. More generally, we might study the penalisation by the functionals (1) h(Vgt ), for h : R+ → R+ belonging to a family of adequately integrable functions. Such a study is developed in [RVY,VII] for α = 1/2 (i.e. : d = 1). For the sake of conciseness, we shall limit ourselves here to the penalisation (1) by (Vgt ≤ x), and we leave to the interested reader the task of extending the arguments from [RVY,VII] to the more general penalisation framework with (1) h(Vgt ). The main result in this section is : Theorem 3.5. Let x > 0 ﬁxed. 1) For every s ≥ 0, and Λs ∈ Fs : lim

t→∞

E(1Λs 1(V (1) ≤x) ) gt

(1)

P (Vgt ≤ x)

exists

(3.50)

(1)

3.3 Penalisation by (Vgt ≤ x)

149

and is equal to : E(1Λs Ms )

(3.51)

(s Ms := 1(V (1) ≤x) M

(3.52)

where : gs

and (s := M

1 R2α + Φ(α) α 2 Γ(α + 1)xα s

R √ s x − As

1As ≤x

(3.53)

We note that the deﬁnition of Ms is left unchanged when replacing in (3.53) 1As ≤x by 1A∗s ≤x . Moreover, (Ms , s ≥ 0) is a positive martingale such that M0 = 1. 2) The formula : Q(1,x) (Λs ) := E(1Λs Ms )

(s ≥ 0, Λs ∈ Fs )

(3.54)

induces a probability Q(1,x) on (Ω, F∞ ). Under this probability Q(1,x) , the canonical process (Xt , t ≥ 0) satisﬁes : (1) α V∞ (1) i) V∞ < ∞ a.s. and is uniformly distributed on [0, 1] (3.55) x ii) A∗∞ = ∞ a.s. iii) Let g := sup{t; Xt = 0}. Then Q(1,x) (0 < g < ∞) = 1

(3.56) (3.57)

and the distribution of g satisﬁes : At∧TxA α (3.58) Q(1,x) (g < t) = E x iv) Lg (= L∞ ) is an exponential variable with expectation 2α Γ(1 + α)xα , i.e. : the density of Lg , fLQg , satisﬁes : 1 P (Vτ(1) ≤ x)1 ≥0 2α Γ(α + 1)xα 1 = α exp − 1 ≥0 2 Γ(α + 1)xα 2α Γ(α + 1)xα

fLQg () =

(3.59) (3.60)

3) Under Q(1,x) i)

The processes (Ru , u ≤ g) and (Rg+u , u ≥ 0) are independent

(3.61)

ii) Conditionally on L∞ = , the process (Ru , u ≤ g)is a Bessel process of dimension d = 2(1 − α), considered up to τ , and conditioned on Vτ(1) ≤x

150

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

iii) The process (Rg+t , t ≥ 0) is a Bessel process with dimension 4 − d = 2(1 + α), i.e. : its inﬁnitesimal generator Lg↑ is given by : 1 1 + 2α f (r) (r ≥ 0) (3.62) Lg↑ f (r) = f (r) + 2 2r (f ∈ Cc2 (0, ∞)) 3 − 2(1 − α) 1 + 2α 4−d−1 = = note that 2 2 2 4) (A universal property). Let 0 ≤ y < x. Then : i)

The process (Au , u ≤ TyA ) is identically distributed under P and under Q(1,x)

ii) The process (Au , u ≤ TyA ) and the r.v. RTyA are independent, both under P and under Q(1,x) ; y α iii) Q(1,x) (g > TyA ) = 1 − (3.63) x iv) The process (Au , u ≤ TyA ) and the event (g > TyA ) are independent under Q(1,x) v) The density of RTyA under Q(1,x) equals : z 2α z z − z2y2 Q(1,x) (α) √ fR A (z) = e +Φ 1z≥0 Ty y 2α Γ(1 + α)xα x−y

(3.64)

Remark. 1) We have denoted by Q(1,x) the probability deﬁned by (3.54) since, in section V below, we deﬁne probabilities Q(n,x) for every integer n, and the probability Q(1,x) deﬁned by (3.54) is the probability Q(n,x) , with n = 1. 2) In Theorem 3.5, we did not include a description of the excursions before g. This description is made, in a more general framework in Theorem 3.8 below, where the interested reader should take n = 1 in point 4) of that Theorem. 3) For α = 1/2, i.e. d = 1, this Theorem 3.5 is Theorem 1.14 of Chapter 1, with the minor change that Theorem 1.14 is relative to Brownian motion, whereas Theorem 3.5, for α = 1/2, is relative to reﬂected Brownian motion. Proof of Theorem 3.5 1) We prove that : E(1(V (1) ≤x) |Fs ) gt

(1)

P (Vgt ≤ x)

−→ Ms

t→∞

a.s.

(3.65)

(1)

3.3 Penalisation by (Vgt ≤ x)

where

Ms := 1(V (1) ≤x) gs

151

1 R2α + Φ(α) 2α Γ(1 + α)xα s

R √ s x − As

1As ≤x

We have

E 1Λs 1(V (1) ) gt ≤x = E 1Λs 1(V (1) ) 1T0 ◦θs >t−s + 1T0 ◦θs <(t−s)∧(x−As ) 1V (1)

t−ds ◦θds ≤x

gs ≤x

E 1(V (1) ≤x) |Fs = (1)t + (2)t

Hence :

(3.66)

gt

We now study successively the asymptotic behavior of (1)t and (2)t . 1a) Asymptotic behavior of (1)t )R (T)0 > t − s) (1)t := 1(V (1) ≤x) E s gs

∼ 1(V (1) ≤x)

t→∞

gs

Rs2α 2α Γ(1 + α)(t − s)α

(by 3.10)

and, thanks to Theorem 3.4, point 1) : Rs2α tα Rs2α −→ 1 (1) (1) gs 2α Γ(1+α)(t−s)α xα t→∞ (Vgs ≤x) 2α Γ(1 + α)xα P (Vgt ≤x) t→∞ (3.67) 1b) Asymptotic behavior of (2)t (2)t := 1(V (1) ≤x) E 1T0 ◦θs <(t−s)∧(x−As ) 1(V (1) ◦θ ≤x) Fs d g (1)t

∼ 1(V (1) ≤x)

s

t−ds

s

∼ 1(V (1) ≤x) − 1(As ≤x) Ht−s (Rs , As ) with : gs α x Hu (r, a) = Er 1(T0 <x−a) by Theorem 3.4, point 1). (u − T0 )α

t→∞

Hence : (2)t (1)

P (Vgt ≤ x)

∼ 1(V (1) ≤x) 1(As ≤x)

t→∞

gs

xα tα ) ERs (T)0 < x − As ) tα xα

(3.68)

where, in the previous expectation, the letters Rs and As are frozen Rs −→ 1(V (1) ≤x) 1(As ≤x) Φ(α) √ (3.69) gs t→∞ x − As by (3.9). The addition of (3.69) and (3.67) implies (3.65).

152

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

2) We prove that (Ms , s ≥ 1) is a martingale 2a) In the case α = 1/2, we have proven in [RVY,VII] that (Ms , s ≥ 0) is a martingale. The result of this proof, obtained in [RVY,VII], relies on Itˆ o’s formula and the fact that (see (3.11) (Rt2α − Lt , t ≥ 0) is a martingale. We invite the interested reader to look up this proof made in [RVY,VII] and we shall now prove that (Ms , s ≥ 0) is a martingale by another method. 2b) Suppose we had proven that : E(Ms ) = 1 |Fs ) (1) (Vg ≤x) t (1) P (Vgt ≤x)

E(1

Then, the fact that

for all s ≥ 0

(3.70)

converges a.s., when t → ∞, towards Ms ,

implies (see [M], Chap. II, T2) that this convergence takes place in L1 and that (Ms , s ≥ 0) is a martingale. Thus, in order to prove (3.50) and the fact that (Ms , s ≥ 0) is a martingale, it suﬃces to prove (3.70). But (3.70) is an immediate consequence of : (1,x)

Proposition 3.6. Let πs

:= E[Ms |Fgs ]. Then :

α As 1) πs(1,x) = 1(V (1) ≤x) sup 1, gs x

(3.71)

2) E(πs(1,x) ) = 1

(3.72)

Indeed, E(Ms ) = 1 follows from (3.71) and (3.72) since : E(Ms ) = E(E(Ms |Fgs )) = E(πs(1,x) ) = 1 . Proof of Proposition 3.6 2c) By the deﬁnition (3.52) and (3.53) of Ms , we have : πs(1,x) := E(Ms |Fgs ) 1 Rs 2α (α) √ E(R |F )+1 E Φ = 1(V (1) ≤x) α |F gs As ≤x gs s gs 2 Γ(α+1)xα x − As α 1 As = 1(V (1) ≤x) α 2α Γ(1 + α)Aα s + 1As ≤x 1 − gs 2 Γ(α + 1)xα x (by (3.18), (3.23) and (3.24))

α α As As + 1As ≤x 1 − gs x x α As = 1(V (1) ≤x) 1 ∨ gs x = 1(V (1) ≤x)

(1)

3.3 Penalisation by (Vgt ≤ x)

153

2d) α As = E 1(V (1) ≤x) 1 ∨ gs x α x + s − TxA = E[1TxA ≥s ] + E 1TxA <s
&

(by (3.9))

α α x + s − TxA x = E[1TxA ≥s ] + E 1TxA <s 1− 1− x x + s − TxA (by (3.18) and (3.24)) = E[1TxA ≥s ] + E[1TxA <s ] = 1 . This ﬁnishes the proof of Proposition 3.6 and of point 1) of Theorem 3.5 3) We prove that, under Q(1,x) ,

(1)

V∞ x

α is uniformly distributed on [0,1]

3a) We have : (1)

≤ x) = lim Q(1,x) (Vg(1) s

t→∞

(1)

P ((Vgs ≤ x) ∩ (Vgt ≤ x)) (1)

P (Vgt ≤ x)

=1

(3.73)

and so Q(1,x) (V∞(1) ≤ x) = lim Q(1,x) (Vg(1) ≤ x) = 1 s s→∞

(3.74)

3b) Let 0 < y < x. We have : Q(1,x) (V∞(1) > y) = Q(1,x) (TyV

(1)

< ∞) = E(MT V (1) ) y

by the deﬁnition of Q(1,x) and the optional stopping Theorem of Doob, where TyV

(1)

= inf{s ≥ 0; Vg(1) > y} s

But : MT V (1) = 1T0 ◦θT A <x−y (3.75) y ⎧ y ⎨ 1 if the length of the ﬁrst excursion longer than y is smaller than x (3.76) = ⎩ 0 elsewhere

154

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Hence : Q(1,x) (V∞(1) > y) = E[1T0 ◦θT A <x−y ] y

= E[ERT A (T0 < x − y)] (by the strong Markov property) y RTyA = E Φ(α) (by (3.9)) x−y α y α y =1− (3.77) = 1− x−y+y x by (3.18) and (3.24). 4) We prove that, under Q(1,x) , g < ∞ a.s. We have : Q(1,x) (g > t) = Q(1,x) (T0 ◦ θt < ∞) = Q(1,x) (t + T0 ◦ θt < ∞) = E(1dt <∞ Mdt ) (by Doob’s optional stopping time Theorem) with dt = t + T0 ◦ θt . But Mdt ≤ 1(V (1) ≤x) (because Rdt = 0). Hence : gt

Q(1,x) (g > t) ≤ P (Vg(1) ≤ x) −→ 0 t t→∞

Observe that g < ∞

Q(1,x) a.s. implies A∗∞ = ∞

Q(1,x) a.s.

5) We now calculate Az´ema’s supermartingale associated with g Proposition 3.7. Let, for t ≥ 0, Zt := Q(1,x) (g > t|Ft ). 1) Then : Zt = =

1V (1) ≤x gt

Mt 1A∗t ≤x Mt

(α)

1At ≤x Φ R (α) √ t Φ x − At

R √ t x − At

2) For any (Fs ) predictable process (Ks , s ≥ 0) : ∞ 1 E Ks 1(V (1) ≤x) dLs EQ(1,x) (Kg ) = α s 2 Γ(1 + α)xα 0 ∞ 1 E (1,x) Ks 1(V (1) ≤x) dLs = α s 2 Γ(1 + α)xα Q 0

(3.78)

(3.79) (3.80)

(1)

3.3 Penalisation by (Vgt ≤ x)

155

Proof of Proposition 3.7 5a) From the deﬁnition (3.54) of Q(1,x) and Doob’s optional stopping Theorem, we have : Q(1,x) (g > t|Ft ) = Q(1,x) (dt < ∞|Ft ) =

E(Mdt |Ft ) Mt

1 )R (T)0 < x − At ) 1 (1) 1A <x E t Mt Vgt ≤x t (by Markov property) 1 Rt = 1A∗t ≤x Φ(α) √ by (3.9) Mt x − At

=

5b) It follows from (3.78) that for any (Ft ) bounded stopping time T : RT EQ(1,x) (g > T ) = E Φ(α) √ 1A∗T ≤x (3.81) x − AT But :

Rs2α + 1As ≤x Φ(α) 2α Γ(1 + α)xα

R √ s x − As

, s≥0

and (Rs2α −Ls , s ≥ 0) are two martingales (by (3.11) and point 1) of Theorem 3.5). Hence : EQ(1,x) [g < T ] =

1 E 2α Γ(1 + α)xα

0

T

1(V (1) ≤x) dLs s

and, from the monotone class theorem, we obtain, for every predictable process (Ks , s ≥ 0) : ∞ 1 E K 1 dL EQ(1,x) [Kg ] = α (1) s s Vs ≤x 2 Γ(1 + α)xα 0 (1)

(1)

which is precisely (3.79). (Note that Vgs = Vs , dLs a.e.). 5c) We now prove (3.80). But (3.80) is a consequence of (3.79). Indeed, for every (Fs ) predictable process (Ks , s ≥ 0) we have : t t t EQ(1,x) [ Ks dLs ] = E Mt Ks dLs = E Ms Ks dLs 0 0 0 t = EP Ks 1(V (1) ≤x) dLs 0

since Ms = 1(V (1) ≤x) , dLs ⊗ dP a.s. s

s

156

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

6) We prove (3.58). We have, by (3.81) Rt Q(1,x) (g > t) = E 1A∗t ≤x Φ(α) √ x − At α At = E 1A∗t ≤x 1 − (by (3.18) and (3.24)) x At∧TxA α , which is (3.58). = E 1− x We also have by (3.79) : Q(1,x) (g < t) = = =

1 E α 2 Γ(1 + α)xα 1 E 2α Γ(1 + α)xα 2α Γ(1

t

1(V (1) ≤x) dLs s

0 A t∧Tx

dLs 0

1 E(Lt∧TxA ) + α)xα

In particular : Q(1,x) (g < ∞) = lim Q(1,x) (g < t) = t→∞

2α Γ(1

1 E(LTxA ) = 1 + α)xα

(3.82)

by point 5) of Proposition 3.2. Note that (3.82) provides another way to obtain (3.57). 7) We prove (3.59) and (3.60). By (3.79), we have, for h a positive Borel function : EQ(1,x) [h(Lg )] = EQ(1,x) [h(L∞ )]

∞ 1 E h(Ls )1(V (1) ≤x) dLs = α s 2 Γ(1 + α)xα 0 ∞ 1 E h()1(V (1) ≤x) d = α τ 2 Γ(1 + α)xα 0

(from (3.79))

(thanks to the change of variable = Ls , using the fact that τ is the right continuous inverse of (Lt , t ≥ 0)) ∞ 1 = α h()dE 1 (1) (Vτ ≤x) 2 Γ(1 + α)xα 0 ∞ 1 = α h()e− xα 2α Γ(1+α) d (3.83) α 2 Γ(1 + α)x 0 by point 6) of Proposition 3.2.

(1)

3.3 Penalisation by (Vgt ≤ x)

157

8) We prove point 3) ii) of Theorem 3.5. For every positive functional F and every h positive Borel function, we have, by (3.79) : EQ(1,x) (F (Ru , u ≤ g)h(Lg )) ∞ 1 E F (Ru , u ≤ s)h(Ls )1(V (1) ≤x) dLs (from (3.79)) = α s 2 Γ(1 + α)xα 0 ∞ 1 E F (Ru , u ≤ τ )h()1(V (1) ≤x) d = α τ 2 Γ(1 + α)xα 0 (after the change of variable = Ls ) ∞ 1 (1) = α E F (R , u ≤ τ )|V ≤ x h()P (Vτ(1) ≤ x)d u τ 2 Γ(1 + α)xα 0 Hence, by (3.16) and (3.17) EQ(1,x) (F (Ru , u ≤ g)h(Lg )) ∞ (1,x) E F (Ru , u ≤ τ )|Vτ(1) ≤ x h()fLQg ()d = 0 ∞ (1,x) = EQ(1,x) (F (Ru , u ≤ g)|Lg = ) h()fLQg ()d

(3.84)

0

We deduce from (3.84) : EQ(1,x) (F (Ru , u ≤ g)|Lg = ) = E F (Ru , u ≤ τ )|Vτ(1) ≤ x

(3.85)

which is point 3) of Theorem 3.5. 9) We prove that (Rg+t , t ≥ 0) is , under Q(1,x) a Bessel process of dimension 4 − d. For this purpose, we shall use the technique of (progressive) enlargement of ﬁltration. Thus, let (Gt , t ≥ 0) denote the smallest ﬁltration containing (Ft ) and such that g becomes a (Gt , t ≥ 0) stopping time. Let us recall that : t Nt := Rt2α − Lt = 2αRs2α−1 dBs 0 1 2α Rs Rs + Φ(α) √ 1As ≤x and Mt := 1V (1) ≤x gt cα x − As

(3.86) (3.87)

(where we have denoted cα = 2α Γ(1 + α)xα to simplify our writing) are two o’s formula : martingales. Thus, on the set (A∗t ≤ x), by Itˆ Rt Rt 1 Rt1−2α √ dNt (3.88) d Φ(α) √ = Φ(α) √ x − At x − At 2α x − At

158

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

and

dMt =

1 dNt + Φ(α) cα

R √ t x − At

1 Rt1−2α √ dNt (3.89) 2α x − At

Thus, according to Girsanov’s theorem, under Q(1,x) : t 1 Rs 1 Rs1−2α 2αRs2α−1 t + √ Bt = B + Φ(α) √ ds (3.90) Ms cα x − As 2α x − As 0 t , t ≥ 0) a ((Ft , t ≥ 0), Q(1,x) ) Brownian motion. with (B We also recall that, from Proposition 3.7: Rt 1 (α) √ Zt := Q(g > t|Ft ) = 1A∗t ≤x Φ Mt x − At

(3.91)

Thus, from (3.88) and (3.89), the martingale part (Zt ) of (Zt ) satisﬁes : R1−2α Φ(α) 1 (α) 1 √ dN dN + Φ dZ = − M 2 cα 2α x − A R1−2α 1 1 √ + dN Φ(α) M 2α x − A and we have :

,

d < Z, B >t = 2αR +

1 M

2α−1

Φ(α) − M2

Φ(α)

1 R1−2α 1 √ + Φ(α) cα 2α x − A 1 R1−2α √ dt 2α x − A

(3.92)

(3.93)

where in (3.92) and (3.93), to simplify our writing, we have denoted : R R (α) (α) (α) (α) √ √ resp. Φ for Φ resp. Φ Φ x−A x−A From the enlargement formulae (see [J]), there exists a (Gt , Q(1,x) ) Brownian motion (Wt , t ≥ 0) such that, for t > g : t = Wt − B

t

g

d < Z, B >s 1 − Zs

(3.94)

Thus, gathering (3.86), (3.90), (3.93) and (3.94), there exists a (Gt , t≥0, Q(1,x) ) Brownian motion (Wt , t ≥ 0) such that, for t ≥ 0, we have : 2α Rg+t

t 2α−1 2αRg+s dWs

= 0

t 2α−1 2αRg+s Λg+s ds

+ 0

(3.95)

(1)

3.3 Penalisation by (Vgt ≤ x)

159

(using the fact that Rg = 0, and that (Lt ) is constant after g) with, from (3.90), (3.93) and (3.94) : ,

1 R1−2α 1 √ + Φ(α) cα 2α x − A R 1 1 M (α) (α) 1 √ − −Φ + Φ 2α x − A M − Φ(α) M 2 cα , 1 2α 2αR2α−1 cα (α) R1−2α √ Φ = R 1 + 2α (M − Φ(α) )M cα x − A cα R 1 √ Φ(α) +Φ(α) − 2α x − A 1 2α 2αR2α−1 (α) = R +Φ M (M − Φ(α) )cα cα 2αR2α−1 2αR2α−1 = = 1 2α (M − Φ(α) )cα cα R cα

Λ = 2αR

2α−1

1 M

i.e. : Λs =

2α Rs

(3.96)

t := Rg+t Plugging (3.96) in (3.95), we obtain, after denoting R t2α = R

0

t

s2α−1 dWs + 2αR

t

s2α−2 ds 4α2 R

(3.97)

0

t = (f (R t ))2α , with : f (u) = u 2α1 , Then, applying Itˆ o’s formula to compute : R 1−2α 1−4α 1 α f (u) = 2α u 2α , f (u) = 1−2α we obtain : 4α2 u

t

1 1−2α 2α−2 ds) 2α−1 dWs + 4α2 R Rs (2αR s s 0 2α t 1 − 2α 1−4α R 4α−2 ds + 4α2 R s s 8α2 0 t 1 − 2α ds = Wt + + 2α s 2 0 R t ds 1 + 2α = Wt + s 2 0 R

t = R

(3.98)

t , t ≥ 0) is a Bessel process with dimension (4 − d) ≡ which proves that (R 2(1 + α).

160

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

The independence of the processes (Rt , t ≤ g) and (Rt , t ≥ g) under Q(1,x) follows classically from the fact that the stochastic diﬀerential equation (3.98) enjoys the strong uniqueness property. 10) We now prove that E(MTyA ) = 1 (0 < y < x) and point 4)i) of Theorem 3.5. 10a) We have, by the deﬁnitions (3.52) and (3.53) of Ms : RTyA 1 2α (α) √ MTyA = α R A +Φ (3.99) 2 Γ(α + 1)xα Ty x−y and

E[MTyA ] = E MTyA |FgT A y

1 (x−y) = α Kψ1,α (y) + Kψ2,α (y) 2 Γ(α + 1)xα (with the notations of point 3 of Proposition 3.2) y α 1 2α Γ(α + 1)y α + 1 − =1 = α α 2 Γ(α + 1)x x by (3.23) and (3.24). Hence : E[MTyA ] = 1

(0 < y < x)

(3.100)

10b) For every positive functional F , we have : EQ(1,x) [F (Au , u ≤ TyA )] = E[F (Au , u ≤ TyA )MTyA ] and, because MTyA depends only on RTyA (by (3.99)), MTyA and FTyA are independent (see (3.20)). Hence : EQ(1,x) [F (Au , u ≤ TyA )] = E[F (Au , u ≤ TyA )]E(MTyA ) = E[F (Au , u ≤ TyA )]

by (3.100)

11) We prove point 4)ii) of Theorem 3.5 and we calculate the density of RTyA . 11a) Let h be a positive Borel function and F a positive functional. We have : EQ(1,x) [F (Au , u ≤ TyA )h(RTyA )] = E[F (Au , u ≤ TyA )h(RTyA )(MTyA )] = E[F (Au , u ≤ TyA )]E[h(RTyA )(MTyA )] (because MTyA only depends on RTyA , and RTyA and FTyA are independent) (x−y)

= E[F (Au , u ≤ TyA )]K(h(ψ1,α + ψ2,α ))(y)

(3.101)

(1)

3.3 Penalisation by (Vgt ≤ x)

161

11b) The formula (3.101), with F ≡ 1, provides the following : (x−y)

EQ(1,x) [h(RTyA )] = K[h(ψ1,α + ψ2,α )](y) and, from the explicit form of K (see (3.19)) and of ψ1,α and ψ2,α (see point 3 of Proposition 3.2) we obtain : (1,x) fRQ A (z) T y

z z2 = e− 2y y

z 2α + Φ(α) α 2 Γ(α + 1)xα

√

z x−y

1z≥0

(3.102)

12) We compute Q(1,x) (g > TyA ) and we prove point 4)iii) of Theorem 3.5. 12a) By point 1 of Proposition 3.7 we have : (1,x)

Q

(g >

TyA |FgT A ) y

=

1A∗ A ≤x Ty

MTyA

(α)

Φ

RT A √ y x−y

(3.103)

But 1A∗ A ≤x = 1 (because y < x) and so : Ty

RT A Q(1,x) (g > TyA ) = E Φ(α) √ y x−y y α (x−y) = K(ψ2,α )(y) = 1 − x

(3.104)

with the notation of point 3) of Proposition 3.2 and by (3.24). 12b) For every positive functional F , one has : EQ(1,x) [F (Au , u ≤ TyA )1g>TyA ] = EQ(1,x) [F (Au , u ≤ TyA )E(g > TyA |FgT A )] y % & R 1 TyA A (α) √ = EQ(1,x) F (Au , u ≤ Ty ) Φ by (3.78) x − y MTyA RT A = E F (Au , u ≤ TyA )E Φ(α) √ y |FgT A y x−y (x−y)

= E[F (Au , u ≤ TyA )]•K(Ψ2,α )(y) by (3.24) = EQ(1,x) [F (Au , u ≤ TyA )]•Q(1,x) (g > TyA ) by point 4)i) of Theorem 3.5 and (3.104). This ends the proof of Theorem 3.5.

162

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

3.4 Penalisation by (Vt(1) ≤ x) and (Vd(1) ≤ x) t In this section, we penalize the original law P (α) with : 1(V (1) ≤x) and with t 1(V (1) ≤x) . Again, we obtain a limiting probability, but the martingale which dt

occurs here is very diﬀerent from the one we obtained in the preceding section. This is explained by Theorem 3.4, which shows a radically diﬀerent behavior, (1) (1) (1) as t → ∞, for P (Vgt ≤ x) on one hand, and P (Vt ≤ x) and P (Vdt ≤ x) on the other hand. 1) Penalisation by 1(V (1) ≤x) . t

Theorem 3.8. We suppose (C) is satisﬁed. Let x > 0 and λ0 (α) be deﬁned by Theorem 3.4. 1) For every s ≥ 0 and Λs ∈ Fs : E(1Λs 1(V (1) ≤x) ) t

lim

(1) P (Vt

t→∞

≤ x)

exists

(3.105)

This limit is equal to lim

E(1Λs 1(V (1) ≤x) ) t

(1) P (Vt

t→∞

≤ x)

= E(1Λs Ms∗ )

(3.106)

where : Ms∗ = 1A∗s ≤x eλ0 (α) x s

Rs2α α 2 Γ(α)

x−As

0

du − Rs2 + λ0 (α)u x e 2u uα+1

(3.107)

(Ms∗ , s ≥ 0) is a positive martingale such that : M0∗ = 1. 2) Let, for s ≥ 0 and Λs ∈ Fs : Q∗ (Λs ) := E(1Λs Ms∗ )

(3.108)

Q∗ induces a probability on (Ω, F∞ ). The process (Rt , t ≥ 0) under Q∗ is such that : i) A∗∞ = x ii)

TxA

a.s.

(3.109)

:= inf{t; At = x} = ∞

a.s.

(3.110)

Proof of Theorem 3.8 1) We prove that

E(1(V (1) ≤x) |Fs )

We have, for s < t

t

(1)

P (Vt

≤ x)

−→ Ms∗

t→∞

a.s.

1(V (1) ≤x) = 1(A∗t ≤x) = 1(A∗s ≤x) •1T0 ◦θs ≤x−As •1(A∗t−ds ◦θds ≤x) t

(3.111)

(1)

3.4 Penalisation by (Vt

(1)

≤ x) and (Vdt ≤ x)

163

with ds = s + T0 ◦ θs . We recall that the law of T0 , for the process (Rt , t ≥ 0) started at r, admits, by (3.6), the density : 1

r2

2α Γ(α)

r2α e− 2u

1 1u≥0 uα+1

(3.112)

Hence, by point 2) of Theorem 3.4, and conditioning (3.111) on T0 ◦ θs = u :

E 1(V (1) ≤x) |Fs t

(1)

P (Vt

≤ x)

1(A∗s ≤x) C

∼

x−As

0

t→∞

λ (α) R2 1 2α − 2us − 0x (t−s−u) du R e s 2α Γ(α) uα+1 t −λ (α) 0 x Ce

−→ Ms∗

t→∞

with : Ms∗

s λ0 (α) x

= 1(A∗s ≤x) e

x−As

0

R2 Rs2α du − 2us +λ0 (α) u x e 2α Γ(α) uα+1

(3.113)

Observe that the probabilities on R+ (indexed by r): r2 du r2α − 2u e 2α Γ(α) uα+1

converge, as r goes to 0, to the Dirac measure at 0, and so : M0∗ = 1 2) We prove that (Ms∗ , s ≥ 0) is a martingale. We ﬁrst observe that we may write Ms∗ , after the change of variable as : ∞ 2 R2 s 1 − v2 +λ0 (α) xvs2 2α−1 ∗ λ0 (α) x e v dv Ms = 1(A∗s ≤x) e 2α+1 Γ(α) √ Rs

Rs2 u

= v2

x−As

We also note that 1(A∗s ≤x) = 1(TxA ≥s) and that, for s < TxA , Ms∗ > 0. Thus, (Ms∗ , s ≥ 0) is stopped at its ﬁrst zero. Moreover, we have : ∞ 2 R2 (x−A ) s 1 − v2 +λ0 (α) s 2 s 2α−1 ∗ λ0 (α) x xRs e v dv 0 ≤ Ms ≤ 1(A∗s ≤x) e 2α+1 Γ(α) √ Rs x−As

Rs 1 x − As since v ≥ √ implies 2 ≤ v Rs2 x − As s

≤ C(α)eλ0 (α) x eλ0 (α)

(3.114)

Thus, to show that (Ms∗ , s ≥ 0) is a martingale, it suﬃces to see that it is a local martingale.

164

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

And, in order to see that (Ms∗ , s ≥ 0) is a local martingale, one only has to apply Itˆ o’s formula. The interested reader can refer to [RVYCR ] for this computation. We note that, in this computation, the fact that θ(λ0 (α)) = 0 (see (3.32)) plays a crucial role. 3) We prove point 2) of Theorem 3.8. 3a) It is clear that A∗∞ ≤ x Q∗ a.s., since, for every s ≥ 0 : Q∗ (A∗s ≤ x) = lim

t→∞

P [(A∗s ≤ x) ∩ (A∗t ≤ x)] =1 P (A∗t ≤ x)

3b) We have, by (3.49), for λ > 0, small enough : A E eλTy =

1 1−

yλ e−λy

1 1−α Φ(1

− α, 2 − α, λy)

(3.115)

and so, by the deﬁnition of λ0 (α) : A E eλTy < ∞

if

λy ≤ λ0 (α)

(3.116)

Hence : A A A λ0 (α) A EQ∗ eλTy = E eλTy MTyA ≤ C (α)E eλTy + x Ty

(3.117)

by (3.114). We deduce from (3.117) and (3.116) : A EQ∗ eλTy < ∞

for

λy +

λ0 (α) y < λ0 (α) x

Hence, there exists λ > 0 such that λ + λ0 (α)y < λ0 (α) (because y < x) and x for such a λ : A and EQ∗ eλTy < ∞ EQ∗ [TyA ] < ∞ , hence TyA < ∞

Q∗ a.s.

Finally : Q∗ (A∗∞ > y) = Q∗ (TyA < ∞) = 1 . 3c) We prove that TxA = ∞ For any y < x, one has :

Q∗ a.s.

Q∗ (A∗t < y) = E 1(TyA >t) Mt∗ −→ E 1(TxA ≥t) Mt∗ = E(Mt∗ ) = 1 y↑x

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1

165

since 1(TxA
which proves that TxA = ∞ , Q∗ a.s. 2) Penalisation by 1(V (1) ≤x) . dt

Theorem 3.9. We suppose (C) is satisﬁed. Let x > 0 and λ0 (α) be deﬁned by Theorem 3.4. 1) For every s ≥ 0 and Λs ∈ Fs : lim

E(1Λs 1(V (1) ≤x) )

t→∞

dt

(1)

P (Vdt ≤ x)

exists

(3.118)

This limit is equal to lim

t→∞

E(1Λs 1(V (1) ≤x) ) dt

(1) P (Vdt

≤ x)

= E(1Λs Ms∗ ) = Q∗ (Λs )

(3.119)

where (Ms∗ , s ≥ 0) is the positive martingale deﬁned by (3.107). In other words, the penalisation by 1(V (1) ≤x) leads to the same probability Q∗ dt

as the penalisation by 1(V (1) ≤x) . t The proof of Theorem 3.9 is very close to the one of Theorem 3.8, via Theorem 3.4.

3.5 Penalisation with (Vg(n) ≤ x) (x > 0), for n ≥ 1 t We shall use the following notation : Δxs = {u ≤ s; Au = x} = sup{p ≥ 1; Hx(p) ≤ s}

(3.120)

and the sequences of ranked lengths in decreasing order will be usually denoted as → − (3.121) V = (V (1) , V (2) , ..., V (n) , ...) Besides the notation (Rt , t ≥ 0) for the d-dimensional Bessel process, we shall also use (rt , t ≤ 1) for the corresponding standard Bessel bridge, for which → − v = (v (1) , v (2) , ..., v (n) , ...) shall denote the ranked sequence in decreasing order of excursion lengths, which add up to 1.

166

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

The main result of this section is Theorem 3.10. 1) For every s ≥ 0, Λs ∈ Fs E 1Λs 1V (n) ≤x gt lim (n) t→∞ P (Vgt ≤ x)

exists

(3.122)

2) This limit induces a probability Q(n,x) on (Ω, F∞ ) such that : Q(n,x) (Λs ) = E(1Λs Ms(n,x) )

(3.123)

where :

Rs2α Rs 1 (α) √ + 1 Φ − 1 A ≤x gs 2α Γ(1 + α)xα n n s x − As x + Δ + 1− s (3.124) n

Ms(n,x) = 1V (n) ≤x

where Φ(α) has been deﬁned by (3.8) and Δxs by (3.120). (n,x) (n,x) (Ms , s ≥ 0) is a positive and continuous martingale such that M0 = 1. 3) Under Q(n,x) , the r.v : g := sup{t, Rt = 0}

is a.s. ﬁnite

(3.125)

and the processes : (Ru , u ≤ g)

and

(Rg+u , u ≥ 0)

are independent.

− → 4) Description of Vg under Q(n,x) − → 1 Vg := (Vg(1) , Vg(2) , ...) and Vg(n) are independent; i) g g (d)

ii) Vg(n) = xU 1/α , where U is uniform on [0,1]; − → Vg under Q(n,x) is characterized by : iii) The law of g % % − & → & → − 1 L1 Vg = α E EQ(n,x) h h(V1 ) (n) g n2 Γ(1 + α) (V1 )α % −→ & Vτ1 1 1 E = α h (n) α n2 Γ(1 + α) τ1 (Vτ1 ) 1 1 → − E h( v ) = nΓ(1 + α)Γ(1 − α) (v (n) )α

(3.126)

(3.127) (3.128)

(3.129) (3.130) (3.131)

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1

167

5) Descriptionof (Ru , u ≤ g) under Q(n,x) (n) R Rug √ , u ≤ 1 , which i) √ug g , u ≤ 1 is independent from Vg , and the law of g does not depend on x, satisﬁes : 1 Rug 1 Q(n,x) E = E F (ru , u ≤ 1) F √ ,u ≤ 1 g nΓ(1 + α)Γ(1 − α) (v (n) )α (3.132) (n)

R

ii) Conditionally on Vg , the process √ug g , u ≤ 1 and the r.v. g are independent. iii) Conditionally on L∞ (= Lg ) = , the process (Ru , u ≤ g) is a Bessel process of dimension d = 2(1 − α), stopped at time τ and conditioned on (n) Vτ ≤ x. (n,x) iv) fLQ∞ , the density of L∞ , is equal to : (n,x)

fLQ∞

() =

1 Γ(n + 1)2α Γ(1 + α)xα

∞

e−u un−1 du

(3.133)

2α Γ(1+α)x

v) The law of g is given by : for every t ≥ 0, 1 E[Lt∧H (n) ] x n2α Γ(1 + α)xα % α α & t 1 x = E ∧ (n) n2α Γ(1 + α)xα τ 1 Vτ1 % & At∧H (1) α x =E (if n = 1) x

Q(n,x) [g < t] =

(3.134)

6) The process (Rg+u , u ≥ 0) is a Bessel process of dimension δ = 4 − d = 2 + 2α started at 0. Proof of Theorem 3.10 (n) 1) An equivalent for P (Vgt ≤ x|Fs ) , t → ∞ (n) To obtain this equivalent, we shall partition the event (Vgt ≤ x) with the values of Δxs . We get, for k ≤ n − 1 : (Δxs = k)∩(Vg(n) ≤ x) = {(As ≤ x)∩(D1 ∪D2 ∪D3 )}∪{(As > x)∩(D2 ∪D4 )} t (3.135) with : D1 = (T0 ◦ θs ≤ x − As ) ∩ (Vg(n−k) ◦ θds ≤ x) t−ds D2 = (T0 ◦ θs ≥ t − s) D3 = (x − As < T0 ◦ θs ≤ t − s) ∩ (Vg(n−k−1) ◦ θds ≤ x) t−ds D4 = (T0 ◦ θs ≤ t − s) ∩ (Vg(n−k) ◦ θds ≤ x) t−ds

168

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

and for k = n : ≤ x) = (As > x) ∩ D2 (Δxs = n) ∩ (Vg(n) t Given that :

r √ Pr (T0 < u) = Φ u 2α r Pr (T0 > t) ∼ α t→∞ 2 Γ(α + 1)tα xα P (Vg(n) ≤ x) ∼ n α t t→∞ t (α)

(3.9) : (3.10) : (3.43) :

we deduce from (3.135) that : E 1V (n) ≤x |Fs gt n−1 Rs Rs2α xα (α) √ ∼ 1(As ≤x) Φ (n − k) α + α t→∞ t 2 Γ(α + 1)tα x − As k=0 Rs xα +(1 − Φ(α) ) √ (n − k − 1) α t x − As 2α α Rs x +1(As >x) α + (n − k) α 2 Γ(α + 1)tα t 2α Rs (3.136) +1(Δxs =n) 1(As >x) α 2 Γ(α + 1)tα or, after reorganizing the diﬀerent terms : E 1V (n) ≤x |Fs gt Rs2α Rs xα (α) √ ∼ 1(V (n) ≤x) α + 1 Φ − 1 (As ≤x) α gs t→∞ 2 Γ(α + 1)tα t x − As x α +(n − Δxs )+ t (n)

hence, since : P (Vgt ≤ x) ∼ n t→∞

lim

x α t

, we obtain :

E 1V (n) ≤x |Fs gt

(n)

P (Vgt ≤ x) Rs2α Rs 1 (α) √ = 1(V (n) ≤x) + 1 − 1 Φ (As ≤x) gs n2α Γ(α + 1)xα n x − As + x Δ + 1− s n t→∞

def

= Ms(n,x)

(3.137)

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1

169

2) Proofs of points 1) and 2) of Theorem 3.10 |Fs (n) Vg ≤x t (n) P (Vgt ≤x)

E 1

We have just seen that (n,x) . Ms

converges a.s., as t → ∞, towards (n,x)

Assuming for a moment that E[Ms ] = 1, Scheﬀ´e’s lemma (see [M], Chap. II, T. 21) implies that the convergence holds in L1 , as well as the con(n,x) sequence that (Ms , s ≥ 0) is a martingale. In [RVY,VII], to which we refer the reader, we have proven this result in a very similar situation. (n,x) Moreover, we will prove, with the help of other arguments, that E(Ms )=1 in the Remark 3.12 placed at the end of this section (see (3.164) and (3.165)). (n,x) As a partial check, one may see, by inspection, that the martingale (Ms ) is continuous (recall that every (Fs ) martingale is continuous), and that this positive martingale remains in 0 after its ﬁrst hitting time of 0, here : dH (n) , x as does every positive supermartingale. 3) Computation of the Az´ema supermartingale Zt := Q(n,x) (g > t|Ft ). Most of the results announced in Theorem 3.10 rely heavily on the explicit knowledge of (Zt ), which we now present (recall that g := sup{t ≥ 0; Rt = 0}): Lemma 3.11. 1) One has : Zt := Q(n,x) (g > t|Ft ) + 1V (n) ≤x Rt 1 Δxt gt (α) √ = 1At ≤x −1 + 1− (3.138) Φ Mt n n x − At 2) For every positive (Fs ), predictable process (Ks , s ≥ 0) : ∞ 1 E K 1 dL (3.139) (n) s (V s s ≤x) n2α Γ(α + 1)xα 0 ∞ 1 E K 1 dL (3.140) = (n,x) (n) s (V s s ≤x) n2α Γ(α + 1)xα Q 0

EQ(n,x) [Kg ] =

(Throughout the sequel, we shall now write Q for Q(n,x) ). Proof of Lemma 3.11 : i) From the deﬁnition (3.123) of Q, and Doob’s optional stopping Theorem, one has : Q(g > t|Ft ) = Q(dt < ∞|Ft ) = Since Mdt = 1 − tion ψ : N → R+ .

Δx dt n

+

E(Mdt |Ft ) Mt

(3.141)

, we are led to compute E[ψ(Δxdt )|Ft ] for any func-

170

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

We get : E[ψ(Δxdt )|Ft ] Rt Rt (α) x (α) x √ √ = 1(At ≤x) Φ ψ(Δt ) + 1 − Φ ψ(1 + Δt ) x − At x − At +1(At >x) ψ(Δxt ) Rt = ψ(Δxt ) + 1(At ≤x) Φ(α) √ (3.142) − 1 (ψ(Δxt ) − ψ(Δxt + 1)) x − At

+ Plugging now (3.142), with ψ(u) = 1 − nu in (3.141), we obtain : E[Mdt |Ft ] + Rt Δxt (α) √ = 1− + 1(At ≤x) Φ − 1 × ... n x − At % + + & Δxt Δxt + 1 ... 1 − − 1− n n + Rt Δx 1 1− t + 1(At ≤x) 1(Δxt ≤n−1) −1 Φ(α) √ n n x − At + Rt 1 Δx = 1− t + 1(V (n) ≤x) 1(At ≤x) −1 Φ(α) √ gt n n x − At

=

since : 1(At ≤x) 1(Δxt ≤n−1) = 1(At ≤x) 1(V (n) ≤x) gt

, = 1(V (n) ≤x) gt

since 1 −

Δx t n

+

1(At ≤x)

+ Rt 1 Δxt (α) √ −1 + 1− Φ n n x − At (n)

= 0 for t ∈ [Hx , dH (n) ]. Hence : x

EQ [g > t|Ft ]=

1(V (n) ≤x)

,

gt

1(At ≤x)

Mt

+Rt 1 (α) Δxt √ −1 + 1− Φ n n x − At

ii) We now prove point 2) of lemma 3.11. It follows from (3.138) that for any bounded stopping time T : ,

% EQ [g>T ] =E 1(V (n) ≤x) gT

1(AT ≤x)

+-& RT 1 (α) ΔxT √ −1 + 1 − Φ n n x − AT

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1

171

But, since Rs2α Rs 1 (α) √ + 1(As ≤x) Φ , s ≥ 0 and (Rs2α −Ls , s ≥ 0) n2α Γ(α + 1)xα n x − As are two martingales, we obtain : 1 E EQ [g < T ] = n2α Γ(α + 1)xα =

1 E n2α Γ(α + 1)xα

1(V (n) ≤x) dLs gs

0

T

T

1(V (n) ≤x) dLs s

0

since Vg(n) = Vs(n) , dLs a.s. s Hence, from the monotone class Theorem, we obtain, for every (Fs ) predictable process Ks ≥ 0 : 1 EQ [Kg ] = α E n2 Γ(α + 1)xα

∞

0

Ks 1(V (n) ≤x) dLs s

which is precisely (3.139). Next, to go from (3.139) to (3.140), we shall write : for every (Fs ) predictable process (Ks , s ≥ 0) :

t

EQ

t Ks dLs = E Mt Ks dLs

0

0

t

=E

Ks Ms dLs 0

t

=E 0

Ks 1(V (n) ≤x) dLs s

since Ms = 1(V (n) ≤x) , dLs ⊗ dP a.s. s Applying the last relation with Ks = Ks 1(V (n) ≤x) and letting t → ∞, we s obtain (3.140). In particular : ∞ ∞ Ks 1(V (n) ≤x) dLs = E Ks 1(V (n) ≤x) dLs EQ (3.143) 0

s

0

s

It easily follows from (3.139) that : Q[g = ∞] = lim Q(g > t) = 0 t→∞

hence, that : g < ∞

Q a.s.

172

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

4) We now prove points 4) i),ii) and iii) of Theorem 3.10 4a) Let h : RN + → R+ measurable, and f : R+ → R+ , Borel. By (3.139) : & % − → Vg f (Vg(n) ) EQ h g % − & → ∞ 1 Vs (n) = E h (3.144) f (Vs ) 1(V (n) ≤x) dLs s n2α Γ(α + 1)xα s 0 % −→ & ∞ 1 Vτ = E h ) 1(V (n) ≤x) d f (Vτ(n) τ n2α Γ(α + 1)xα τ 0 (after the change of variable : = Ls ) % −→ & ∞ Vτ1 1 E h ) 11/α V (n) ≤x d f (1/α Vτ(n) = 1 τ1 n2α Γ(α + 1)xα τ1 0 (by scaling) % −→ &- , x α 1 Vτ1 1 α−1 E h f (y)y dy (3.145) = (n) n2α Γ(α + 1) τ1 xα 0 (Vτ1 )α (n)

(after the change of variable 1/α Vτ1 = y). Formula (3.145) exhibits the independence of

− → Vg g

(n)

and of Vg ; it also shows

(n) (d)

that Vg

= x U 1/α , with U uniform on [0,1], and it entails : % −→ & % − → & Vτ1 Vg 1 1 E EQ h h = α (n) α g n2 Γ(α + 1) τ1 (Vτ1 )

(3.146)

4b) To recover (3.129) from (3.146), we use the main result in[PY 4 ], (see also −−→ → − Vτ1 [PY6 ], Corollary, p. 289), which is that V1 is distributed as τ1 , in other terms : τ1 is admissible (see [PY6 ] for this terminology). Consequently, one has : −→ (n) α → − Vτ1 1 Vτ1 (d) (n) , , = (L1 , V1 , (V1 )α ) , hence : α τ1 τ1 τ1 α −→ → − 1 Vτ1 (d) L1 , = , V1 , and ﬁnally : (n) (n) α τ 1 Vτ1 (V1 ) % −→ & → − 1 L1 Vτ1 E =E h h(V1 ) . (n) α (n) τ1 (Vτ1 ) (V1 )α

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1

173

4c) We prove (3.131). We now come back to (3.144), which we write (taking f ≡ 1) : % − → & Vg EQ h g − → ∞ 1 Vs = E h 1(V (n) ≤x) dLs s n2α Γ(α + 1)xα s 0 ∞ 1 → − E(dL ) E h( v ) 1 = (n) ≤ x ) s (v s n2α Γ(α + 1)xα 0 (after conditioning on Rs = 0) ∞ 1 → − α sα−1 ds = E h( v ) 1 (n) ≤ x ) (v s nΓ(1 − α)Γ(1 + α)xα 0 (from (3.13) with m = 1) x 1 1 α sα−1 ds → E h(− v ) (n) α = nΓ(1 − α)Γ(1 + α) xα (v ) 0 1 1 → E h(− v ) (n) α = nΓ(1 − α)Γ(1 + α) (v ) (recall (see, e.g. : [Leb]) that : Γ(1 − α)Γ(1 + α) = 5) We now prove the independence under Q of

πα sin(πα)

Rug √ ,u g

(0 < α < 1)).

(n) ≤ 1 and of Vg , and

we show (3.132) From (3.139), we have : Rug EQ F √ , u ≤ 1 h(Vg(n) ) g ∞ Rus 1 (n) √ = , u ≤ 1 h(V E F ) 1 dL (n) s s (Vs ≤x) n2α Γ(α + 1)xα s 0 (after conditioning on Rs = 0) ∞ 1 (n) α−1 = E F (ru , u ≤ 1) h(sv ) 1(sv(n) ≤x) α s ds nΓ(1 − α)Γ(1 + α)xα 0 (from (3.13) with m = 1) x 1 1 α α−1 = E F (ru , u ≤ 1) h(y)y dy nΓ(1 − α)Γ(1 + α) xα 0 (v (n) )α (n) Vg Rug √ ,u ≤ 1 6) We prove that under Q conditionally on , the process g g and g are independent 6a) Let F denote a positive functional, and h : R+ × R+ → R+ a Borel function.

174

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Using again (3.139), we have : Rug (n) EQ F √ , u ≤ 1 h(Vg , g) g ∞ Rus 1 (n) = E F √ , u ≤ 1 h(Vs , s) 1(V (n) ≤x) dLs s n2α Γ(α + 1)xα s 0 ∞ 1 (n) α−1 = E F (ru , u ≤ 1)h(sv , s)1(sv(n) ≤x) αs ds nΓ(1 − α)Γ(1 + α)xα x 0 1 α 1 y α−1 = E F (ru , u ≤ 1) (n) α • α h y, (n) y dy nΓ(1 − α)Γ(1 + α) (v ) x v 0 (3.147) 6b) Let ϕ and θ denote two R+ valued Borel functions. Deﬁne, for ﬁxed x: Λϕ(a) :=

x a

α sα−1 ϕ(s)ds

(3.148)

0

We now apply (3.147) with h(u, v) = ϕ(v)θ uv . We obtain : % & (n) Rug Vg EQ F √ , u ≤ 1 ϕ(g) θ g g =

1 E[F (ru , u ≤ 1) θ(v (n) ) Λϕ(v (n) )] nΓ(1 − α)Γ(1 + α)xα

y (after the change of variable v(n) = s). Making ϕ ≡ 1 in (3.149) : % (n) & Rug Vg EQ F √ , u ≤ 1 θ g g 1 1 (n) E F (ru , u ≤ 1) θ(v ) (n) α = nΓ(1 − α)Γ(1 + α) (v )

(3.149)

(3.150)

α

since Λ1(v (n) ) = (vx(n) )α . Comparing (3.150) and (3.149), we have, since (3.149) depends only on (ru , u ≤ 1) and v (n) : & % (n) Rug Vg EQ F √ , u ≤ 1 ϕ(g) θ g g % & (n) (n) (n) α Rug Vg Vg Vg 1 = EQ F √ , u ≤ 1 θ Λϕ g g g xα g (3.151)

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1

So, making F ≡ 1 in (3.151) : & % (n) Vg(n) 1 Vg = α Λϕ EQ ϕ(g) g x g

175

(n)

Vg g

α

(3.152)

Now, we write (3.151) in the following form : (3.151) %

% && (n) Vg(n) Rug Vg = EQ θ EQ F √ , u ≤ 1 ϕ(g) g g g % % & (n) & (n) (n) α (n) Vg Rug V V V 1 g g g θ = EQ EQ F √ , u ≤ 1 Λϕ g g g g g xα && % % % (n) & (n) Vg Vg(n) Rug Vg EQ F √ , u ≤ 1 EQ ϕ(g) = EQ θ g g g g which yields (with θ ≡ 1): & % Vg(n) Rug EQ F √ , u ≤ 1 ϕ(g) g g & % % (n) & Vg Vg(n) Rug = EQ F √ , u ≤ 1 EQ ϕ(g) g g g 7) We now prove point 5) iii) of Theorem 3.10 and (3.133) 7a) We apply once more (3.139) in Lemma 3.11; since Lg = L∞ : EQ [F (Ru , u ≤ g) h(L∞ )] ∞ 1 E F (R , u ≤ s) h(L ) 1 dL = (n) u s s (Vs ≤x) n2α Γ(α + 1)xα 0 ∞ 1 = E F (Ru , u ≤ τ ) h() 1(V (n) ≤x) d τ n2α Γ(α + 1)xα 0 (after making the change of variable Ls = ) ∞ (n) 1 Vτ ≤ x h() P (Vτ(n) ≤ x)d = E F (R , u ≤ τ ) u n2α Γ(α + 1)xα 0 (3.153) On the other hand: EQ [F (Ru , u ≤ g) h(L∞ )] ∞ EQ [F (Ru , u ≤ g)Lg = ] fLQg () h() d = 0

(where fLQg () is the density of Lg ≡ L∞ )

176

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

1 = α n2 Γ(α + 1)xα from (3.153)

∞

0

E[F (Ru , u ≤ τ )Vτ(n) ≤ x] h() P (Vτ(n) ≤ x)d

Then, since h is arbitrary : EQ [F (Ru , u ≤ g)Lg = ] = E[F (Ru , u ≤ τ )Vτ(n) ≤ x] •

(n)

1 P (Vτ ≤ x) • α α n2 Γ(α + 1)x fLQg () (3.154)

Applying (3.153) with F ≡ 1, we obtain : fLQg () =

1 P (Vτ(n) ≤ x) n2α Γ(α + 1)xα

(3.155)

Hence : ≤ x] EQ [F (Ru , u ≤ g)Lg = ] = E[F (Ru , u ≤ τ )Vτ(n) 7b) Using (3.155) and point 6) of Proposition 3.2, we have : 1 1 1 Q fLg () = P > α α α (n) n2 Γ(α + 1)x x (Vτ )α 1 P γn > α α = n2α Γ(α + 1)xα x 2 Γ(α + 1)

(3.156)

which is our relation (3.133). Note, that for n = 1, Lg is an exponential variable with mean xα 2α Γ(α + 1). 8) We characterize the law of g under Q 8a) We calculate EQ [h(g)] using (3.127), (3.128) and (3.130) : % & g (n) EQ [h(g)] = EQ h V (n) g Vg % & x α g = α EQ h y y α−1 dy by (3.127) and (3.128) (n) x 0 Vg & % x α τ1 1 = EQ h y y α−1 dy (n) α (n) n2α Γ(α + 1)xα 0 (Vτ1 ) Vτ1 (by (3.130))

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1

177

In particular, with h(u) = 1[0,t] (u) : 1 Q[g < t] = α E n2 Γ(α + 1)xα

%

t x ∧ (n) τ1 Vτ1

α&

(3.157)

Letting t → ∞, we get : xα Q[g < ∞] = α E n2 Γ(α + 1)xα

%

1

α&

(n)

=1

(by point 6) of Prop. 3.2)

Vτ1

8b) We calculate Q[g < t] using point 1 of Lemma 3.11 in the case n = 1 α At Q[g > t] = E 1A∗t ≤x 1 − , hence : x & % At∧H (1) α x . In particular : Q[g < t] = E x Q[g < ∞] =

1 E((AH (1) )α ) = 1 . x xα

8c) We calculate Q[g < t] using point 2 of Lemma 3.11 (relation (3.139)): t 1 Q[g ≤ t] = E 1 dL (n) s (Vs ≤x) n2α Γ(α + 1)xα 0 ⎞ ⎛ t∧ Hx(n) +T0 ◦θ (n) 1 Hx E⎝ dLs ⎠ = n2α Γ(α + 1)xα 0 (n) t∧Hx 1 = E dLs n2α Γ(α + 1)xα 0 since dLs = 0 on ]Hx(n) , Hx(n) + T0 ◦ θH (n) [ x

1 = E(Lt∧H (n) ) x n2α Γ(α + 1)xα

(3.158)

In particular : Q[g < ∞] =

1 n2α Γ(α

+ 1)xα

E(LH (n) ) = 1 x

(by point 5) of Prop. 3.2).

8d) We show directly the equality between (3.157) and (3.158). In other words, we give another proof of (3.134), i.e. : % α& t x E(Lt∧H (n) ) = E ∧ (n) x τ 1 Vτ1

178

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

The left-hand side equals : ∞ E 1(s≤t)∩(s≤H (n) ) dLs x 0 ∞ =E 1(s≤t)∩(V (n) ≤x) dLs s 0 ∞ =E d 1(τ ≤t)∩(V (n) ≤x) τ 0 ∞ = d E 1(1/α ≤(t/τ1 ))∩(1/α ≤ 0

=E

%

t τ1

∧

x (n)

α&

x (n) Vτ 1

)

(by scaling)

(by Fubini).

Vτ1

9) We prove that (Ru , u ≤ g) and (Rg+u , u ≥ 0) are independent and we give the law of (Rg+u , u ≥ 0) This proof is very similar to the one made in point 9) of the proof of Theorem 3.5. This ﬁnishes the proof of Theorem 3.10. Remark 3.12. In the preceding, we used in an essential way the martingale (n,x) , s ≥ 0) : (Ms Rs2α Rs 1 (α) √ 1 Ms(n,x) = 1V (n) ≤x + Φ − 1 A ≤x gs n2α Γ(1 + α)xα n s x − As x + Δ (3.159) + 1− s n Of course, it is interesting to compute the projection of this martingale on the “small” ﬁltration (Fgs , s ≥ 0). 1) Let us show : Π(n,x) := E[Ms(n,x) Fgs ] s % α & Δxgs + 1 + As 1 = 1V (n) ≤x + sup 1 , 1− (3.160) gs n n x In particular, for n = 1 : := E[Ms(1,x) Fgs ] Π(1,x) s α As = 1V (1) ≤x sup 1 , gs x (see (3.71)).

(3.161)

(n)

3.5 Penalisation with (Vgt ≤ x) (x > 0), for n ≥ 1

179

Let us prove (3.160). From (3.18), (3.19), (3.23) and (3.24), we deduce : α Rs2α 1 As Fg s = E (3.162) n2α Γ(1 + α)xα n x and α Rs As 1 1 E 1(As ≤x) Φ(α) √ (3.163) − 1 Fgs = − 1(As ≤x) n n x x − As And, as : Δxgs Δxgs + 1 Δxs 1− = 1(As ≤x) 1 − + 1(As >x) 1 − n n n we obtain : Π(n,x) := E[Ms(n,x) Fgs ] s α α Δxgs 1 As As 1 = 1(V (n) ≤x) − 1(As ≤x) + 1(As ≤x) 1 − gs n x n x n + x Δgs + 1 +1(As >x) 1 − n + α Δxgs + 1 1 As = 1(V (n) ≤x) 1 − + gs n n x + α As 1 + 1(As ≤x) 1 − n x α x Δgs + 1 As 1 = 1(V (n) ≤x) 1 − sup 1 , + (3.164) gs n n x + 2) We have already proven (see point 2d) of the proof of Proposition 3.6) that (1,x) E[Πs ] = 1 (n,x)

3) Let us prove that E[Πs ] = 1 for all n ≥ 2 (1,x) We have just seen that E[Πs ] = 1. Let us calculate : % % α && Δxgs + 1 + 1 As (n,x) sup 1 , 1− + E[Πs ] = E 1V (n) ≤x gs n n x We have : α As 1 E 1V (n) ≤x sup 1 , gs n x α As 1 = E 1s
180

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

%

α & n As 1 = E 1d (k−1) ≤s
Hence :

%

E[Π(n,x) ] s

+ &

α As 1 =E + E 1V (n) ≤x 1 ∨ g s n x & % + Δxgs + 1 Δxgs + 1 ∧1 =1 +E =E 1− n n Δxgs + 1 1− n

(n,x)

4) Now, we can easily verify that (Πs , s ≥ 0) is a (Fgs , s ≥ 0, P ) martingale Indeed, between the stopping times dH (k) and dH (k+1) (which are (Fgs , s ≥ 0) x x x Δgs +1 is constant. Then, we only need to prove stopping times), the process n (1,x)

that (Πs write :

, s ≥ 0) is a (Fgs , s ≥ 0) martingale which is easy to see when we α As (1,x) = 1s≤H (1) + 1H (1) <s
then, for s < t : (1,x)

(1,x)

= (1s≤H (1) )[1T0 ◦θs <x−As [1sds Πt−ds ◦ θds ]] x α As + (t − s) +(1s≤H (1) ) 1T0 ◦θs >x−As 1t−s<x−As + 1x−As
Πt

and ﬁnally with the help of the projection formulae of Proposition 3.2 : (1,x) E(Πt Fgs ) = Π(1,x) (s ≤ t) s Finally, let us note from the previous point 3) : E[Ms(n,x) ] = E[E(Ms(n,x) Fgs )] = E[Π(n,x) ]=1 s

(3.165)

3.6 Weak Convergence of the Penalised Laws Q(n,x) , as n → ∞

181

3.6 Weak Convergence of the Penalised Laws Q(n,x) , as n → ∞; a Commutative Diagram ∞ 1) Let h : R+ → R+ be a Borel function such that : 0 h(x)dx = 1. In [RVY,V], we studied the penalisation of the d-dimensional Bessel process, for 0 < d < 2 by h(Lt ), where (Lt ) denotes the local time at 0. Precisely, we have shown that : E[1Λs h(Lt )] (h ) := Q h (Λs ) −→ E(1Λs M s E[h(Lt )] t→∞

(3.166)

(sh := h(Ls )Rs2α + 1 − H(Ls ), where : H(y) = y h()d (y ≥ 0). with M 0 In the particular case where, for x > 0 being ﬁxed, we choose : hx () =

2α Γ(α

1 1[0,2α Γ(α+1)xα ] () + 1)xα

(3.167)

we obtain : E[1Λs hx (Lt )] (s(x) ) := Q (x) (Λs ) −→ E(1Λs M E[hx (Lt )] t→∞

(3.168)

with : ((x) := M s

+ 1 1 2α 1[Ls ≤2α Γ(α+1)xα ] Rs + 1 − α Ls 2α Γ(α + 1)xα 2 Γ(α + 1)xα (3.169)

2) It is shown in ([PY6 ], Proposition 2.2, p. 289) that, a.s. : n1/α V (n) (t) −→

n→∞

1 Lt 2α Γ(α + 1)

1/α (3.170)

with this convergence taking place also in every Lp , p ≥ 1. This result motivated us to prove the following Theorem 3.13. Let x > 0 be ﬁxed. Then, the sequence of (P, Fs (s ≥ 0)) (n,x n−1/α ) martingales : (Ms , n ≥ 1) converges a.s., as n → ∞, uniformly on (sx . every compact set (in time s) towards M p This convergence also holds in every L (p ≥ 1). Consequently, for any s ≥ 0, and Λs ∈ Fs : Q(n,x n

−1/α

)

(x) (Λs ) (Λs ) −→ Q n→∞

(3.171)

182

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

We comment upon Theorem 3.13 by remarking that the diagram :

1(V (n) ≤ t

(n,

x 1 nα

Penalisation of−→ P Q −−−−−−−−−−−

)

x 1 nα

)

(t→∞)

⏐ ⏐ ⏐ ⏐n → ∞ 3

⏐ ⏐ ⏐ (n → ∞) ⏐ (a.s.) 3 1( α Lt ≤xα ) 2 Γ(α+1)

Penalisation of−→ P −−−−−−−−−−−

(x) Q

t→∞

commutes, which shows, in some sense, the continuity at n = ∞ of the nth order penalisations. Proof of Theorem 3.13 (n,x) , (see Theorem 3.5) one has : According to the deﬁnition of Ms Ms(n,x n

−1/α

)

%

= 1n α1 V (n) ≤x gs

& Rs 1 Rs2α (α) α + 1As ≤x n−1/α Φ

√ −1 n x − As 2α Γ(α + 1)n x n−1/α −1/α

Δx n + 1− s n

+

(3.172)

The ﬁrst term on the right hand-side of (3.172) converges a.s. as n → ∞ from (3.170) (and the fact that 0 ≤ |Φα − 1| ≤ 1) towards : 1(Ls ≤2α Γ(α+1)xα )

Rs2α 2α Γ(α + 1)xα

Theorem 3.13 will then be proven if one establishes that : + + −1/α Δxs n 1 1− −→ 1 − α Ls a.s. n→∞ n 2 Γ(α + 1)xα

(3.173)

Now, for all 0 ≤ a ≤ 1 : −1/α

(Δxs n

≤ [na]) =

Vs[na] ≥ [x n−1/α ] 1

1

= ([an] α Vs[na] > [xa α ]) (where [z] denotes the integer part of z) −→ 1(Ls ≥2α Γ(α+1)xα a) (from (3.170))

n→∞

(n)

3.7 Penalisations by (Vt

(n)

≤ x) and (Vdt ≤ x), for n ≥ 2

183

Hence : −1/α

Δxs n n

−→

n→∞

1 Ls 2α Γ(α + 1)xα

a.s. hence, (3.173) is satisﬁed.

We now end this section by showing that the probabilities Q(n,x) Fg constitute a “projective” family of probabilities. More precisely, we show : Proposition 3.14. For every m and n integers, with n ≥ m ≥ 1, one has : n 1V (m) ≤x Q(n,x) Fg Q(m,x) Fg = m g

(3.174)

Consequently : Q(n,x) (Vg(m) ≤ x) =

m n

and, on Fg

Q(n,x) ( • Vg(m) ≤ x) = Q(m,x) (•) (3.175)

Proof of Proposition 3.14 With the notation of Lemma 3.11, one has : ∞ 1 EQ(m,x) [Kg ] = E Ks 1(V (m) ≤x) dLs s m2α Γ(α + 1)xα 0 ∞ n 1 = E Ks 1(V (m) ≤x) 1(V (n) ≤x) dLs s s m n2α Γ(α + 1)xα 0 (since 1(V (m) ≤x) 1(V (n) ≤x) = 1(V (m) ≤x) s

=

s

n (n,x) EQ Kg 1(V (m) ≤x) g m

s

(n ≥ m) .

Both formulae (3.174) and (3.175) follow.

3.7 Penalisations by (Vt(n) ≤ x) and (Vd(n) ≤ x), for n ≥ 2 t We shall now show that both probabilities obtained by penalisations by (n) (n) (Vt ≤ x) and (Vdt ≤ x) are equal to the probability Q(n−1,x) deﬁned in Theorem 3.10.

184

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Theorem 3.15. We suppose (C) is satisﬁed. Let x > 0 be ﬁxed, and n ≥ 2. 1) For every s > 0, and Λs ∈ Fs , one has : E[1Λs 1(V (n) ≤x) ] t

(n) P (Vt (n−1,x)

where (Ms 2) Similarly :

≤ x)

−→ E(1Λs Ms(n−1,x) ) = Q(n−1,x) (Λs )

t→∞

(3.176)

, s ≥ 0) is the positive martingale deﬁned by (3.124).

E[1Λs 1(V (n) ≤x) ] dt

(n)

P (Vdt ≤ x)

−→ E(1Λs Ms(n−1,x) ) = Q(n−1,x) (Λs )

t→∞

(3.177)

Thus, the canonical process (Rt , t ≥ 0) under the probability deﬁned by the (n) (n) penalisation by (Vt ≤ x) (resp. (Vdt ≤ x)) is the Q(n−1,x) process studied in Section 3.5. We recall that the penalisation by 1(V (1) ≤x) and by 1(V (1) ≤x) has been studt

dt

ied in Section 3.4 of this Chapter. This leads (under (C)) to a very diﬀerent martingale than those we obtained here. This is due to the exponential (n) equivalent (see (3.34) and (3.35)), instead of polynomial, for P (Vt ≤ x) (n) and P (Vdt ≤ x) as t → ∞. Proof of Theorem 3.15 (n) (n) We give the proof only for (Vt ≤ x), since that of the case (Vdt ≤ x) is very similar. As for the proof of point 1) of Theorem 3.10, we write for k ≤ n − 2 : (n)

(Δxs = k)∩(Vt

≤ x) = {(As ≤ x)∩(D1 ∪D2 ∪D3 )}∪{(As > x)∩(D2 ∪D4 )} (n−k)

with : D1 = (T0 ◦ θs ≤ x − As ) ∩ (Vt−ds ◦ θds ≤ x) D2 = (T0 ◦ θs ≥ t − s) (n−k−1)

D3 = (x − As < T0 ◦ θs ≤ t − s) ∩ (Vt−ds

◦ θds ≤ x)

(n−k)

D4 = (T0 ◦ θs ≤ t − s) ∩ (Vt−ds ◦ θds ≤ x) and for k = n − 1 : (n)

(Δxs = n − 1) ∩ (Vt

≤ x) = 1(As ≤x) 1(T0 ◦θs >t−s) + 1(As >x) 1(T0 ◦θs >t−s) +1(As >x) 1(T0 ◦θs
t−ds ◦θds ≤x)

The computation of the limit, as t → ∞, of E[1Δxs =k 1(V (n) ≤x) |Fs ] t

(n)

P (Vt

≤ x)

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

185

may be obtained in the same manner as for point 1) of Theorem 3.10, the only change being the contribution of the terms of the form 1(V (p) ◦θ ≤x) , thus t−ds

ds

yielding, for p = 1 and under (C), an exponentially small contribution (from (n) (3.39)), which, consequently, vanishes when one divides by P (Vt ≤ x) ∼ x α (n − 1) t . Thus, we obtain : |Fs ] (n) (Vt ≤x) (n) P (Vt ≤x)

E[1

−→

t→∞

n−2 k=0

1As ≤x

Rs (n − 1 − k) Φ(α) √ −1 n−1 x − As

2α

Rs + (n−1)Γ(α+1)x α (n−k−2) (α) √ Rs + 1−Φ n−1 x−As (n−1−k) Rs2α +1As >x 2α Γ(α+1)(n−1) + n−1

+1Δxs =n−1 1As >x

Rs2α (n−1)2α Γ(α+1)xα

(n−1,x)

= Ms

Penalisations of a BES(d) Process, Involving Several Ranked Lengths of its Excursions 3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions of a BES(d) Process 3.8.1 In Section 3.5 of this Chapter, we studied the penalisation of the Bessel process (Rt , t ≥ 0) with the weight process (1(V (n) ≤x) , t ≥ 0). Our gt aim is now to study the penalisation of (Rt , t ≥ 0) by the weight process : 1(V (1) ≤x ,...,V (n) ≤x ) , for any given n, and x1 ≥ x2 ≥ ... ≥ xn , i.e : we shall 1 n gt gt study the limit, as t → ∞, of E (α) 1(V (1) ≤x ,...,V (n) ≤x ) 1Λs 1 n gt gt (3.178) (1) (n) P (α) (Vgt ≤ x1 , ..., Vgt ≤ xn ) The study is achieved in the next Section 3.9. For now, as a beginning of our study, we shall examine the denominator of (3.178), i.e : we shall describe the asymptotic behavior of : P (α) (Vg(1) ≤ x1 , ..., Vg(n) ≤ xn ) t t

as t → ∞

186

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

3.8.2 We ﬁx some notation. We denote by S ↓ the set S ↓ = {s = (s1 , s2 , ..., sn , ...); s1 ≥ s2 ... ≥ sn ≥ ... ≥ 0}

(3.179)

and for every integer n, we denote by Sn↓ the set Sn↓ = {s = (s1 , s2 , ..., sn ); s1 ≥ s2 ... ≥ sn ≥ 0}

(3.180)

For any integer i, W i denotes the ith coordinate on S ↓ , or on Sn↓ , for i ≤ n : W i ((s1 , s2 , ...)) = si

(3.181)

In the following : •

(ρ2 , ρ3 , ..., ρn , ...) denotes a sequence of independent r.v’s such that, for (law)

every n, ρn = U 1/n , where U is uniform on [ 0, 1 ], so that ρn follows •

the beta (n, 1) distribution; (3.182) (Ti , i ≥ 1) denotes the increasing sequence of jumps times of a standard Poisson process. In other terms, (T1 , T2 − T1 , ..., Tn − Tn−1 , ...) is an i.i.d sequence of standard exponential variables. (3.183)

Before stating the main result of this section, we draw the reader’s attention to the fact that we shall be working with a σ-ﬁnite ≥ 0 measure Π on S ↓ , and that we need to be careful about the notion of independence : in particular, we shall say that, under Π, X is distributed as a σ-ﬁnite (inﬁnite) measure ν(dx), and is independent from Y , which is “distributed” as a probability measure p(dy), if : Π(f (X)g(Y )) = ν(dx)f (x) p(dy)g(y) (*) for every f, g ≥ 0, Borel; but, note that, if ν(dx) = ∞,we cannot take f ≡ 1 in (*) in order to “deduce” to p(dy)g(y). that Π(g(Y )) is equal Indeed, on the “contrary”, if ν(dx) = ∞ and p(dy)g(y) > 0, then : Π(g(Y )) = ∞. On the other hand, the notion of conditioning with respect to X is meaningful, and we deduce from (*) that : ∀g ≥ 0, Borel, Π(g(Y )|X = x) = p(dy)g(y) which now may be taken as an equivalent form of independence between X and Y . Theorem 3.16. i)There exists a positive, σ-ﬁnite measure Π on S ↓ , which does not depend on α ∈ (0, 1), such that, for every F : S ↓ → R+ , bounded,

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

187

Borel, and with compact support in the ﬁrst variable : Π[F (W 1 , W 2 , ...)] := lim tα E (α) [F ((Vgit )α ; i = 1, 2, ...)] t→∞ ∞ T2 T2 T2 = E F x, x ,x , ..., x , ... dx T3 T4 Tn+1 0 ∞ = dx E[F (x, xρ2 , xρ2 ρ3 , ..., xρ2 ...ρn , ...)] (3.184) 0

with the notations of (3.182) and (3.183). ii) The measure Π enjoys the following properties : a) Under Π, W 1 is distributed as Lebesgue measure on R+ , and is independent from : 2 T2 W Wk T2 (law) , ..., 1 , ... = , ..., , ... (3.185) W1 W T3 Tk+1 b) More generally, under Π, for any n, W n is distributed as n times Lebesgue’s measure on R+ and is independent of : Tn+1 W n+k Tn+1 W n+1 (law) , ..., , ...) = , ..., , ... ( Wn Wn Tn+2 Tn+k+1 (law)

= (ρn+1 , ρn+1 ρn+2 , ..., ρn+1 ...ρn+k , ...) (3.186)

c) Under Π, the density of (W 1 , ..., W n ) is : fn (s1 , ..., sn ) =

(n!)(sn )n−1 1(0≤sn ≤...≤s1 ) (s1 s2 ...sn−1 )2

(3.187)

d) Shifting the sequence (W k , k ≥ 1) into (W n+k , k ≥ 1), for any given n ≥ 1, has the following eﬀect on Π : & % p+1 n−1 n+p W n n+p 1 p+1 )] = Π h(W , ..., W ) Π[h(W , ..., W p+1 W1 (3.188) for any h : Rp+1 → R+ , Borel. + Proof of Theorem 3.16 We shall give two proofs for this theorem. The ﬁrst one is “self-contained”, whereas the second one hinges on a result of Pitman-Yor ([PY5 ],[PY6 ]) which we shall recall, when needed.

188

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

First proof : 1) To prove the existence of Π and the form (3.187) of the density of Π, it suﬃces to see that, for every sequence x1 ≥ x2 ≥ ... ≥ xn ≥ 0, one has : α t(α) P (α) [(Vg(1) ) ≤ x1 , ..., (Vg(n) ) ≤ xn ] −→ Pn (xα 1 , ..., xn ) t t t→∞

(3.189)

where Pn : Rn+ → R+ is a sequence of functions, which do not depend on α, and such that : ∂ n Pn (x1 , ..., xn ) n! xn−1 = 2 n2 (3.190) ∂x1 ...∂xn x1 ...xn−1 Indeed, from (3.189), we have : P (α) [Vg(1) ≤ x1 , ..., Vg(n) ≤ xn ] t t (n) α α )α ≤ xα = P (α) [(Vg(1) 1 , ..., (Vgt ) ≤ xn ] ∼ t

1 α Pn (xα 1 , ..., xn ) tα

Thus, for every sequence y1 ≥ y2 ≥ ... ≥ yn ≥ 0 : tα P (α) [(Vg(1) )α ≤ y1 , ..., (Vg(n) )α ≤ yn ] → Pn (y1 , ..., yn ) t t Hence, for every function F : Rn+ → R+ , Borel, with compact support in the ﬁrst variable : tα E (α) [F ((Vg(i) )α ; i = 1, 2, ..., n)] t ∂ n Pn (s1 , ..., sn ) F (s1 , ..., sn ) ds1 ...dsn −→ t→∞ S ↓ ∂s1 ...∂sn n = F (s1 , ..., sn ) dΠ(s1 , ..., sn ) (from (3.190)) ↓ Sn

2) Let us prove that (3.187) implies that, under W 1 is distributed as Π, n W2 Lebesgue measure on R+ , is independent from W 1 , ..., W , ... and let us W1 prove (3.184) According to (3.187), and for every function F : Rn+ → R+ , Borel, with compact support in the ﬁrst variable : 2 n! sn−1 Wn s2 sn n 1 W Π F W , 1 , ..., 1 F s1 , , ..., ds1 ...dsn = ↓ W W s1 s1 (s1 ...sn−1 )2 Sn After the change of variables : s1 = u1 , s2 = s1 u2 , ..., sn = s1 un : 2 Wn 1 W Π F W , 1 , ..., 1 W W 1 un−1 ∞ n! unn−1 du1 ... F (u1 , ..., un ) du2 ...dun = (u2 ...un−1 )2 0 0 0

(3.191)

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

189

2 Wn hence the independence of W 1 and W , ..., , ... and the fact that W 1 , W1 W1 under Π, is distributed as Lebesgue measure on R+ . Now, let us write (3.191) for a function F which does not depend on the ﬁrst variable, and a test function f : R+ → R+ , for which we denote : ∞ f = 0 f (w)dw : 2 W Wn Π f (W 1 )F , ..., W1 W1 1 un−1 ... F (u2 , ..., un ) = f n! 0

0

unn−1 2 (u2 ...un−1 )2

du2 ...dun

After the change of variables : u2 = v2 , u3 = v2 v3 , ..., un = v2 v3 ...vn , we obtain : 2 W Wn 1 , ..., 1 Π f (W )F W1 W 1 1 (v2 v3 ...vn )n−1 ... F (v2 , v2 v3 , ..., v2 v3 ...vn ) 2 = f n! v2 (v2 v3 )2 ...(v2 v3 ...vn )2 0 0

1

= f n!

v2n−2 v3n−3 ...vn−1 dv2 ...dvn

1

F (v2 , v2 v3 , ..., v2 v3 ...vn ) v2 v32 ...vnn−1 dv2 ...dvn

... 0

0

= f E[F (ρ2 , ρ2 ρ3 , ..., ρ2 ...ρn )] T2 T2 T2 = fE F , , ..., , ... T3 T4 Tn+1

(3.192) (3.193)

according to the classical properties of the beta-gamma algebra. 2 Wn , ..., , ... we From (3.192), (3.193) and the independence of W 1 and W 1 1 W W easily deduce (3.184). Formulae (3.186) and (3.188) can also be easily deduced from (3.187) with the help of closely related arguments. In other words, it suﬃces to prove (3.189) and (3.190) to achieve the proof of Theorem 3.16. (n) 3) Let, for every x > 0, Hx denote the nth -time at which the age process attains level x : Hx(n) := dH (n−1) + TxA ◦ θd

(n−1) Hx

x

Hx(1) := TxA = inf{t ≥ 0; At ≥ x}

(n ≥ 2)

(3.194) (3.195)

For every sequence (x) = (x1 , ..., xn ), with x1 ≥ x2 ≥ ... ≥ xn ≥ 0, we deﬁne : (n)

H (x) :=

inf

i=1,2,...,n

Hx(i)i

(3.196)

190

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Lemma 3.17. For every sequence (x) = (x1 , ..., xn ), with x1 ≥ x2 ≥ ... ≥ xn ≥ 0, one has : ) ≤ x1 , ..., (Vg(n) ) ≤ xn ] −→ t(α) P (α) [(Vg(1) t t

t→∞

1 E (α) (LH (n) ) (x) 2α Γ(1 + α)

(3.197)

Proof of Lemma 3.17 We proceed in a very similar manner to that of the proof of point 1) of Theorem 3.4. Indeed, it is clear that, for every t ≥ 0 : ≤ xk ) = (gt ≤ Hx(k) ) (Vg(k) t k hence : ≤ x1 , ..., Vg(n) ≤ xn ) = (Vg(1) t t

n 4

(n)

(gt ≤ Hx(k) ) = (gt ≤ H (x) ) k

(3.198)

k=1 (n)

from the deﬁnition of H (x) . Let now Sβ denote an exponential variable with parameter β, independent from (Rt , t ≥ 0). Replacing t by Sβ in (3.198), we obtain : (n)

P (α) (Vg1

Sβ

≤ x1 , ..., Vg(n) ≤ xn ) = P (α) (gSβ ≤ H (x) ) Sβ

= P (α) (Sβ ≤ dH (n) ) = 1 − E (α) (exp(−βdH (n) )) (x)

(x)

(from the deﬁnition of Sβ ) (n)

= 1 − E (α) [exp −β(H (x) + T0 ◦ ΘH (n) )] (x)

1−E

(α)

(n) ) (α) (exp(−β T)0 ))] [exp −βH (x) ]E (α) [E R (n) H

(3.199)

(x)

(from the Markov property and (3.20)) But, from (3.7) and (3.22) : (n)

(n)

E (α) [exp −βH (x) ] = 1 − βE (α) (H (x) ) + o(β) β→0

(3.200)

) (α) (exp −β T)0 )] = 1 − Γ(1 − α) β α E (α) (R2α(n) ) + o(β) (3.201) E (α) [E R (n) H (x) β→0 2α Γ(1 + α) H (x) Plugging (3.200) and (3.201) in (3.199), we obtain : P (α) (Vg1

Sβ

Γ(1 − α) α (α) 2α β E (RH (n) ) 2α Γ(1 + α) (x) Γ(1 − α) α (α) ∼ β E (LH (n) ) (3.202) β→0 2α Γ(1 + α) (x)

≤ x1 , ..., Vg(n) ≤ xn ) ∼ Sβ

β→0

(n)

since (Rt2α − Lt , t ≥ 0) is a martingale and, since H (x) ≤ TxA1 , we may ap(n)

ply Doob’s optional stopping Theorem to this martingale at the time H (x) .

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

191

Finally, since : P (α) (Vg1 ≤ x1 , ..., Vg(n) ≤ xn ) Sβ Sβ ∞ exp(−βt)P (α) [(Vg(1) ) ≤ x1 , ..., (Vg(n) ) ≤ xn ]dt =β t t 0

one has, from (3.202) : ∞ exp(−βt)P (α) [(Vg(1) ) ≤ x1 , ..., (Vg(n) ) ≤ xn ]dt t t 0

∼

β→0

Γ(1 − α) α−1 (α) β E (LH (n) ) (x) 2α Γ(1 + α)

(3.203)

and (3.197) is then a consequence of (3.203) and of the Tauberian Theorem. 4) As we already observed, in order to prove Theorem 3.16, it suﬃces to show (3.189) and (3.190). However, these two relations are immediate consequences of the following : Proposition 3.18. Let n ≥ 1, and y1 ≥ y2 ≥ ... ≥ yn ≥ 0, a decreasing sequence of n real numbers. Then (all functions introduced below do not depend on α) : 1) There exist functions Pn of n variables such that : P (α) [(Vg1t )α ≤ y1 , ..., (Vgnt )α ≤ yn ] ∼

t→∞

1 Pn (y1 , ..., yn ) tα

(3.204)

2) For any p ≥ 2, there exists a polynomial Qp in p variables, which is homogeneous of degree (p − 2), such that : Pn (y1 , ..., yn ) = yn +

n p=2

ynp

1 1 − yn yp−1

Qp

1 1 1 1 , , ..., , y1 y2 yp−1 yn

(3.205) In particular, Q2 ≡ 1, Q3 (x1 , x2 , x3 ) = x3 + x2 − 2x1 , Q4 (x1 , x2 , x3 , x4 ) = x23 + x3 x4 + x24 − 3x3 x1 − 3x22 − 3x1 x4 + 6x1 x2 . 3) The polynomials (Qn , n ≥ 2) satisfy the following recurrence relation Qn (x1 , ..., xn )

=

n−2 n−1 p=1

p

(xn − xn−1 )p−1 (xn−1 − xn−p−1 )

Qn−p (x1 , x2 , ..., xn−p−2 , xn−p−1 , xn−1 ) + (xn − xn−1 )n−2 4) The coeﬃcient of x1 x2 ...xn−2 in Qn (x1 , ..., xn ) is equal to : (−1)n (n − 1)!

(3.206)

192

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

5) The nth mixed partial derivative of Pn , with respect to all variables is : ∂ n Pn n! ynn−1 (y1 , ..., yn ) = ∂y1 ...∂yn (y1 y2 ...yn−1 )2

(3.207)

for yn < yn−1 < ... < y1 . Proof of Proposition 3.18 5) We ﬁrst prove point 1) of Proposition 3.18. 5a) We ﬁrst recall that (τ , ≥ 0), the right-continuous inverse of the local time (Lt , t ≥ 0) is a subordinator, whose L´evy measure ν α (dt) equals : ν α (dt) =

dt 2α Γ(α)tα+1

(3.208)

Let, for u ≥ 0, Δτu := τu − τu− , and deﬁne, for every Borel set A(⊂ R+ ) the Poisson process : NA := 1(Δτu ∈A) ( ≥ 0) (3.209) u≤

It is quite well-known that :

dt • for every A ∈ BR+ , such that : ν α (A) ≡ A 2α Γ(α)t α+1 < ∞, and every A , N is Poissonian with parameter ν α (A). In particular, if A = [a, b], then : 1 1 − α ν α ([a, b]) = α (3.210) 2 Γ(1 + α) aα b

• if A1 , ..., An are n disjoint Borel sets, with ν α (Ai ) < ∞, then the Poisson processes (NAi )≥0 ; i = 1, 2, ..., n are independent. (3.211) 5b) One has : E (α) (LH (n) ) = (x)

∞

0 ∞

=

0

∞

= 0

P (α) (LH (n) ≥ )d (x)

(n)

P (α) (H (x) ≥ τ )d P (α) (Vτ(i) ≤ xi ; i = 1, ..., n)d

(3.212)

from (3.198). Now, since the set of jumps of the subordinator (τ , ≥ 0) labels exactly the set of the excursions of (Rt , t ≥ 0) away from 0, we get : [x1 ,∞[

(Vτ(i) ≤ xi ; i = 1, 2, ..., n) = (N

[xn ,xn−1 [

...N

[x2 ,x1 [

= 0, N

[x2 ,x1 [

+ ...N

[x3 ,x2 [

≤ 1, N

[x2 ,x1 [

+ N

≤2

≤ n − 1) (3.213)

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

[xi+1 ,x1 [

Since, from (3.210) and (3.211), the r.v’s N variables with respective parameters :

2α Γ(1 + α)

193

are independent Poisson 1 1 − α (according xα xi i+1

to (3.210) and (3.211)), we get : P (Vτ(i) ≤ xi ; i = 1, 2, ..., n) ki n−1 5 1 1 1 = exp − α α − α α α Γ(1 + α) xn 2 Γ(1 + α) 2 x x k i! i+1 i i=0 K

(3.214) ⎞ k0 , ..., kn−1 where K = ⎝ k0 + ... + ki ≤ i ⎠ (and with the convention x1α = 0). 0 ∀i = 0, 1, ..., n − 1 5c) Plugging then (3.214) in (3.212) and using (3.197), we obtain : ⎛

lim t(α) P (α) [Vg(1) ≤ x1 , ..., Vg(n) ≤ xn ] t t

t→∞

1 E (α) (LH (n) ) (x) 2α Γ(1 + α) ∞ 1 = α P (α) (Vτ(i) ≤ xi ; i = 1, 2, ..., n)d (from (3.212)) 2 Γ(1 + α) 0 ∞ 1 = exp − α α × ... 2α Γ(1 + α) 0 xn 2 Γ(1 + α) K n−1 ki 5 1 1 1 − α ... α α Γ(1 + α) 2 x x k i! i+1 i i=0 =

d

(from (3.214)) =u: which yields, after the change of variables : α α xn 2 Γ(1 + α) ki n−1 ∞ 5 1 1 1 α α = du (3.215) xn exp(−u) − α xn u α x x k i! 0 i+1 i i=0 K n−1 ki 5 α(1+' n−1 1 1 (k0 + ... + kn−1 )! i=0 ki ) (3.216) = xn − α α x x k0 !...kn−1 ! i+1 i i=0 K

Thus, replacing in (3.216) xα i by yi (i = 1, ..., n), we get : lim t(α) P (α) [(Vg(1) )α ≤ y1 , ..., (Vg(n) )α ≤ yn ] t t n−1 k (1+' n−1 5 1 1 i (k0 + ... + kn−1 )! i=0 ki ) = yn − yi+1 yi k0 !...kn−1 ! i=0 t→∞

K

:= Pn (y1 , y2 , ..., yn )

(3.217)

194

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

We note that, when writing (3.216), we made the convention x0 = +∞, ie : 1 xα = 0. We also note that the functions Pn satisfy : 0

Pn (y1 , y2 , ..., yn−1 , yn−1 ) = Pn−1 (y1 , y2 , ..., yn−1 )

(3.218)

6) Point 2) of Proposition 3.18 follows from (3.217) by rearranging the order of the terms. For example : 'n−1 • the coeﬃcient of yn in (3.217) is obtained for i=0 ki = 0; this coeﬃcient is then equal to : 0 n−1 5 1 0! 1 =1 − yi+1 yi 0! i=0 • the coeﬃcient of yn2 in (3.217) is cient is then equal to : 1 1 1 − + yn yn−1 yn−1 1 1 1 = − − = yn y1 yn

obtained for

−

1

'n−1

i=0

ki = 1; this coeﬃ1 1 − y2 y1

+ ... + yn−2 1 Q2 since Q2 ≡ 1 y1

• the coeﬃcient of yn3 in (3.217) is obtained for

n−1

ki = 2; this coeﬃcient

i=0

is then equal to :

1 1 1 1 1 1 − + − + ... + − yn yn−1 yn−1 yn−2 y2 y1 2 2 1 1 1 1 = − − − yn y1 y2 y1 1 1 1 1 1 1 2 = − + − + − yn y1 y1 y2 yn y2 y1

2 −

1 1 − y2 y1

2

7) We now prove (3.206) For this purpose, for every integer n, we shall say that a sequence of integers (k0 , ..., kn−1 ) is admissible if : i

kj ≤ i

for every i = 0, ..., n − 1

(3.219)

j=0

We denote by An the set of admissible sequences (of length n). We note, for every sequence (k0 , ..., kn−1 ) = (k) : deg(k) =

n−1 i=0

ki

(3.220)

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

195

It is clear that, if (k) ∈ An , then : deg(k) ≤ (n − 1). With this notation, formula (3.217) writes : n−1

Pn (y1 , y2 , ..., yn ) =

ν k0 ν k1 ...ν kn−1 0 1 n−1 k0 !k1 !...kn−1 !

ynp+1 p!

p=0

(3.221)

(k)∈An deg(k)=p

with νi :=

1 yi+1

−

1 yi

(i = 0, ..., n − 1).

We shall now prove point 3) of the Proposition 3.18, with the help of a recurrence argument bearing upon n. For this purpose, we observe that, if (k) ∈ An , and deg(k) = n − 1, and if kn−1 = p, then (k0 , k1 , ..., kn−2 ) is an admissible sequence (of length (n − 2)) and of degree (n − 1 − p). Conversely, if (k0 , k1 , ..., kn−2 ) is an admissible sequence (of length (n − 2)) and of degree (n − 1 − p), then the sequence (k0 , k1 , ..., kn−2 , p) is an admissible sequence (of length (n − 1)). Thus, the term in ynn in (3.221) equals : ynn (n − 1)!

(k)∈An deg(k)=n−1

= ynn (n − 1)!

kn−1 ν0k0 ...νn−1 k0 !...kn−1 !

n−1 p ν

n−1

p=1

p!

(= ynn νn−1 Qn (y1 , ..., yn ))

(k0 ,k1 ,...,kn−2 )∈An−1

kn−2 ν0k0 ...νn−2 k0 !...kn−2 !

deg(k0 ,...,kn−2 )=n−1−p

= ynn νn−1

n−1 p=1

p−1 νn−1 (n

− 1)!

p!

(k0 ,k1 ,...,kn−2 )∈An−1

kn−2 ν0k0 ...νn−2 k0 !...kn−2 !

deg(k0 ,...,kn−2 )=n−1−p

Now, the term which appears above : (k0 ,k1 ,...,kn−2 )∈An−1

kn−2 ν0k0 ...νn−2 k0 !...kn−2 !

deg(k0 ,...,kn−2 )=n−1−p

is equal to : 1 × [the coeﬃcient of ynn−p in Pn−1 (y1 , ..., yn−1 )] (n − 1 − p)! from (3.221). This result ends the recurrence argument, thanks to (3.205) which deﬁnes the Qr ’s. 8) Let us show that in Qn (x1 , x2 , ..., xn ) the coeﬃcient of the term x1 x2 ...xn−2 equals (−1)n (n − 1)!

196

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Indeed, in (3.206), the term x1 x2 ...xn−2 can only be obtained from the summation index p = 1. Let λn be the coeﬃcient of x1 x2 ...xn−2 in Qn (x1 , x2 , ..., xn ). Then, we get : n−1 × [– the coeﬃcient of x1 x2 ...xn−3 in Qn−1 (x1 , ..., xn−1 )] λn = 1 (because, in (3.206), the coeﬃcient before Qn−1 (x1 , x2 , x3 , ..., xn−1 ) is (xn−1 − xn−2 ), and in the computation of this coeﬃcient, the term xn−1 has no incidence). Thus, by recurrence on n, λn = (−1)n (n − 1)!

(3.222)

9) We now end the proof of Theorem 3.16 by showing that : ∂ n Pn (y1 , ..., yn ) n! y n−1 = 2 n 2 1yn
1 yn−1

1 (−1)n (n − 1)! y1 ...yn−2

Thus : (−1)n−1 (n − 1)!ynn ∂ n Pn (y1 , ..., yn ) ∂n = ∂y1 ...∂yn ∂y1 ...∂yn y1 y2 ...yn−2 yn−1 n−1 n! y = 2 2 n 2 1y1 ≥y2 ≥...≥yn y1 y2 ...yn−1 This ends the ﬁrst proof of Theorem 3.16. We shall now give a second proof of Theorem 3.16 which is, this time, essentially obtained from the description made by Pitman and Yor on the structure of excursions of a Bessel process of dimension d = 2(1 − α) (0 < α < 1) in terms of Poisson - Dirichlet distributions Pα,0 and Pα,α . 10) We gather in the following Theorem 3.19, the results of Pitman and Yor ([PY1 ],[PY2 ],[PY3 ]) which we are going to use. By scaling, the law of : 1 (1) (2) (V , Vt , ...) (3.223) t t

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

197

does not depend on t, it has some remarkable properties, and is found naturally in a number of probabilistic studies. Its distribution, the PoissonDirichlet distribution with parameter (α, 0), is denoted by Pα,0 in ([PY2 ]). Pα,0 is also (see ([PY1 ])) the distribution of : 1 (V (1) , ..., Vτ(n) , ...) τ τ

(3.224)

The identity in law between (3.223) and (3.224) was obtained in ([PY1 ]); see also ([PY3 ]) where other random times σ than τ are exhibited, such that the sequence (3.224) with σ instead of τ still has the same distribution. These results may be considered as a reinforcement of L´evy’s result for the time spent in R+ by a Brownian motion (Bt , t ≥ 0) : if A+ t :=

t

1(Bs >0) ds, then both 0

1 + 1 A and A+ are arc-sine distributed. t t τ τ

Likewise, if in (3.223), we replace t by gt := sup{s ≤ t : Rs = 0} then, the variable : 1 (V (1) , ..., Vg(n) , ...) t gt gt

(3.225)

is independent from gt , and is distributed as : (v (1) , v (2) , ..., v (n) , ...)

(3.226)

the sequence of ranked excursions of the standard BES(d) bridge; its distribution, the Poisson-Dirichlet distribution with parameter (α, α) is denoted by Pα,α in ([PY2 ]). Here is a description of Pα,0 and Pα,α on the canonical space S ↓ of decreasing sequences s = {s1 ≥ s2 ≥ ... ≥ sn ≥ ... ≥ 0}, where we denote W i (s) = si the sequence of coordinates. Theorem 3.19. ([PY2 ]) : Let {Ti = ε1 + ε2 + ... + εi , i = 1, 2, ...} denote the sequence of jump times of a standard Poisson process with parameter 1, i.e : the εi are i.i.d. standard exponential variables. Then : i) Under Pα,0 , the sequence (W i = (V (i) )α ; i = 1, 2, ...) is distributed as ⎛, ⎞ α -−1 ∞ −1/α ⎝ Ti Tm ; i = 1, 2, ...⎠ (3.227) m=1

198

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Consequently, L := limn→∞ nW n exists Pα,0 a.s. (compare with (3.170)), ' −α −1/α ∞ T . In other and is distributed (jointly with the W i ’s) as m m=1 terms : ∞ −α 1 Wi (law) −1/α ; i = 1, 2, ... = Tm , ; i = 1, 2, ... (3.228) L, L Ti m=1 ii) A closely related and useful description of Pα,0 is : (d) 2 3 n+1 1 T1 1 T1 1 T1 (W , W , ..., W , ...) = W ,W , ..., W , ... T2 T3 Tn+1 (d)

= (W 1 ρ1 , W 1 ρ1 ρ2 , ..., W 1 ρ1 ρ2 ...ρn , ...) (3.229)

n where ρn = TTn+1 , n ≥ 1 is a sequence of independent variables, and ρn is beta(n, 1) distributed. Note that, from (3.229), one has : (W 1 )1/α +

∞

(W 1 )1/α (ρ1 ...ρn )1/α = 1

(3.230)

n=1

so that W 1 is determined from the ρi ’s. iii) Pα,α is absolutely continuous with respect to Pα,0 , with : Pα,α = Cα L•Pα,0

(3.231)

where : Cα = B(1 + α, 1 − α) = Γ(1 + α)Γ(1 − α) =

πα sin(πα)

(3.232)

11) We show with the help of Theorem 3.19 that : α ) ; i = 1, 2, ...)] = lim tα E (α) [F ((Vg(i) t

t→∞

T2 T2 T2 E F x, x , x , ..., x , ... dx T3 T4 Tn+1 (3.233)

∞ 0

for every function F : Rn+ → R+ , Borel, with compact support in the ﬁrst variable. Evidently, as in points 1) and 2) of the ﬁrst proof of Theorem 3.16, proving (3.233) implies Theorem 3.16. Let us prove (3.233). Let us recall that W i = (V (i) )α ; i = 1, 2, .... We have :

=

tα E (α) [F ((Vg(i) )α ; i = 1, 2, ..., n)] t & % (i) α V gt α (α) α ; i = 1, 2, ..., n F gt t E gt

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

=

tα B(α, 1−α)

199

1

Eα,α (F (tα xα (v (i) )α ); i = 1, 2, ..., n)xα−1 (1−x)−α dx

0 (law)

=

=

(from (3.225), (3.226) and the fact that gt = tβα,1−α ) 1 tα Cα Eα,0 (LF (tα xα (V (i) )α ); i = 1, 2, ..., n)xα−1 (1−x)−α dx B(α, 1−α) 0 (from (3.231)) 1 α uEα,0 (LF (uxα (V (i) )α ); i = 1, 2, ..., n)xα−1 (1 − x)−α dx 0

(letting tα = u and since Cα = Γ(1 + α)Γ(1 − α) = αB(α, 1 − α)

=

from (3.232)) % u(V (1) )α dz L Eα,0

z α1 1 α (V (1) )α 0 1− u V (1) (2) α (n) α V V F z, z , ..., z V (1) V (1)

& (3.234)

(from Fubini and with the change of variables uxα (V (1) )α = z) (2) α (n) α ∞ V V L Eα,0 F z, z , ..., z dz (3.235) = (1) α (1) tα =u→∞ (V ) 0 V V (1) because F is bounded and with compact support in the ﬁrst variable. But, (2) α (n) α V V L from (3.228) the law under Pα,0 of , , ..., equals (V (1) )α V (1) V (1) T1 T1 T1 . Hence : the law of T1 , , , ..., T2 T3 Tn tα E (α) [F ((Vgit )α ; i = 1, 2, ..., n)] ∞ T1 T1 T1 −→ E T1 F z, z , ..., z , ..., z dz t→∞ T2 T3 Tn 0 ∞ T2 T2 T2 = E F z, z , ..., z , ..., z dz T3 T4 Tn+1 0

(3.236)

The last equality in (3.236) follows from : ∞ T2 T2 T2 F z, z , ..., z , ..., z dz E T3 T4 Tn+1 0 x1 + x2 x1 + x2 = F z, z ,z , ..., x1 + x2 + x3 x1 + x2 + x3 + x4 Rn+2 + z

x1 + x2 x1 + x2 + ... + xn+1

exp −(x1 + ... + xn+1 )dz dx1 ...dxn+1

200

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

= Rn+1 +

exp(−x3 − ... − xn+1 )dx1 dx3 ...dxn+1 dz × . . .

∞

y y y ,z , ..., z y + x3 y + x3 + x4 y + x3 + ... + xn+1 x1 (after the change of variables x1 + x2 = y) y y y = F z, z ,z , ..., z y + x3 y + x3 + x4 y + x3 + ... + xn+1 Rn+1 + y dx1 exp −(x3 − ... − xn+1 − y)dx3 ...dxn+1 dzdy ...

F

z, z

(exp −y)dy

0

(from Fubini) = yF z, z Rn+1 +

y y y ,z , ..., z y + x3 y + x3 + x4 y + x3 + ... + xn+1

exp −(y − x3 − ... − xn+1 )dzdydx3 ...dxn+1 ∞ T1 T1 T1 = E T1 F z, z , z , ..., z dz T2 T3 Tn 0

(3.237)

This ends the second proof of Theorem 3.16. (2)

(n+1)

12) Asymptotic behavior of the law of (Vt , ..., Vt ) as t → ∞ We now consider the asymptotic behavior of the laws of the n-uplet : (2)

(n+1)

(Vt , ..., Vt

)

(3.238) (1)

as t → ∞. Let us note that in (3.238), we omit the term Vt . The reason for (1) this is the fact that, under (C), the quantity P (α) (Vt ≤ x) is exponentially (n) small as t → ∞ (see Theorem 3.4 here after) while P (α) (Vt ≤ x) has a polynomial behavior as t → ∞ for n ≥ 2. The main result of this indent is Theorem 3.20 which, in some way, is the “vector” version of Theorem 3.15. Theorem 3.20. For every function F : Rn+ → R+ , bounded, and with compact support in the ﬁrst variable, we get : (i)

lim tα E (α) [F ((Vt )α ; i = 2, 3..., n + 1)] = Π[F (W 1 , W 2 , ..., W n )]

t→∞

(1)

(3.239)

(2)

In other terms, the sequence (Vt , Vt , ...), after being shifted of one index, (1) (2) has, in law, the same asymptotic behavior as (Vgt , Vgt , ...). Proof of Theorem 3.20 (1) (2) Let (Vt , Vt , ...), denote as usual, the decreasing ranked sequence of the excursions of the Bessel process (Ru , u ≤ t). Let us recall that the “end excursion interval” t − gt features in this list. We deﬁne the r.v. Nt with values in N by : (j)

{Nt = j} = {(t − gt ) = Vt },

j = 1, 2, ...

(3.240)

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

201

Then, Theorem 3.20 is an immediate consequence of the : Proposition 3.21. For every n ≥ 1, and every sequence x1 ≥ x2 ≥, ..., ≥ xn ≥ 0 : (2)

lim tα P (α) [Vt

1)

t→∞

(n+1)

≤ x1 , ..., Vt

≤ xn , Nt = 1]

= lim tα P (α) [Vg(1) ≤ x1 , ..., Vg(n) ≤ xn ] t t

(3.241)

t→∞

(2)

lim tα P (α) [Vt

2)

t→∞

(n+1)

≤ x1 , ..., Vt

≤ xn , Nt ≥ 2] = 0

(3.242)

12a) Proof of (3.241) We have : (2)

tα P (α) [Vt α

(α)

[Vg(1) t

(n+1)

≤ x1 , ..., Vt ≤

≤ xn , Nt = 1]

≤ xn , t − gt > Vg(1) ] t

x1 , ..., Vg(n) t

=

t P

=

α (n) α α (1) tα P (α) [tα (Vg(1) )α ≤ xα 1 , ..., t (Vg1 ) ≤ xn , 1 − g1 > Vg1 ] 1 (by scaling)

=

α (n) α (1) uP (α) [ug1α (v (1) )α ≤ xα ) ≤ xα 1 , ..., ug1 (v n , 1 − g1 > g1 v ] (from (3.226), by letting tα = u)

=

α (n) α (1) uPα,α [ug1α (v (1) )α ≤ xα ) ≤ xα 1 , ..., ug1 (v n , 1 − g1 > g1 v ] (where, in this expression, g1 is a βα,1−α r.v. independent from

=

(v (1) , ..., v (n) )) Cα uEα,0 L1 g

=

(from (3.231)) 1 1+V (1) uE αxα−1 (1 − x)−α L1(uxα (V (1) )α ≤xα ,...,uxα (V (n) )α ≤xα ) dx

1<

1 1+V (1)

1

(ug1α (V (1) )α ≤xα1 ,...,ug1α (V (n) )α ≤xαn )

1

0

n

(law)

=

(since g1 = βα,1−α and g1 is independent from (V (1) , ..., V (n) )) % u V (1) α dz L 1+V (1) α Eα,0

1 (1) α (V ) 0 1 − uz α V1(1) & 1

z<xα 1 ,z

V (2) V (1)

α

≤xα 2 ...z

V (n) V (1)

α

≤xα n

(after the change of variables uxα (V (1) )α = z) ∞ L dz (2) α (n) α −→ E 1 α,0 V V z<xα ≤xα ≤xα tα =u→∞ n (V (1) )α 0 1 ,z V (1) 2 ,...,z V (1) =

(3.243)

lim tα P (α) [Vg(1) ≤ x1 , ..., Vg(n) ≤ xn ] t t

t→∞

(from (3.236) in which we take F (a1 , ..., an ) = 1{a1 ≤x1 ,...,an ≤xn } )

202

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

12b) We now show (3.242) We have : (2)

≤ ≤

(n+1)

tα P (α) [Vt

≤ x1 , ..., Vt

(2) t P [Vt tα P (α) [Vg(2) t

≤ x1 , Nt ≥ 2]

α

(α)

(2)

(since Vt ⎛

≤ xn , Nt ≥ 2]

≤ x1 , (t − gt ) ≤ Vg(1) ] t ≥ Vg(2) ) t

⎜ L ≤ Eα,0⎝ (1) α (V )

⎞

u(V (1) )α

u

V (1) 1+V (1)

α

1−

dz

z α1

α 1

V (1)

u

z

1

V (2) V (1)

α

⎟ ⎠ ≤xα 1

(3.244)

(with : u = tα , from a similar computation to that made for the the preceding point) α V (2) ≤ xα → 0 since z 1 u→∞ V (1)

∩

z>u

V (1) 1 + V (1)

α =∅

for u large enough. We note that Theorem 3.20 does not depend on Conjecture (C). 13) Theorem 3.20 invites to estimate, as t → ∞, the diﬀerence : ≤ x1 , ..., Vg(n) ≤ xn ] Λt := |tα P (α) [Vg(1) t t (2)

−tα P (α) [Vt

(n+1)

≤ x1 , ..., Vt

≤ xn ]|

(3.245)

We obtain : Proposition 3.22. Under (C), for every β < α, Λt = o(t−β ) Proof of Proposition 3.22 We write :

(1)

(3.246)

(2)

Λ t ≤ Λ t + Λt

(3.247)

where : (1)

Λt = |tα P (α) [Vg(1) ≤ x1 , ..., Vg(n) ≤ xn ] t t (2)

−tα P (α) [Vt (2) Λt

α

=t P

(α)

(2) [Vt (1)

(n+1)

≤ x1 , ..., Vt

≤

(n+1) x1 , ..., Vt

≤ xn , Nt = 1]|

≤ xn , Nt ≥ 2]

(2)

and we study successively Λt and Λt .

3.8 Asymptotics for the Distribution of Several Ranked Lengths of Excursions

203

(1)

13a) A majorant of Λt From (3.243) and (3.234), we obtain, with tα = u : (1)

≤ x1 , ..., Vg(n) ≤ xn ] Λt = tα P (α) [Vg(1) t t (2)

(n+1)

−tα P (α) [Vt ⎛ ⎜ = Eα,0⎝

L (V (1) )α

⎛ ⎜ ≤ Eα,0⎝L ⎡

≤ x1 , ..., Vt

u(V (1) )α

V (1) 1+V (1)

u

u(V (1) )α u

V (1) 1+V (1)

α

1−

1(z≤xα1 )

dz

z α1

V (1) 1+V (1)

α 1

1 V (1)

u

⎞ z≤xα 1 ,...,z

α/q 1

V (1) 1+V (1)

tα

≤

x1 t

V (n) V (1)

α

≤xα n

⎞

u(V

(1) α

0

)

⎞1/p⎤ dz ⎟ ⎥ ⎥

z α1 pα ⎠ ⎦ V (1) − u

1 1 (from Hlder’s inequality, with + = 1) p q ≤ Eα,0 Lx1

dz ⎟

z α1 α ⎠ (1) V − u

⎛ 1/q ⎜ α xα ⎝ ≤ 1

⎢ α ≤ Eα,0⎢ ⎣L x1 1

α

≤ xn , Nt = 1]

•(B(α, 1

− pα))1/p u1/p

1 (V

(1) )(

p−1 p

)α

because, with pα < 1,

u(V (1) )α

dz 1

z α1 pα = B(α, 1 − pα)u (V (1) )(p−1)α V − u L α/q 1/p α/p ≤ x1 (B(α, 1 − pα)) t Eα,0 1 V (1) ≤ x1 p−1 α ( ) (1) t p 1+V (1) (V )

0

(1)

% ≤ c(α, p, x1 )t

α/p

E

L (V (1) )α

2(p−1) p •

2/p

L

&1/2 1/2 V (1) x1 P < t 1 + V (1)

1/2 V (1) x1 ≤ c (α, p, x1 )t P < t 1 + V (1) L (law) (because L and = γ1 have moments of all orders) (V (1) )α x1 (1) ≤ c (α, p, x1 )P (α) Vt ≤ (by scaling) 1 − xt1

α/p

≤ c (α, p, x1 )P (α) (Vt

(1)

≤ 2x1 )1/2 ≤ c exp(−at)

(for some a > 0)

⎟

⎠

204

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

for t big enough, and from (C). Thus : (1)

Λt = O(exp(−at))

(for some a > 0)

(3.248)

(2)

13b) A majorant of Λt Formula (3.244) leads to : (2)

(2)

(n+1)

Λt = tα P (α) [Vt ≤ x1 , ..., Vt ≤ xn , Nt ≥ 2] ⎛ u(V (1) )α L dz ⎜ ≤ Eα,0 ⎝ (1) α 1 (2) α

z α1 (V (1) )α z VV (1) ≤xα1 u V (1) 1− u

1+V

1−ε V (1) x1 V (1) ≤ ≤ c (ε, α, p, x1 )tα/p Pα,0 1 + V (1) tV (2)

⎞ 1

⎟ α ⎠

V (1)

(3.249)

(for some ε > 0, by taking up again some computations which are quite similar to those made in the preceding point) ≤ c (ε, α, p, x1 )tα/p [Pα,0 (tV (2) ≤ x1 (1 + V (1) ))]1−ε

(3.250)

Now, for every δ > 0, we have : Pα,0 [tV (2) ≤ x1 (1 + V (1) )]

= Pα,0 [tV (2) ≤ x1 (1 + V (1) ), V (1) > δ ] + Pα,0 [tV (2) ≤ x1 (1 + V (1) ), V (1) < δ ] % & V (2) 1 (2) ≤ Pα,0 [tV ≤ x1 (1 + δ )] + Pα,0 t (1) ≤ x1 1 + % ≤ P ∼

t→∞

(α)

(2) [Vt ≤ x1 (1

+ δ )] + P

δ

V

T1 T2

1/α

x1 ≤ t

1+

1

&

(3.251)

δ

c tα

(3.252)

with the help of (C) for the ﬁrst term in (3.251) and by a direct computation for the second one. Proposition 3.22 now follows by the gathering of (3.247), (3.249), (3.250) and (3.252). (2)

(n+1)

) as t → ∞ 14) Asymptotic estimate of the distribution of (Vdt , ..., Vdt We now consider the asymptotic behavior of the laws of the n-tuple : (2) (n+1) ) as t → ∞. Here again, and for the same reasons as in sub(Vdt , ..., Vdt section 3.8.12 (see also Theorem 3.4 and (C) above), we have omitted the (1) (2) (n+1) ). term Vdt in (Vdt , ..., Vdt Theorem 3.20 admits the following companion.

(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

205

Theorem 3.23. For every function F : Rn+ → R+ , bounded, and with compact support in the ﬁrst variable, we get : (2)

(n+1) α

lim tα E (α) [F ((Vdt )α , ..., (Vdt

t→∞

) )] → Π[F (W 1 , ..., W n )]

(3.253)

The proof is quite similar to that of Theorem 3.20; hence, we omit it.

(n) 3.9 Penalisation by (Vg(1) ≤ x1 , ..., Vgt ≤ xn) t

Before presenting the main result of this section, which is the penalisation (1) (n) Theorem by (Vgt ≤ x1 , ..., Vgt ≤ xn ), we need to introduce a number of deﬁnitions and remarks. 1) Let n ≥ 2 a ﬁxed integer and (x) = (x0 > x1 > x2 > ... > xn > xn+1 ) a ﬁxed sequence such that : x0 = ∞ and xn+1 = 0. We denote Ik = [xk+1 , xk [

for every

k = 0, 1, ..., n .

→ − Let = (1 ≥ 2 ≥ ... ≥ p ≥ p+1 ...) denote a decreasing sequence of positive reals such that ∞ = 0. We deﬁne the counting variables : → − nk ( ) = {j ≥ 1, xk+1 ≤ j < xk } = {j ≥ 1; j ∈ Ik }

(3.254)

where k = 0, 1, 2, ..., n − 1. → − We say that a sequence belongs to S(x1 ,...,xn ) if : 1 ≤ x1 , ..., n ≤ xn

(3.255)

→ − We say that a sequence is (x1 , ..., xn )-admissible, or, more simply, admissible if : → − → − → − → − n0 ( ) = 0, n1 ( ) ≤ 1, n1 ( ) + n2 ( ) ≤ 2, → − → − ...n1 () + n2 ( ) + ... + nn−1 ( ) ≤ n − 1

(3.256)

Clearly, the following equivalence holds : → − ( ∈ S(x1 ,...,xn ) ) ⇐⇒ ( is admissible)

(3.257)

→ − A sequence is said to be maximal if → − → − → − → − n0 ( ) = 0, n1 ( ) = 1, n2 ( ) = 1, ..., nn−1 ( ) = 1

(3.258)

206

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

→ − A sequence is said to be saturated if it is admissible and if n−1

→ − ni ( ) = n − 1

(3.259)

i=0

→ − 2) Deﬁnition of sp → − → − To any admissible sequence , we associate a new sequence sp( ) (the → − sequence spread towards the top) by using the following algorithm : → − → − → − • if for every k ∈ 2, n − 1, nk ( ) ≤ 1, then : sp( ) = → − • if for some k ∈ 2, n − 1, nk ( ) ≥ 2, we modify the value of one j ∈ Ik which then becomes )j - by increasing the value of j such that )j ∈ Ik−1 . The new sequence thus deﬁned is still admissible. Then, we iterate this operation as many times as necessary so that all the → − + nk ( ) become smaller than or equal to 1. Thus, we obtain a sequence, de→ − noted as sp( ), after a ﬁnite number of operations, and it is clear that : → − → − n0 (sp( )) = 0, nk (sp( )) ≤ 1 for every k = 1, ..., n − 1, and → − sp( ) is an admissible sequence. Here are some examples which illustrate this deﬁnition.

I1 I2

In−1

x1 x2

− → ××

x1 x2 →

xn−1 ×× xn × × ×× 0

× ×

→ − sp( ) × ×

x1 x2 →

×× xn × × ×× 0

× × xn × × ×× 0

Example 1

I1 I2

x1 x2 x3

− → ×× ×

x1 x2 x3 →

×

× In−1

xn−1 xn 0

××× ××

× × ×

xn 0

××× ×× Example 2

x1 x2 x3

xn 0

→ − sp( ) × × × × × × × ××

(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

I1 I2

x1 x2 x3

− → × ××

In−1

xn 0

×× × ×××

x1 x2 x3 →

xn 0

207

× × ×

→

×× × ×××

x1 x2 x3 x4 x5 x6

xn 0

→ − sp( ) × × × × × × ×××

Example 3 In these examples, the number of crosses per line gives the number of indices → − j such that j belongs to the corresponding interval. Clearly, if is saturated → − (and admissible), then sp( ) is maximal. This is the case in Example 2, with n = 8. → − 3) Deﬁnition of the set of vacant indices for an admissible sequence → − → − Let be an admissible sequence. We deﬁne λ( ), the set of vacant indices → − of , by : → − → − (3.260) λ( ) = {j ∈ 1, ..., n; nj (sp( )) = 0} ∪ {n} → − If Card(λ( )) = p, we note : → − → − → − λ( ) = (λ1 ( ), ..., λp ( )) ∈ 1, ..., n

(3.261)

→ − → − → − → − where λ1 ( ) < λ2 ( ) < ... < λp ( ). In particular, λ1 ( ) is the smallest → − vacant indice in the sequence . → − • In Example 1, with n = 6, λ( ) = (3, 6) → − • In Example 2, with n = 8, λ( ) = (8) → − • In Example 3, with n = 9, λ( ) = (2, 5, 9) → − → − If the sequence is saturated (and admissible), thus λ( ) = {n}, since → − sp( ) is maximal. 4) Mixing two sequences → − → − Let and k denote two decreasing sequences (with ∞ = k∞ = 0). The → − − → → − sequence ∨ k is the sequence obtained from all elements of and all the → − elements of k , ranked in decreasing order. The following proposition follows clearly from the deﬁnitions.

208

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

→ − → − Proposition 3.24. Let and λ( ) be deﬁned as above. We assume that → − → − → − is admissible, i.e : ∈ S(x1 ,...,xn ) . Let k denote a second sequence. Then, → − − → the sequence ∨ k is admissible (i.e : it belongs to S(x1 ,...,xn ) ) if, and only → − → − − → → − → − → − if, denoting p = Card(λ( )), and λ( ) = (λ1 ( ), ..., λp ( )), k is a λ( )→ − admissible sequence (i.e : k ∈ S(xλ (−→ ) ,...,xλ (−→ ) ) ), that is, in other terms, if p 1 and only if : → , ..., kp ≤ x → − (3.262) k1 ≤ xλ (− ) λ () 1

p

→ − We shall use the following notation : if a is a positive real, and if is a → − decreasing sequence of reals, ∨ (a) is the sequence which is obtained by → − ranking in decreasing order the sequence made of the elements of and of a. 5) We now come back to the excursions of the Bessel process (Rt , t ≥ 0) −→ We denote, as usual, by Vgt , the sequence : −→ Vgt = (Vg(1) , Vg(2) , ..., Vg(n) , ...) t t t We note : λ(s) = (λ1 (s), ..., λp (s)) (3.263) −→ the sequence of vacant indices in the sequence Vgs . If a ∈ Ij (j ≥ 1), we note λ(s, j) = (λ1 (s, j), ..., λk (s, j)) the sequence of −→ vacant indices in the sequence Vgs ∨ a (: we add the element a to the sequence −→ Vgs , and we consider the new list of vacant indices obtained in this manner). We remark that this notation makes sense since the list of vacant indices in −→ the sequence Vgs ∨ a depends only whether or not a belongs to Ij , and not about the precise value of a. We also observe that k = (p − 1) if j ≤ (n − 1), whereas k = p if j ≥ n. We recall - see the above Proposition 3.24 - that for every sequence x1 > x2 > ... > xn > 0 we have : 1 (α) (1) α P (Vgt ≤ x1 , ..., Vg(n) ≤ xn ) −→ Pn (xα 1 , ..., xn ) t t→∞ tα

(3.264)

where the functions Pn are deﬁned in Proposition 3.24. Theorem 3.25. Let n and x1 > x2 > ... > xn > 0 be ﬁxed. Then, for every s ≥ 0 and Λs ∈ Fs : E (α) 1Λs 1(V (1) ≤x ,...,V (n) ≤x ) 1 n gt gt exists (3.265) lim t→∞ P (α) (V (1) ≤ x , ..., V (n) ≤ x ) gt gt 1 n

(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

209

and equals Msx1 ,...,xn ,n , with : Ms(x1 ,...,xn ,n) =

1(V (1) ≤x ,...,V (n) ≤x

,

Rs2α 2α Γ(1 + α) % n k R (α) ! s 1As ∈Ik Φ + x j − As k=λ1 (s) j=λ1 (s) & Rs (α) α α ! −Φ Prj (xλ1 (s,j) , ..., xλr (s,j) ) j xj+1 − As gt

1

gt

n)

α Pn (xα 1 , ..., xn )

(3.266) (x1 ,...,xn ,n)

Moreover, (Ms In (3.266) :

, s ≥ 0) is a ((Fs , s ≥ 0), P (α) ) martingale.

−→ • λ1 (s) is the ﬁrst vacant index in the sequence Vgs ; −→ • (λ1 (s, j), ..., λrj (s, j)) is the set of vacant indices in the sequence Vgs ∨a, when a ∈ Ij . Of course : rj = {λ(s)} − 1 rj = {λ(s)}

if if

j ≤n−1 j=n

(3.267)

• In the second sum of (3.266) and for the index j = k, the term Rs Φ(α) ! is equal to 0, since if As ∈ Ik , xj+1 − As < 0 we xj+1 − As get : R Rs def s Φ(α) ! = Φ(α) ! xj+1 − As (xj+1 − As )+ Rs (3.268) = Φ(α) = Φ(α) (∞) = 0 0 • By switching, in (3.266), the order of summations in j and in k, one can write : , 1(V (1) ≤x ,...,V (n) ≤x ) Rs2α 1 n gs gs (x1 ,...,xn ,n) Ms = α α α Pn (x1 , ..., xn ) 2 Γ(1 + α) % n R (α) ! s + 1As ≤xj Φ xj − As j=λ1 (s) & Rs (α) α α ! −Φ Prj (xλ1 (s,j) , ..., xλr (s,j) ) j xj+1 − As (3.269)

210

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Proof of Theorem 3.25 5a) We begin by showing that E 1(V (1) ≤x ,...,V (n) ≤x ) |Fs gt

(1) P (Vgt

≤

1

gt

n

(n) x1 , ..., Vgt

≤ xn )

−→ Ms(x1 ,...,xn ,n)

t→∞

a.s.

(3.270)

where Msx1 ,...,xn ,n is deﬁned in (3.266). We ﬁrst recall that : r2α (see (3.10)) t→∞ tα 2α Γ(1 + α) r Pr(α) (T0 < u) = Φ(α) √ (see (3.9)) u Pr(α) (T0 > t) ∼

and

(3.271) (3.272)

and we write, for s < t : −→ −−−→ −→ (Vgt ) = (Vgs ∨ ((As + T0 ◦ θs )1s+T0 ◦θs ds )

(3.273)

where, in (3.273) : −→ −→ • Vgs ∨ ((As + T0 ◦ θs )1s+T0 ◦θs ds is the sequence 0 if t < ds and is the sequence Vgt−ds ◦θds if t > ds . Let Θt denote the event : Θt := (Vg(1) ≤ x1 , ..., Vg(n) ≤ xn ) t t

(3.274)

Then, ω ∈ Θt ∩ (As ∈ Ik ) if and only if : (1)

(n)

i) ω ∈ (Vgs ≤ x1 , ..., Vgs ≤ xn ) and ii) either, T0 ◦ θs (ω) > t − s, and then : (1) (n) (1) (n) (Vgt (ω), ..., Vgt (ω)) = (Vgs (ω), ..., Vgs (ω)).

•

From (3.273), conditionally on Fs , this event happens with a probability R2α , which is equivalent, as t → ∞, to t−α α s 2 Γ(1 + α) • or, λ1 (s) ≤ k, and for every j ∈ λ1 (s), k, we may come back to 0 after s between the times (s+xj+1 −As )and (s+xj −As ), which happens, conRs Rs −Φ(α) ! , ditionally on Fs , with probability Φ(α) ! xj − As xj+1 − As

(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

211

−−−→ from (3.272), then it is necessary that the sequence Vgt−ds ◦ θds belongs to Sλ1 (s,j),...,λrj (s,j) . Of course, from (3.264), the probability of this event satisﬁes : −−−→ P (α) (Vgt−ds ◦ θds ∈ Sλ1 (s,j),...,λrj (s,j) ) ∼

t→∞

1 Pr (xα , ..., xα λrj (s,j) ) (3.275) tα j λ1 (s,j)

The assertion (3.270) now follows easily from (3.273), (3.264) and (3.275). Indeed : E (α) 1(V (1) ≤x ,...,V (n) ≤x ) |Fs Nt 1 n gt gt := (1) (n) (α) D t P (Vgt ≤ x1 , ..., Vgt ≤ xn ) and, considering the previous description of Θt ∩ (As ∈ Ik ), we get : %

,

Nt = E 1(V (1) ≤x ,...,V (n) ≤x gs

1

n)

gs

1(T0 ◦θs >t−s) +

n

1As ∈Ik × . . .

k=λ1 (s)

⎫ ⎬ −−−−→ F ... 1T0 ◦θs ∈[xj −As ,...,xj+1 −As ] 1− Vgt−d ◦θds ∈Sxλ (s,j) ,...,xλ (s,j) s ⎭ s rj 1 j=λ1 (s) , Rs2α ∼ 1(V (1) ≤x ,...,V (n) ≤x ) α 1 n gs gs t→∞ 2 Γ(1 + α)tα % n n 1 Rs + α 1As ∈Ik Φ(α) ! t xj − As k=λ1 (s) j=λ1 (s) & Rs (α) α α ! −Φ Prj (xλ1 (s,j) , ..., xλr (s,j) ) j xj+1 − As k

hence (3.270) since Dt ∼

1 α α α Pn (x1 , ..., xn ). t→∞ t (x ,...,x ,n)

n , s ≥ 0) is a martingale 5b) We now show that (Ms 1 As we already noted several times, it suﬃces to prove that

E (α) (Msx1 ,...,xn ,n ) = 1

(3.276)

In order to prove (3.276), we shall use the following Lemma 3.26. Consider the functions Pn introduced in Proposition 3.18. Then, for every n ≥ 1 : Pn (y1 , y2 , ..., yn ) = yn +

n−1 j=1

yn

1 yj+1

1 − yj

Pn−1 (y1 , ..., yˇj , ..., yn ) (3.277)

212

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

where we note : Pn−1 (y1 , ..., yˇj , ..., yn ) = Pn−1 (y1 , ..., yj−1 , yj+1 , ..., yn )

(3.278)

Before we engage in the proof of Lemma 3.26, we remark that the relation (3.277), once translated thanks to (3.205) in terms of polynomials Qn (y1 , ..., yn ), deﬁned in Proposition 3.18 becomes : Qn (y1 , ..., yn ) = (yn − yn−2 )Qn−1 (y1 , ..., yˇn−1 , ..., yn ) +

n−2

(yj+1 − yj )Qn−1 (y1 , ..., yˇj , ..., yn )

(3.279)

j=1

i) We begin by showing that (3.279) is equivalent to (3.277), although we shall not need it in the sequel. We write (3.277) and, in the RHS of (3.277), we replace Pn−1 by its value given in (3.205). We obtain :

= ...

...

...

...

Pn (y1 , ..., yn ) 1 1 yn + yn − × ... y2 y1 1 1 1 2 3 − Q2 + yn yn + yn yn y2 yn 1 1 +yn − × ... y3 y2 1 1 1 2 3 − Q2 + yn yn + yn yn y1 yn 1 1 +yn − × ... y4 y3 1 1 1 2 3 − Q2 + yn yn + yn yn y1 yn ... 1 1 − +yn × ... yn yn−1 1 1 1 − Q2 + yn3 yn + yn2 yn y1 yn

1 − y3

1 − y3

1 − y2

−

1 y2

Q3

Q3

Q3

Q3

1 1 1 , , y2 y3 yn

1 1 1 , , y1 y3 yn

1 1 1 , , y1 y2 yn

1 1 1 , , y1 y2 yn

+ ...

+ ...

+ ...

+ ...

Then, by summing up these diﬀerent relations, we obtain : Pn (y1 , ..., yn )

= yn + yn2

1 1 − yn y1

+ yn3

+yn4 A4 + yn5 A5 + ...

1 1 − yn y2

1 1 2 + − yn y2 y1

(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

with

213

1 1 1 1 1 1 1 − , , − A4 = Q3 yn y2 y1 y2 yn yn y3 1 1 1 1 1 1 1 + − − , , Q3 yn y3 y3 y2 y1 y3 yn 1 1 1 1 1 1 1 + − − , , Q3 yn y3 y2 y1 y2 y3 y5

Using once more (3.205), we obtain : 1 1 1 1 1 1 A4 = − , , , Q4 yn y3 y1 y2 y3 yn i.e :

Q4 (y1 , y2 , y3 , y4 ) = (y4 −y2 )Q3 (y1 , y2 , y4 ) + (y3 −y2 )Q3 (y1 , y3 , y4 ) + (y2 −y1 )Q3 (y2 , y3 , y4 )

and, more generally : Qn (y1 , ..., yn ) = (yn − yn−2 )Qn−1 (y1 , ..., yˇn−1 , yn ) +

n−2

(yj+1 − yj )Qn−1 (y1 , ..., yˇj , ..., yn )

j=1

ii) We now prove (3.277) For this purpose, we write, from (3.197) and (3.204): Pn (y1 , ..., yn ) =

1 2α Γ(1

+ α)

E (α) (LH (n) )

(3.280)

(x)

with yi = xα i for every i = 1, ..., n and (x) = (x1 , ..., xn ). Hence, since : 1 E (α) (LH (n) ) (x) 2α Γ(1 + α) ⎧ n−1 ⎨ 1 1Ad− = α LH (n) + 2 Γ(1 + α) ⎩ (x) (1) H j=1

∈[xj+1 ,xj ] LH (n−1)

xn

(x1 ,...,x ˇ j ,...,xn )

◦ θd

⎫ ⎬ ⎭

(1) Hxn

(3.281)

we have, taking the expectation of (3.281) after having noticed that : E[LH (1) ] = 2α Γ(1 + α)xα n xn

and that P

Ad−

(1) Hxn

(this is formula (3.28))

xα ∈ [xj+1 , xj ] = 1 − nα xj

−

1−

xα n xα j+1

214

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

(this relation follows from (3.24)) : Pn (y1 , ..., yn ) 1 E(LH (n) ) = α (x) 2 Γ(1 + α) n−1 1 = xα xα − n + n α xj+1 j=1 n−1 1 α α xn − = xn + α xj+1 j=1 = yn +

n−1 j=1

yn

1 yj+1

−

1 α 2 Γ(1 + α)

1 xα j

E

1 xα j

Pn−1 (y1 , ..., yˇj , ..., yn ) (from (3.280))

1 yj

LH (n−1)

(x1 ,...,x ˇ j ,...,xn )

◦ θd

(1) Hxn

Pn−1 (y1 , ..., yˇj , ..., yn )

We now prove that

(x ,...,xn ,n) E(Ms 1 )

= 1.

To begin with, we prove that : E M

(x1 ,...,xn ,n) (1)

s∧Hxn

= 1.

Indeed : (x ,...,x ,n) (x1 ,...,xn ,n) E M 1 (1) n = E Ms(x1 ,...,xn ,n) 1s
xn

Hxn

xn

:= (1) + (2) (1) = E(Ms 1s
(where, to simplify writing, we note M instead of M x1 ,...,xn ,n ) % Rs Rs2α 1 (α) √ 1 +Φ =E Pn (y1 , ..., yn ) (1) Pn (y1 , ..., yn ) s
(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

215

that is, from Lemma 3.26 : (1) =

Pn (y1 , ..., yn )

+ =

(2) =

%

1

xn

1 Pn (y1 , ..., yn ) n−1

+

&

Aα 1− s yn

Pn (y1 , ..., yn )

E 1s
,

xn

%

E 1H (1) ≤s

⎛

⎣Φ(α) ⎝

j=1

Aα s

Pn (y1 , ..., yn )

⎡

Aα s

E 1s
Pn (y1 , ..., yn ) − yn yn

1

+

xn

RH (1)

xn

α xα j − xn

R2α(1) Hxn

2α Γ(1 + α) ⎞ ⎛ ⎠ −Φ(α) ⎝

⎞⎤

RH (1)

xn

α xα j+1 − xn

&⎠⎦ Pn−1 (y1 , ..., yˇj , ..., yn )

(1)

Using then the independence between RH (1) and Hxn as well as (3.23) and xn (3.24), we get : , % n−1 1 1 1 α α E 1H (1) ≤s xn + xn − α (2) = α xn Pn (y1 , ..., yn ) x x j+1 j j=1 &Pn−1 (y1 , ..., yˇj , ..., yn ) 1 E 1H (1) ≤s Pn (y1 , ..., yn ) xn Pn (y1 , ..., yn ) = E 1H (1) ≤s

=

(from 3.26)

xn

Finally : (x ,...,x ,n) = (1) + (2) = E 1s
We now show that

xn

⎛ E ⎝M

xn

(3.282)

⎞ (x1 ,...,xn ,n) ⎠ (1) s∧ d (1) ∧Hxn−1 Hxn

(x1 ,...,xn ,n) = 1. We already observe that E M (1) Hxn

=1

(3.283)

216

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Indeed, (from (3.269)) : (x1 ,...,xn ,n) E M (1) Hxn ,% R2α(1) 1 Hxn E + = Pn (y1 , ..., yn ) 2α Γ(1 + α) &% & n−1 RH (1) RH (1) xn xn (α) (α) Φ −Φ Pn−1 (y1 , ..., yˇj , ..., yn ) √ √ xj − xn xj+1 − xn j=1 ⎡ ⎤ n−1 1 1 1 ⎣xα = xα − α Pn−1 (y1 , ..., yˇj , ..., yn )⎦ n + n α Pn (y1 , ..., yn ) x xj j+1 j=1 = 1 (from Lemma 3.26) For our further use, we note that a computation close to the preceding implies : (x1 ,...,xn ,n) for every y ≤ xn , E M (1) =1 (3.284) Hy

(1)

To prove (3.283), it now suﬃces to observe that, for s between the times Hxn (1) (x ,...,xn ,n) and dH (1) ∧ Hxn−1 , Ms 1 is a linear combination of the martingales : xn

(1)

a) (Rs2α , s ≥ 0) (since Rs2α − Ls is a martingale, and between the times Hxn (1) and dH (1) ∧ Hxn−1 , R never vanishes) xn

and b)

(α)

Φ

R ! s xj − As

Thus,

−Φ

(α)

!

Rs xj+1 − As ⎞

⎛ E ⎝M

(x1 ,...,xn ,n) ⎠ (1) s∧ d (1) ∧Hxn−1

,s ≥ 0

(see Theorem 3.5)

(x1 ,...,xn ,n) =0 − E M (1) Hxn

Hxn

(x1 ,...,xn ,n) and we have seen previously that E M (1) = 1. This proves (3.284). Hxn

Finally, to prove (3.276), we iterate the preceding arguments along the sequence of stopping times which consist of the successive passage times of the age process in the levels xn , xn−1 , ..., x2 , and of the returns to 0 of the process (Rs , s ≥ 0) after these passage times. Remark 3.27. 1) Taking n = 1 in (3.269) we obtain Ms(x1 ,1) =

1(V (1) ≤x ) gs

xα 1

1

Rs2α + 1As ≤x1 Φ(α) α 2 Γ(1 + α)

R √ s x1 − As

xα 1

(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

217

α since P1 (xα 1 ) = x1 and also because λ1 (s) = 1 (see (3.260) and (3.263) for the deﬁnition of λ(s)); hence α Pr1 (xα λ1 (s,j) , ..., xλr

j

(s,j) )

= xα 1

where Ms has been deﬁned in (3.52) and (3.53). Thus, our formula (3.269) is coherent with (3.52) and (3.53), which corresponds to the case n = 1. 2) Given formula (3.266), it is clear that : Ms(x1 ,...,xn−1 ,xn−1 ,n) = Ms(x1 ,...,xn−1 ,n−1)

(3.285)

This follows from the fact that Pn (y1 , ..., yn−1 , yn−1 ) = Pn−1 (y1 , ..., yn−1 ) from (3.218) and that, if one takes xn−1 = xn in (3.266), the terms 1As ∈In−1 and R Rs s Φ(α) √ − Φ(α) √ disappear. xn−1 − As xn − As 3) Some moments thoughts are necessary to verify that formula (3.266) (or (3.269)) deﬁnes indeed a continuous process. In fact, it suﬃces to verify this continuity property at the times of : • hitting of a level xi by the age process (As ), and then the coeﬃcient R s Φ(α) √ equals Φ(α) (∞) = 0 xi − As • hitting of 0 by the process (Rs , s ≥ 0) after the age process has reached a level xi : the continuity at this time follows from the fact that Φ(α)

R ! s xj − As

− Φ(α)

and

(α)

Φ

!

Rs xj+1 − As

R √ s xi − As

=1−1=0

=1

α and from the deﬁnition of the function Prj (xα λ1 (s,j) , ..., xλr

j

(s,j) ).

4) We have already observed, as we stated in Theorem 3.13, that the penalisation mechanism enjoys some continuity property. It is also the case in the following situation : we shall now show that : Ms(x1 ,...,xn ,n) (n,xn )

where Ms

→

x1 ,...,xn−1 →∞

has been deﬁned in (3.124) :

Ms(n,xn )

(3.286)

218

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

Rs2α Rs 1 (α) √ 1 + Φ − 1 A ≤x n gs 2α Γ(1 + α)nxα n s xn − As n + xn Δ + 1− s (3.287) n

Ms(n,x) = 1V (n) ≤x

Let us show (3.286). First of all, we observe that the recurrence formula (3.277) implies easily that : α α Pn (xα (3.288) lim 1 , ..., xn ) = nxn x1 ,...,xn−1 →+∞

and that this formula is coherent with point 1) of Theorem 3.4. On the other hand, from (3.287), we have : xn • on the set (As ≤ xn ) ∩ (Δs = p) (with p ≤ n), we have : Rs2α Rs 1 (α) √ Ms(n,x) = 1V (n) ≤x + − 1 Φ n gs 2α Γ(1 + α)nxα n xn − As n p (3.289) + 1− n •

whereas on the set : (As > xn ) ∩ (Δxs n = p), we have : Rs2α p (n,x) Ms = 1V (n) ≤x + 1 − n gs 2α Γ(1 + α)nxα n n (x1 ,...,xn ,n)

We now study the limit as x1 , ..., xn−1 → +∞ of Ms that : 1V (1) ≤x ...1V (n) ≤x → 1V (n) ≤x 1

gs

gs

n

gs

(3.290)

. We ﬁrst note

n

and α α Pn (xα 1 , ..., xn ) → nxn

and on the set (As ≤ x) ∩ (Δxs = p) : Ms(x1 ,...,xn ,n)

→

x1 ,...,xn−1 →∞

because : Φ

(α)

!

1V (n) ≤x

Rs2α Rs (α) α √ +Φ (n − p)xn nxα 2α Γ(1 + α) x − As n n Rs 1 + α 1 − Φ(α) √ (n − p − 1)xα n nxn xn − As

Rs xj − As

gs

n

→

xj →∞

1, and Prj → (n − p − 1)xα n on the set (x1 ,...,xn ,n)

(As ≤ xn ) ∩ (Δxs n = p) (see (3.266) for the deﬁnition of Ms

).

(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

219

Hence, on the set (As ≤ xn ) ∩ (Δxs n = p) : Rs2α Rs 1 (α) (x1 ,...,xn ,n) √ + Φ Ms → 1 (n) x1 ,...,xn−1 →∞ Vgs ≤xn 2α Γ(1 + α) n xn − As n−p−1 (3.291) + n Thus, comparing (3.291) and (3.289) : →

Ms(x1 ,...,xn ,n)

x1 ,...,xn−1 →+∞

Ms(n,xn )

on the set (As ≤ x) ∩ (Δxs n = p)

On the other hand, on the set (As > xn ) ∩ (Δxs = p) : Ms(x1 ,...,xn ,n)

→

x1 ,...,xn−1 →∞

1V (n) ≤x gs

n

nxα n

Rs2α + (n − p)xα n 2α Γ(1 + α)

(3.292)

Rs → 1 (j ≤ n − 1), and Prj = (n − p)xα n on the xj − As xj →∞ set (As ≥ xn ) ∩ (Δxs n = p) (the only index j = n − 1, in the sum (3.269) (x ,...,xn ,n) which deﬁnes Ms 1 gives a non-zero contribution). Comparing (3.292) and (3.290), we have:

since Φ

(α)

!

Ms(x1 ,...,xn ,n)

→

x1 ,...,xn−1 →+∞

Ms(xn ,n)

on the set (As > x) ∩ (Δxs n = p)

This proves (3.286). (x ,...,x ,n)

n We now deﬁne, as we did previously many times, a probability Qs 1 on (Ω, F∞ ) via the formula : (x1 ,...,xn ,n) 1 ,...,xn ,n) Q(x (Λs ∈ Fs ) (Λ ) = E 1 M (3.293) s Λ s s s

and we shall now describe the properties of the canonical process (Ω, Rt , t ≥ 0) under Q(x1 ,...,xn ,n) . Theorem 3.28. (Description of the Q(x1 ,...,xn ,n) process) With respect to Q(x1 ,...,xn ,n) (to simplify, we note Q instead of Q(x1 ,...,xn ,n) ), one has : 1) Let g := sup{t; Rt = 0}. Then : 0 < g < ∞ a.s. The r.v. Lg (= L∞ ) admits as a density : fLQg () =

1 (1) (n) P V ≤ x , ..., V ≤ x 1 n τ τ α 2α Γ(1 + α)Pn (xα 1 , ..., xn )

2)i) The processes (Ru , u ≤ g) and (Rg+u , u ≥ 0) are independent. ii) The process (Rg+u , u ≥ 0) is a Bessel process starting from 0 and with dimension 4 − d = 2 + 2α.

220

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

iii) Conditionally upon Lg (= L∞ ) = , the process (Rt , t ≤ g) is a Bessel process with dimension d = 2(1 − α), stopped at time τ and conditioned by (1) (n) (Vτ ≤ x1 , ..., Vτ ≤ xn ). Proof of Theorem 3.28 This proof hinges essentially on the computation of Az´ema supermartingale : EQ (g > t|Ft ) This supermartingale is described in the following Proposition 3.29. 1) EQ (g > t|Ft ) = 1

+

1

1 (1) • (n) (x ,...,xn ,n) (Vgt ≤x1 ,...,Vgt ≤xn ) P (xα , ..., xα ) n 1 n Mt 1 % , n Rs 1At ≤xj Φ(α) ! xj − As j=λ (t) 1

−Φ

(α)

Rs ! xj+1 − As

&

-

α Prj (xα λ1 (s,j) , ..., xλrj (s,j) )

(3.294)

2) For every positive predictable process (Ks , s ≥ 0), we have : 1 E EQ (Kg ) = α α 2 Γ(1 + α)Pn (xα 1 , ..., xn )

=

Ks 1(V (1) ≤x ,...,V (n) ≤x ) dLs 1

s

0

1 EQ α 2α Γ(1 + α)Pn (xα 1 , ..., xn )

∞

0

n

s

∞

Ks 1(V (1) ≤x ,...,V (n) ≤x s

1

s

n

(3.295) dLs ) (3.296)

Proof of Proposition 3.29 It is quite similar to that of Lemma 3.11. We now give some main points. We have : (x ,...,xn ,n) E(Mdt 1 |Ft ) Q(g > t|Ft ) = Q(dt < ∞|Ft ) = (x1 ,...,xn ,n) Mt Now : (x ,...,xn ,n)

Mdt 1

1(V (1) ≤x ,...,V (n) ≤x 1

gt

=

gt

α Pn (xα 1 , ..., xn )

(α)

(from (3.269), since Φ

!

Rdt xj − Adt

n)

α {Prn (xα λ1 (dt ,n) , ..., xλrn (dt ,n) )}

= 1, because Rdt = 0.)

(1)

(n)

3.9 Penalisation by (Vgt ≤ x1 , ..., Vgt ≤ xn )

221

Hence : (x ,...,x ,n)

n E(Mdt 1 |Ft ) , % n R (α) ! t = 1At ≤xj Φ x j − At j=λ (t)

−Φ

(α)

1

α . . . Prj (xα λ1 (t,j) , ..., xλr

since, conditionally on Ft :

j

Rt ! xj+1 − At -

& × ...

(t,j) )

Rt Rt (α) ! ! T0 ∈ [xj+1 , xj ] with the probability Φ −Φ xj − At xj+1 − At α α α and the functions Prj (xα , ..., x ) and P (x , ..., x rn λ1 (t,j) λr (t,j) λ1 (dt ,n) λr (dt ,n) ) are (α)

j

n

the same if Ad−t ∈ [xj , xj+1 ]. This proves (3.294). To go from (3.294) to (3.295) (and to (3.296)), we operate exactly as in the proof of Lemma 3.11 : all hinges here upon the fact that (Rt2α − Lt , t ≥ 0) (x ,...,xn ,n) and (Mt 1 , t ≥ 0) are two martingales. The end of the proof of Theorem 3.28 is now quite similar to that of Theorem 3.10. For example, for every functional F ≥ 0, and every positive Borel function h : EQ [F (Ru , u ≤ g)h(Lg )] ∞ 1 E F (Ru , u ≤ s)h(Ls )1(V (1) ≤x ,...,V (n) ≤x ) dLs = α α 1 n s s 2 Γ(1 + α)Pn (xα 0 1 , ..., xn ) (from (3.295)) ∞ 1 = α E F (Ru , u ≤ τ )h()1(V (1) ≤x ,...,V (n) ≤x ) d α 1 n τ τ 2 Γ(1 + α)Pn (xα 0 1 , ..., xn ) (after the change of variable s = τ .) Thus, taking F ≡ 1 in the preceding point : fLQg (u) =

1 E 1 (1) (n) α (Vτu ≤x1 ,...,Vτu ≤xn ) 2α Γ(1 + α)Pn (x1 , ..., xα n)

and : EQ [F (Ru , u ≤ g)h(Lg )] (3.297) ∞ 1 (1) (n) E F (R , u ≤ τ )|V ≤ x , ..., V ≤ x = α u 1 n τ τ α 2 Γ(1 + α)Pn (xα 1 , ..., xn ) 0 •

P (Vτ(1) ≤ x1 , ..., Vτ(n) ≤ xn )h()d

(3.298)

222

3 Penalisations of a Bessel Process with Dimension d (0 < d < 2)

On the other hand :

EQ [F (Ru , u ≤ g)h(Lg )] = 0

∞

EQ [F (Ru , u ≤ g)|Lg = ] h()fLQg ()d

(3.299) Comparing (3.298) and (3.299), we deduce : EQ [F (Ru , u ≤ g)|Lg = ] = E F (Ru , u ≤ τ )|Vτ(1) ≤ x1 , ..., Vτ(n) ≤ xn which is point 2)iii) of Theorem 3.28. Remark 3.30. Using the fact that E(M

(x1 ,...,xn ,n) (1)

Hy

) = 1, for every y < xn

(see (3.284)), we easily obtain, with the same arguments as those used in the proof of point 4) of Theorem 3.5 that : (1)

i) the process (Au , u ≤ Hy ) is equally distributed under P and under Q(x1 ,...,xn ,n) ; (1) ii) the process (Au , u ≤ Hy ) and the r.v. RH (1) are independent, under P y

and under Q(x1 ,...,xn ,n) ; iii) Q(g > Hy(1) ) = 1 −

yα

α Pn (xα 1 , ..., xn )

.

Bibliography [D-M,RVY] C. Donati-Martin, B. Roynette, P. Vallois and M. Yor. On constants related to the choice of the local time at 0 and the corresponding Itˆ o measure for Bessel processes with dimension d = 2(1 − α), 0 < α < 1. Studia Sci. Math. Hungarica, 44(4):207–221, 2008. [D-M,Y2 ] C. Donati-Martin and M. Yor. Some explicit Krein representations of certain subordinators, including the Gamma process. Publ. RIMS, vol.42, p.879–895, Kyoto, 2006. [DRVY] B. De Meyer, B. Roynette, P. Vallois and M. Yor. On independent times and positions for Brownian motions. Revista Math. IberoAmericana 18(3):541– 586, 2002. [Fel] W. Feller. An introduction to Probability Theory and its Applications. Vol. II, Second Edition, Wiley, 1971. [G] R.K. Getoor. The Brownian escape process. Ann. Prob. 7, p.864–867, 1979. [J] T. Jeulin. Semi-martingales et grossissement d’une ﬁltration. LNM 833, Springer, 1980. [Leb] N.N. Lebedev. Special functions and their applications. Dover Publications Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman, Unabridged and corrected republication.

Bibliography [M]

[PY1 ] [PY2 ] [PY3 ]

[PY4 ] [PY5 ] [PY6 ]

[RVYCR ]

[RVYJ ]

[RVY,I]

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[RVY,V]

[RVY,VI] [RVY,VII] [Wat] [Y]

223 P.A. Meyer. Probabilit´es et potentiel. Publications de l’Institut de Math´ematique de l’Universit´e de Strasbourg, N◦ XIV. Actualit´es Scientiﬁques et Industrielles, N◦ 1318. Hermann, Paris, 1966. J. Pitman and M. Yor. Asymptotic laws of planar Brownian motion. Ann. Probab., 14(3):733–779, 1986. J. Pitman and M. Yor. Further asymptotic laws of planar Brownian motion. Ann. Probab., 17(3):965–1011, 1989. J. Pitman and M. Yor. Bessel processes and inﬁnitely divisible laws. In Stochastic integrals (Proc. Sympos., Univ. Durham, Durham, 1980), LNM 851, pages 285–370. Springer, Berlin, 1981. J. Pitman and M.Yor. Arc sine laws and interval partitions derived from a stable subordinator. Proc. London Math. Soc., (3), 65, p.326–356, 1992. J. Pitman and M.Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Prob. 25(2):855–900, 1997. J. Pitman and M.Yor. On the relative lengths of excursions derived from a stable subordinator. S´em. Probas. XXXI, LNM 1655, p.287–305, Springer, 1997. B. Roynette, P. Vallois and M.Yor. Asymptotics for the distribution of lengths of excursions of a d-dimensional Bessel process (0 < d < 2). Comptes Rendus Acad. Sci. Paris, vol.343, fasc.3, p.201–208, Aug. 2006. B. Roynette, P. Vallois and M.Yor. Some penalisations of the Wiener measure. Survey article; Japanese Journal of Mathematics, vol.1, p.263–290, 2006. B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I. Studia Sci. Math. Hungar., 43(2):171–246, 2006. B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time. II. Studia Sci. Math. Hungar., 43(3):295–360, 2006. B. Roynette, P. Vallois, and M. Yor. Penalizing a BES (d) process (0 < d < 2) with a function of its local time. V. Studia Sci. Math. Hungar., 45(1):67–124, 2008. B. Roynette, P. Vallois, and M. Yor. Penalisations of multi-dimensional Brownian motion. VI. To appear in ESAIM P.S., 2009. B. Roynette, P. Vallois, and M. Yor. Brownian penalisations related to excursion lengths. VII. To appear in Annales de l’Inst. H. Poincar´e., 2009. G.N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge, 1944. M. Yor. On certain exponential functionals of real-valued Brownian Motion. J. App. Prob. 29, p.202–208, 1992. Also : Paper 1 In Exponential functionals of Brownian Motion and Related Processes. Springer-Finance, 2001.

Chapter 4

A General Principle and Some Questions about Penalisations

Abstract In the preceding chapters, we have shown that, in a large class of frameworks, certain families {P (t) , t ≥ 0} of probabilities deﬁned on a ﬁltered space {Ω, (Fs )s≥0 } converge, as t → ∞, to a probability Q, at least in the sense that : ∀ s > 0, , ∀Λs ∈ Fs , P (t) (Λs ) −→ Q(Λs ) t→∞

The typical question which we consider in this chapter is, roughly : to which extent can we replace Λs ∈ Fs by Λ ∈ F∞ ? The answer is, to say the least, subtle, and not every Λ ∈ F∞ is suitable. This led us to detect some non-uniform integrability results, as well as the convergence of certain (Hellinger, total variation...) distances between P (t) and Q|Ft , as t → ∞. We also prove, in the Brownian framework, the convergence in law of 1 (t) √ := X ; u ≤ 1 , under P (t) or Q, as t → ∞. The limits in law X t tu are diﬀerent, and may be expressed in terms of the BES(3) process or the Brownian meander. Keywords Brownian motion and Bessel processes · penalisations · local times · unilateral maximum process · uniform integrability · convergence in law of processes

4.1 Introduction 4.1.1 Let Ω = C([0, ∞[→ R) and let (Xt , t ≥ 0) denote the coordinate process on this space : Xt (ω) = ω(t), ω ∈ Ω.

B. Roynette, M. Yor, Penalising Brownian Paths, Lecture Notes in Mathematics 1969, DOI 10.1007/978-3-540-89699-9 4, c Springer-Verlag Berlin Heidelberg 2009

225

226

4 A General Principle and Some Questions about Penalisations

We denote by (Ω, (Xt , Ft )t≥0 , F∞ =

>

Ft , Px (x ∈ R)) a canonical process

t≥0

which will be mostly, either the canonical Brownian motion, or the canonical Bessel process with dimension d = 2(1 − α), (0 < d < 2, or 0 < α < 1). 4.1.2 We also consider (Γt , t ≥ 0) a process deﬁned on (Ω, F∞ ) which may not be (Ft ) adapted and which takes its values in a measurable space (E, E); let furthermore h : E → R+ be measurable and such that : 0 < Ex [h(Γt )] < ∞

for every x ∈ R , t ≥ 0

(4.1)

As we have seen in the preceding chapters, studying the penalisations of the probabilities (Px , x ∈ R) by the process (h(Γt ), t ≥ 0) consists in the study of (t) the limit, as t → ∞, in a sense to be made precise, of the probabilities Px deﬁned by : h(Γt ) • Px (4.2) Px(t) := Ex [h(Γt )] (t)

Thus, these probabilities Px are the probabilities Px , weighted by:

h(Γt ) . Ex [h(Γt )]

4.1.3 The prototype of the Theorems we have obtained in the preceding chapters is the following : Generic Theorem Under certain adequate hypotheses on (Γt , t ≥ 0), namely : • there exists a function λ : R+ −→ R+ , such that λ(t) −→ ∞ t→∞

(4.3)

• for every x ∈ R, there exists a measure ν x on (E, E), which is σ-ﬁnite and positive such that : for every function h : E −→ R+ measurable, which satisﬁes : h(y)ν x (dy) < ∞

0<

(4.4)

E

the following holds : 1)

λ(t)Ex [h(Γt )] −→

t→∞

2) for every s ≥ 0 :

Ex

h(y)ν x (dy)

(4.5)

E

h(Γt ) Fs −→ Msh t→∞ Ex [h(Γt )]

with the convergence in (4.6) taking place both a.s. and in L1 .

(4.6)

4.1 Introduction

227

Given (4.5), the assertion 2) is equivalent to the next assertion : 2’) for every s ≥ 0 : λ(t)Ex [h(Γt )|Fs ] −→ h(y)ν x (dy) • Msh t→∞

(4.7)

E

and this convergence takes place both a.s. and in L1 . 3) (Msh , s ≥ 0) is a (Fs , s ≥ 0, Px ) positive martingale such that M0h = 1 (hence : Ex (Msh ) = 1 for every s ≥ 0). We note that this martingale (Msh , s ≥ 0) depends on x = X0 , but we omit to indicate this dependency. 4) One easily deduces from (4.6) that, for every s and every Λs ∈ Fs (resp. for every r.v. Fs which is Fs measurable and bounded), one has :

Ex [1Λs h(Γt )] −→ Ex [1Λs Msh ] := Qhx (Λs ) Ex [h(Γt )] t→∞ resp.

(4.8)

Ex [Fs h(Γt )] −→ Ex [Fs Msh ] := Qhx (Fs ) Ex [h(Γt )] t→∞

(4.8’)

The formula : Qhx (Λs ) = Ex [1Λs Msh ] induces a probability Qhx on (Ω, F∞ ) We have also studied in detail, in many examples, the canonical process (Ω, (Xt , Ft )t≥0 ) under the family of probabilities (Qhx , x ∈ R). Deﬁnition 4.1. A process (Γt , t ≥ 0) such that the conclusions of the Generic Theorem apply shall be said to enjoy the penalisation principle. 4.1.4 The Generic Theorem invites to raise several natural questions : i) Is the martingale (Msh , s ≥ 0) uniformly integrable ? More generally, h , s ≥ 0)uniformly for which stopping times T is the martingale (Ms∧T integrable ? h a.s. to 0 as s → ∞ ? In the opposite direction, does Ms converge h(Γt ) , t ≥ 0 uniformly integrable ? ii) Is the family of r.v.’s Ex [h(Γt )] Does it converge to 0 - in a sense to be made precise - as t → ∞ ? h(Γt ) iii) The relation (4.8) indicates that the r.v. is, in a weak sense, Ex [h(Γt )] h close to Mt , as t → ∞. Does this closeness hold in a stronger sense, e.g. (t) : does the total variation of the measures Px − (Mth •Px ) converge to 0, as t → ∞ ? In other terms, is is true that : h(Γt ) (4.9) − Mth = 0 ? lim Ex t→∞ Ex [h(Γt )] iv) Can we replace, in (4.8), the generic set Λs ∈ Fs by a set Λ ∈ F∞ ? In other terms, for which sets Λ ∈ F∞ , is it true that : lim

t→∞

Ex [1Λ h(Γt )] = Qhx (Λ) Ex [h(Γt )]

?

(4.10)

228

4 A General Principle and Some Questions about Penalisations

We note immediately that this question (4.10) has a somewhat “rough” character. Indeed, let us assume for a moment that Mth −→ 0 a.s. Then, t→∞

the probabilities Px and Qhx are singular on F∞ and there exists Λ ∈ F∞ such that : Px (Λ) = 1 and Qhx (Λ) = 0 (4.11) Under these hypotheses, we have (see Theorem 4.14 below) : Ex [1Λ h(Γt )] ≡1 Ex [h(Γt )]

whereas

Qhx (Λ) = 0

(4.12)

This remark leads us to a “softer” formulation of the question (4.10). Rather than replacing brutally in (4.8) the sets Λs ∈ Fs , for ﬁxed s, by any Λ ∈ F∞ , we shall, given the scaling property of Brownian (t) motion or Bessel processes, consider the test functions F (X• ), where 1 (t) (Xu ; 0 ≤ u ≤ 1) := √ Xut ; 0 ≤ u ≤ 1 and F is any functional : t C([0, 1] → R) → R+ , which is bounded and continuous. Thus, our modiﬁed question is now : v) Does the quantity h(Γt ) Ex F (X•(t) ) converge, as t → ∞ ? (4.13) Ex [h(Γt )] We also consider the question : does

Qhx [F (X•(t) )]

converge, as t → ∞ ?

(4.14)

Furthermore, assuming that these two limits exist, are they equal ? 4.1.5 The aim of this Chapter is to answer - sometimes only partially - these ﬁve questions. Throughout our work on penalisations, we have not been able - unfortunately - to obtain a “uniﬁed” proof of the Generic Theorem. In other terms, we have only been able to prove this theorem by providing, for each example, a speciﬁc proof which may diﬀer a lot from one example to another. Our aim here is to answer the preceding questions in the most uniﬁed possible manner, that is to provide an answer which is valid for a large class of examples, thus liberating ourselves from a case to case study. We have reached this goal in a satisfying way for a part of the questions i), ii), iii) and iv) (see Theorems 4.5, 4.9 and 4.10 below). On the contrary, our answer to question v) remains dependent on the explicit form of the examples which we consider (see Theorems 4.16 and 4.18 below). 4.1.6 Before we answer the questions 4.1.4 i),...,v), we begin by recalling some examples of processes (Γt , t ≥ 0) which satisfy the penalisation principle.

4.1 Introduction

229

Example 4.2. i) (Xt ) denotes the canonical Brownian motion, and (Γt , t ≥ 0) = (St := sup Xs , t ≥ 0)

(4.15)

s≤t

Then, we have obtained :

πt , ν x (dy) = 1[x,∞[ (y)dy 2 ∞ h Ms = h(Ss )(Ss − Xs ) + h(y)dy λ(t) =

(for x = 0)

(4.16)

Ss

(see [RVY,II] or Chapter 1, Section 1.3, Case I). ii) (Xt ) still denotes the canonical Brownian motion, and (Γt , t ≥ 0) = (Lt , t ≥ 0) is the (continuous) local time at level 0. Then, we have obtained : πt , ν x (dy) = |x|δ0 (dy) + 1[0,∞[ (y)dy λ(t) = 2 ∞ h Ms = h(Ls )|Xs | + h(y)dy (for x = 0) (4.17) Ls

(see [RVY,II] or Chapter 1, Item B ). This example may be extended by taking for (Xt ) the Bessel process with dimension d = 2(1 − α) (0 < α < 1); we obtained : λ(t) = tα Γ(1 + α)2α , ν x (dy) = x2α δ0 (dy) + 1[0,∞[ (y)dy ∞ Msh = h(Ls )Xs2α + h(y)dy (for x = 0)

(4.18)

Ls

The Brownian case corresponds to the preceding situation with α = 1/2 (see [RVY,V] or Chapter 1, Theorem 1.10). iii) (Xt ) still denotes canonical Brownian motion. Let a < b and (Γt , t ≥ 0) = (Dta,b , t ≥ 0) where Dta,b denotes the number of down-crossings of X, before time t, on the (space) interval [a, b]. More precisely, let : σ1 := inf{t > 0; Xt ≥ b} ; σ2 := inf{t ≥ σ1 , Xt < a} σ2n+1 := inf{t ≥ σ2n ; Xt ≥ b} ; σ2n+2 := inf{t ≥ σ2n+1 , Xt < a} and : Dta,b :=

n≥1

1σ2n ≤t

t≥0

(4.19)

230

4 A General Principle and Some Questions about Penalisations

We have obtained : , 1 |x − b| πt , ν x (h) = 2(b − a) + h(n) + h(0) λ(t) = 2 2 2(b − a) n≥1 (4.20) The reader may refer to [RVY,II] or Chapter 0, Example 0.3, concerning the explicit form of Msh in this case. iv) (Xt ) still denotes the canonical Brownian motion. Let : gt := sup{s ≤ t; Xs = 0}

(4.21)

and (Γt , t ≥ 0) = (Sgt := sups≤gt Xs , t ≥ 0). We have obtained : 1 πt , ν 0 (dy) = 1[0,∞[ (y)dy λ(t) = 2 2 ∞ 1 h + Ms = h(Sgs )|Xs | + h(Ss )(Ss − Xs ) + h(y)dy 2 Ss

(4.22)

(see [RY,VIII]). Example 4.3. i) (Xt ) still denotes the canonical Brownian motion. Let q be a positive Radon measure on R such that : 0 < (1 + |x|)q(dx) < ∞ (4.23) R

and (Γt , t ≥ 0) =

(q)

At :=

∞

−∞

Lyt q(dy), t ≥ 0

(4.24)

where (Lyt ; t ≥ 0, y ∈ R) denotes the jointly continuous family of the local times of (Xt ). We have obtained : √ λ(t) = t , ν x is the measure on R+ which is characterized by : ∞ μ e− 2 y ν x (dy) = ϕμ q (x) (4.25) 0

where ϕμ q is the unique solution of :

ϕ = μ q • ϕ , ϕ (+∞) = −ϕ (−∞) =

2 π

(4.26)

(see [RVY,I] or Chapter 2). (q) In this example, since (At , t ≥ 0) is an additive functional of Brownian motion, i.e : it satisﬁes : (q)

(q)

At = A(q) s + At−s ◦ θs

(s ≤ t)

4.1 Introduction

231

then the martingale (Msh , s ≥ 0) takes the form : ∞ h(A(q) Msh = s + y)ν Xs (dy)

(4.27)

0

∞

(under Px , with :

h(y)ν x (dy) = 1).

In the particular case where h(z) := exp − μ2 z , we obtain : 0

Msh =

μ ϕμ q (Xs ) exp − A(q) ϕμ q (x) 2 s

(under Px )

As an extension of this example, we may replace the canonical Brownian motion (Xt ) by the Bessel process with dimension d = 2(1 − α), with 0 < α < 1, which led us to the following results : ∞ Let q : R+ → R+ be measurable, and satisfy 0 < 0 (1 + x)q(x)dx < ∞.

t (q) Let : At := 0 q(Xs )ds, and consider h(z) := exp − z2 . Then, we have obtained : ϕ(α) (Xs ) 1 s α h λ(t) = t , Ms = (α) q(Xu )du (under Px ) exp − 2 0 ϕ (x) (4.28) where the function ϕ(α) may be deﬁned by : 1 t ϕ(α) (x) = lim tα Ex exp − q(Xu )du (4.29) t→∞ 2 0 ii) We come back once again to the Brownian set up, but we consider now (Γt , t ≥ 0) the local time process, taking values in E = C(R → R+ )and deﬁned by : Γt (X) := (Lyt ; y ∈ R) Let H : E → R+ . We have obtained under some additional conditions on H : √ y H()Λx (d) (4.30) λ(t) = t and λ(t)Ex [H(Lt ; y ∈ R)] −→ t→∞

E

where the measure Λx is described in detail in Chapter 2. Furthermore : MsH = H(L•s + )ΛXs (d) (4.31) E

The preceding example i) - in the Brownian set up - is a particular case of that considered here in ii), by choosing : μ μ ∞ (y)q(dy) ( ∈ E) H() = exp − < q, > := exp − 2 2 −∞ (4.32)

232

4 A General Principle and Some Questions about Penalisations

and where the measure ν x is the image of Λx by the application : i(q) : E → R+ which is deﬁned by i(q) () :=< q, >. Example 4.4. (see Chapter 3). (Xt ) now denotes the Bessel process with dimension d = 2(1 − α), with 0 < −→ (1) (2) (n) α < 1. Let gt := sup{s ≤ t ; Xs = 0} and Vgt := (Vgt , Vgt , ..., Vgt , ...) denote the sequence of lengths of the excursions before gt , ranked in decreasing order. Let now : −→ Γt = Vgt α := ((Vg(1) )α , (Vg(2) )α , ..., (Vg(n) )α , ...) t t t

(4.33)

(Γt , t ≥ 0) is a process taking values in S ↓ := {σ = (σ1 , σ2 , ..., σn , ...); σ1 ≥ σ2 ≥ ... ≥ 0} Then, we have obtained, for λ(t) = tα and H : S ↓ → R+ , satisfying some adequate conditions that : −→ λ(t)E0 [H(Vgt α )] −→ H(σ)Π(dσ) (4.34) t→∞

S↓

where the measure Π and the explicit form of the martingale (MsH , s ≥ 0) are described in Chapter 3.

4.2 Asymptotic Studies of (Mth, t ≥ 0), h(Γt ) h and of Mt − Ex [h(Γt )] , t ≥ 0

h(Γt ) ,t Ex [h(Γt )]

≥0

4.2.1 Throughout this section 4.2, we take for (Γt , t ≥ 0) a process (At , t ≥ 0) with real values and we assume the following hypotheses : • H1) The process (At , t ≥ 0) satisﬁes the penalisation principle. • H2) (At , t ≥ 0) is an increasing process, which is Ft adapted and such that A∞ = ∞ , Px a.s., for every x. The hypotheses H1) and H2) are satisﬁed in Examples 4.2 i), ii), iii) and iv), 4.3 i). This is also the case, using the notation in Example 4.4, for the process : (n) (At , t ≥ 0) = ((Vgt )α , t ≥ 0), n ﬁxed, n ≥ 1, as well as (see Chapter 3) for : (n) α (At , t ≥ 0) = ((Vt ) , t ≥ 0) for n ≥ 2. Theorem 4.5. We assume that H1) and H2) are satisﬁed. Then, with the notation of the Generic Theorem, and with Γt = At , one has : 1) Mth −→ 0 Px a.s. for every x; t→∞

thus, the family (Mth , t ≥ 0) is not uniformly integrable.

4.2 Asymptotic Studies of (Mth , t≥0),

2) λ(t)h(At ) −→ 0 t→∞

h(Γt ) Ex [h(Γt )] , t≥0

t) and Mth − Exh(Γ [h(Γt )] , t≥0

233

in Px probability for every x;

consequently, the family (λ(t)h(At ), t ≥ 0) is not uniformly integrable. Proof of Theorem 4.5 1) We ﬁrst show that : Mth −→ 0 a.s. t→∞ Without loss of generality, we may assume that x = 0, and : h(y)ν 0 (dy) = 1

(4.35)

R

Since the measure ν 0 is σ-ﬁnite, there exists a function ψ : R → R+ which is increasing, and which goes to +∞, as y → +∞, such that : h(y)ψ(y)ν 0 (dy) < ∞ (4.36) R

Let, for every p > 0, Up := inf{s ≥ 0, As > p}

(4.37)

From H2), Up is an a.s. ﬁnite stopping time, and Up −→ ∞ a.s. We shall p→∞ prove : C E[MUhp ] ≤ (4.38) ψ(p) It follows from (4.38) that Mth −→ 0 a.s. Indeed, since (Mth , t ≥ 0) is a posit→∞

tive martingale, it converges a.s. as t → ∞, and (4.38) implies that this limit equals 0. We now prove (4.38) From (4.7), since R h(y)ν 0 (dy) = 1, we have λ(t)E0 [1Λs h(At )] −→ E0 [1Λs Msh ] t→∞

(4.39)

We apply (4.39) with Λs = (Up < s) (note that : (Up < s) ∈ Fs from H2)). We obtain : ls,t := λ(t)E0 [1(Up <s) h(At )] −→ E0 [1(Up <s) Msh ] t→∞

(4.40)

Now, with s < t, the left hand side ls,t of (4.40) satisﬁes : ls,t ≤ λ(t)E0 [1(Up
(4.41)

234

4 A General Principle and Some Questions about Penalisations

from (4.5). Gathering (4.41) and (4.40), we obtain : E0 [1(Up <s) Msh ] = E0 [1(Up <s) MUhp ] ≤

1 ψ(p)

R

h(y)ψ(y)ν 0 (dy)

(4.42)

where the ﬁrst equality in (4.42) follows from Doob’s optional stopping Theorem. It remains to let s tend to +∞ in (4.42) to obtain (4.38), after observing that Up < ∞ P0 a.s. from H2). 2) (Mth , t ≥ 0) is not uniformly integrable Indeed, if it were, then we would have : h |Ft ] = 0 Mth = E0 [M∞

which is absurd. 3) We prove that λ(t)h(At ) −→ 0 in Px probability t→∞

As previously, we may take x = 0 and we may assume that With the same notation as above, we have, for δ > 0

R

h(y)ν 0 (dy) = 1.

P0 [λ(t)h(At ) > δ] = P0 [λ(t)h(At )ψ(At ) > ψ(At )δ] = P0 [(λ(t)h(At )ψ(At ) > ψ(At )δ) ∩ (ψ(At ) < M )] +P0 [(λ(t)h(At )ψ(At ) > ψ(At )δ) ∩ (ψ(At ) > M )] (4.43) (for every M > 0) ≤ P0 [ψ(At ) < M ] + P0 [λ(t)h(At )ψ(At ) > M δ] λ(t) E0 [h(At )ψ(At )] ≤ P0 [ψ(At ) < M ] + Mδ

(4.44)

Thus, from (4.5), we obtain : lim P0 [λ(t)h(At ) > δ] ≤ lim P0 [ψ(At ) < M ] +

t→∞

t→∞

R

h(y)ψ(y)ν 0 (dy) Mδ

However, since : At → ∞ a.s. and ψ(y) → ∞ we have : y→∞

lim P0 [ψ(At ) < M ] = 0

t→∞

Hence :

lim P0 [λ(t)h(At ) > δ] ≤

t→∞

R

h(y)ψ(y)ν 0 (dy) Mδ

which implies the result, since M is arbitrary.

(4.45)

4.2 Asymptotic Studies of (Mth , t≥0),

h(Γt ) Ex [h(Γt )] , t≥0

t) and Mth − Exh(Γ [h(Γt )] , t≥0

235

4) We show that the family (λ(t)h(At ), t ≥ 0) is not uniformly integrable Indeed, if it were, then the convergence in probability to 0 would imply the convergence in L1 (to 0) and we would have : λ(t)E0 [h(At )] −→ 0 t→∞

which is contradictory with (4.5). Remark 4.6. Under the hypothesis of Theorem 4.5 and with the same notation one has, for any γ ∈]0, 1[ : h(At ) γ (4.46) Ex Mth − −→ 0 t→∞ Ex [h(At )] h(At ) γ Indeed, the family Mth − , t ≥ 0 is uniformly integrable, as it Ex [h(At )] is bounded in L1/γ since : % 1/γ & h(At ) γ h(At ) h Ex = Ex Mth − Mt − Ex [h(At )] Ex [h(At )] h(At ) h ≤ Ex Mt + =2 Ex [h(At )] On the other hand, from Theorem 4.5, Mth −

h(At ) γ converges to 0 in Ex [h(At )]

probability as t → ∞, hence (4.46) follows. We shall show below (see Theorems 4.10 and 4.12) that it is not possible in general to take γ = 1 in (4.46). 4.2.2 Here are some other hypotheses which yield the convergence to 0 in probability, of λ(t)h(At ), as t → ∞. Theorem 4.7. Let (Γt , t ≥ 0) denote a process which satisﬁes the penalisation principle. Let h be such that : 0< h(y)ν x (dy) < ∞ (4.47) E

We further assume, with the notation of the Generic Theorem that : Mth −→ 0 t→∞

Px a.s.

(4.48)

Then : λ(t)h(Γt ) −→ 0 t→∞

in Px probability.

(4.49)

236

4 A General Principle and Some Questions about Penalisations

Proof of Theorem 4.7 To simplify, we assume that x = 0 and E h(y)ν 0 (dy) = 1. To prove (4.49), it is equivalent to show : λ(t)h(Γt ) E0 −→ 0 1 + λ(t)h(Γt ) t→∞

(4.50)

x is concave on R+ (indeed its second 1+x −3 derivative equals : −2(1 + x) ), Jensen’s inequality implies, for s < t : E0 [λ(t)h(Γt )|Fs ] λ(t)h(Γt ) ≥ E0 (4.51) Fs 1 + λ(t)E0 [h(Γt )|Fs ] 1 + λ(t)h(Γt ) Now, since the function : x →

Hence, taking expectations on both sides of (4.51), and using (4.5), we obtain : λ(t)h(Γt ) Msh lim E0 (4.52) ≤ E0 t→∞ 1 + λ(t)h(Γt ) 1 + Msh from Lebesgue’s dominated convergence Theorem. Now, since Msh → 0 a.s., s→∞

then, by letting s tend to +∞ in (4.52), and applying Lebesgue’s dominated convergence again, we obtain : λ(t)h(Γt ) lim E0 =0 t→∞ 1 + λ(t)h(Γt ) Remark 4.8. Let us observe that Theorem 4.7 admits no converse statement. Let f : R → R+ be a Borel function such that ∞ f (x)dx = 1 (4.53) −∞

Let us penalise Wiener’s measure Px with the weight process (f (Xt )). We have, in succession : 2 ∞ 1 y i) Ex [f (Xt )] = √ f (x + y) exp − 2t 2πt −∞ ∞ 1 1 (4.54) (from (4.53)) ∼ √ f (x + y)dy = √ t→∞ 2πt −∞ 2πt √ √ ii) 2πtEx [f (Xt )|Fs ] = 2πtEXs [f (Xt−s )] −→ 1 (4.55) t→∞

from (4.54). Consequently, in this situation, the Generic Theorem applies with λ(t) = √ 2πt and Msf ≡ 1 for every s. Thus, the penalisation eﬀect is trivial, and ν 0 is Lebesgue measure on R.

4.2 Asymptotic Studies of (Mth , t≥0),

h(Γt ) Ex [h(Γt )] , t≥0

t) and Mth − Exh(Γ [h(Γt )] , t≥0

237

√ On the other hand, tf (Xt ) tends to 0 in probability, as t → ∞. Indeed, we have : √ ∞ f (x) exp − x2 dx 2t tf (Xt ) 1 √ √ E0 =√ 1 + tf (Xt ) 1 + tf (x) 2π −∞ which goes to 0, as t → ∞, by Lebesgue’s dominated convergence Theorem. Of course, the fact that nonetheless (Mth ) does not converge to 0 a.s., as t → ∞, does not contradict Theorem 4.5, since the process (Xt , t ≥ 0) is obviously not increasing. 4.2.3 All along our works on penalisation, we have noticed the importance of the scaling property. Here is yet another illustration of this, which generalizes the preceding remark. Theorem 4.9. Let (Γt , t ≥ 0) denote a weight process taking values in Rk such that : • H1) (Γt , t ≥ 0) satisﬁes the penalisation principle (law)

• H3) there exists α > 0 such that under P0 , Γt = tα Γ1 and Γ1 admits a positive density fΓ1 , which is bounded and such that fΓ1 (0+ ) > 0

(4.56)

Then, with the notation of the Generic Theorem, one has : λ(t) = tαk

1)

and

ν 0 (dy) = fΓ1 (0+ )dy (4.57)

2) For every function h : R → R+ , which is Lebesgue integrable, one has : k

tαk h(Γt ) −→ 0 t→∞

in P0 probability

(4.58)

We observe that the hypothesis H3) is satisﬁed in many of our examples : (0)

(1)

(2)

(3)

Γt = Xt , Γt = Lt , Γt = St , Γt = Sgt , and so on... Proof of Theorem 4.9 1) We ﬁrst prove (4.57) For every h : Rk → R+ integrable, one has : ∞ E0 [h(Γt )] = E0 [h(tα Γ1 )] = h(tα y)fΓ1 (y)dy k R x dx h(x)fΓ1 α αk = t t Rk

(4.59)

We then apply the dominated convergence Theorem, and H3), to obtain : αk t E0 [h(Γt )] −→ fΓ1 (0+ ) h(x)dx (4.60) t→∞

Rk

238

4 A General Principle and Some Questions about Penalisations

2) We now prove point 2) of Theorem 4.9, which amounts to showing : αk t h(Γt ) (4.61) E0 −→ 0 1 + tαk h(Γt ) t→∞ By scaling, we have : αk t h(Γt ) tαk h(tα x) E0 fΓ1 (x)dx = αk α 1 + tαk h(Γt ) Rk 1 + t h(t x) tαk h(tα x)dx ≤C since fΓ1 is bounded by C αk α Rk 1 + t h(t x) h(y)dy (changing variables : y = tα x) ≤C αk Rk 1 + t h(y) which converges to 0, by another application of the dominated convergence Theorem. 4.2.4 Asymptotic study of the family − λ(t)h(Γt )|, t ≥ 0) : a case study We have shown, in Theorem 4.5, that under some reasonable hypotheses, h(Γ ) t , t ≥ 0 is uniformly integrable. What neither (Mth , t ≥ 0), nor Ex(h(Γt )) h(Γt ) h can be said about the family Mt − , t ≥ 0 ? Ex (h(Γt )) We shall ﬁrst answer this question for an example (see Theorem 4.10) then we shall consider a more general framework (see Theorem 4.12). In the framework of the following Theorem 4.10, we consider : (|Mth

• (Xt , t ≥ 0) Brownian motion, starting from 0, • (Γt , t ≥ 0) = (St := sup Xs , t ≥ 0), s≤t ∞ • h : R+ → R+ such that h(y)dy = 1. 0

Thus, with the notation of the Generic Theorem, we have : πt , ν 0 (dy) = 1[0,∞[ (y)dy λ(t) = 2 ∞ Msh = h(Ss )(Ss − Xs ) + h(y)dy

(4.62)

Ss

Under these conditions, the following holds : Theorem 4.10. 1) There is the limit lim E0 [|λ(t)h(St ) −

t→∞

Mth |]

=

∞

e 0

u du 1 − 2 π

−u

(4.63)

4.2 Asymptotic Studies of (Mth , t≥0),

h(Γt ) Ex [h(Γt )] , t≥0

t) and Mth − Exh(Γ [h(Γt )] , t≥0

239

2) None of the three families : (Mth , t ≥ 0) , (λ(t)h(St ), t ≥ 0) , (|λ(t)h(St ) − Mth |, t ≥ 0) is uniformly integrable. Ampliﬁcation of Theorem 4.10 : Hellinger distances (t) Instead of computing the limit as t → ∞, of the total variation between P0 (h) and Q0|Ft , we may also compute the limit as t → ∞ of the Hellinger distance H between these two probabilities. We have : ! (t) H(P0 , Qh0|Ft ) := E[( λ(t)h(St ) − Mth )2 ] More generally, for α ∈]0, 1[, let us consider : (t)

Hα (P0 , Qh0|Ft ) := E[((λ(t)h(St ))α − (Mth )α )1/α ] This expression is equivalent to : E[|(λ(t)h(St ))α − (h(St )(St − Xt ))α |1/α ] α 1/α √ √ π √ α h( tS1 ) = tE − (h( tS1 )(S1 − X1 )) 2 1/α π α √ 1 √ = t du E h( tuR1 ) − (1 − u)α R1α 2 0 α √t 1/α π α/2 x α = dxE h(xR1 ) − 1− √ R1 2 t 0 1/α 1 π α/2 ∼ E − R1α t→∞ R1 2 α π 1 1/α = E 1 − 2 R1 We have used (see Chapter 1) the fact that : (law)

(S1 , S1 − X1 ) = (U R1 , (1 − U )R1 ) where on the RHS, U and R1 are independent, U is uniform on [0, 1] and R1 is the value of a 3-dimensional Bessel process at time 1, starting from 0. Hence, no matter which distance Hα (0 < α ≤ 1) we consider, the proba(t) bilities P0 and Qh0|Ft remain far apart as t → ∞. Furthermore, this limiting distance is precisely (3)

Hα (P0 , M) , (3)

where P0 resp. M is the law of the 3-dimensional Bessel process resp. the Brownian meander, both processes being indexed by t ∈ [0, 1].

240

4 A General Principle and Some Questions about Penalisations

Proof of Theorem 4.10 We shall give two proofs for point 1). 1. First proof Denote by ρt the quantity of interest here : ρt := E0 [|λ(t)h(St ) − Mth |]

(4.64)

∞ Since : E0 [ St h(y)dy] −→ 0, it suﬃces to study ρt , which is deﬁned by : t→∞

ρt :=

πt E0 [h(St )|1 − (E0 (h(St )))(St − Xt )|] 2

Since the density ft of the pair (Xt , St ) is well known to be : 2 (2b − a)2 1(a0) , ft (a, b) = (2b − a) exp − 3 πt 2t

(4.65)

(4.66)

we obtain :

∞ b 2 (2b − a)2 × ... db da(2b − a) exp − ρt = π t3 0 2t −∞ . . . h(b)|1 − E0 (h(St ))(b − a)| (4.67) 2 √ ∞ b (2b − a) × ... = t db da(2b − a) exp − 2 0 −∞ √ √ . . . h(b t)|1 − E0 (h(St ))(b − a) t| (4.68) √ √ (after making the change of variables : b = b t, a = a t) 2 c/√t ∞ 2c 1 2c √ − a × ... h(c)dc da √ − a exp − = 2 t t 0 −∞ √ c (4.69) t . . . 1 − E0 (h(St )) √ − a t √ (after making the change of variables : c = b t) πt 2

Now, we have : E0 (h(St ))

√ √ c c 2 2 √ −a √ −a a t ∼ t −→ − t→∞ t→∞ πt π t t

(4.70)

Hence : 2 a2 ρt −→ h(c)dc da(−a) exp − 1 + a t→∞ 2 π 0 −∞ 2 ∞ ∞ 2 x a = a exp − exp(−x)1 − 2 da = dx 1 − a 2 π π 0 0 a2 (after making the change of variables : x = ) 2

∞

0

4.2 Asymptotic Studies of (Mth , t≥0),

h(Γt ) Ex [h(Γt )] , t≥0

t) and Mth − Exh(Γ [h(Γt )] , t≥0

241

We note that :

∞

0

x π 1 exp(−x)1 − 2 dx = E 1 − π 2 R1

(4.71)

where R1 denotes the value at time 1 of a 3-dimensional Bessel process starting from 0. The reader may ﬁnd it interesting to refer to Theorem 4.18 below in order to understand better how this constant appears. 2. Second proof This time, we shall use L´evy’s equivalence Theorem (see Chapter 1, Item B ) : (law)

((St , St − Xt ), t ≥ 0) = ((Lt , |Xt |), t ≥ 0) where (Lt , t ≥ 0) denotes the local time at 0 of the Brownian motion (Xt , t ≥ 0). Thus, we obtain :

πt E0 [h(St )|1 − (E0 (h(St )))(St − Xt )|] 2 πt E0 [h(Lt )|1 − (E0 (h(Lt )))|Xt ||] = 2

ρt =

Moreover, we have : √ √! (law) √ √ √ (Lt , |Xt |) = ( t g1 2e , t 1 − g1 2e )

(4.72)

Concerning formula (4.72), the reader may refer to Chapter 1, Item G , which is relative to the Brownian meander, or to Chapter 2, 2.2.3. In formula (4.72), e, e and g1 denote three independent r.v.’s, e and e are two standard exponential variables and g1 follows the arc sine law. Thus : ρt =

πt 2

∞

e−x dx

0

1

0

! dγ ! [h( 2xtγ) × . . . π γ(1 − γ)

! . . . |1 − E0 (h(Lt )) t(1 − γ)2e |] 1 = √ 2π

0

∞

e−x dx

t 0

√ du

[h( 2xu) × . . . u 1 − ut

! . . . |1 − E0 (h(Lt )) (t − u)2e |] (after making the change of variables : u = γt)

(4.73)

242

4 A General Principle and Some Questions about Penalisations

It is easily shown that : !

Hence :

√ ∞ 2 t 2 (t − u)E0 (h(Lt )) ∼ √ h(u)du = t→∞ π 2πt 0

(4.74)

∞ ∞ 1 du √ 2 √ −x √ h( 2xu)E 1 − ρt −→ √ e dx 2e t→∞ π u 2π 0 0 2 ∞ e−x dx e √ = E 1 − 2 π 0 π 2x √ (after making the change of variables : 2xu = v) ∞ √ 1 x −x − 2 dx since Γ = = π e 1 π 2 0

3. We now show that none of the three families : (Mth , t ≥ 0), (λ(t)h(St ), t ≥ 0), (|λ(t)h(St ) − Mth |, t ≥ 0) is uniformly integrable. Concerning the two ﬁrst families, this results from Theorem 4.5. Concerning the third family (|λ(t)h(St ) − Mth |, t ≥ 0), assuming it were uniformly integrable then, from Theorem 4.5, it would converge to 0 in L1 , which is contradictory with point 1) of Theorem 4.10. Remark 4.11. 1) One may compare point 1) of Theorem 4.10 with the relation (4.46) of Remark 4.6. 2) It may be shown directly - i.e : without using Theorem 4.5 - that the family h (|λ(t)h(S t ) − Mt |, t ≥ 0) is not uniformly integrable. Indeed, the family ∞ ( St h(y)dy, t ≥ 0) is uniformly bounded, hence it is uniformly integrable; consequently, it suﬃces to prove that for any c > 0 : lim E[h(St )|(St − Xt ) − γt |1h(St )|(St −Xt )−γt |>c ] > 0

t→∞

where : γt :=

1 ∼ E0 (h(St )) t→∞

πt 2

(4.75)

(4.76)

We now show (4.75) Let δt := E0 [h(St )|(St − Xt ) − γt |1h(St )|(St −Xt )−γt |>c ] √ √ π √ = tE0 h( tS1 )(S1 − X1 ) − (4.77) 1h( tS1 )|(S1 −X1 )−√ π |> √c 2 t 2 from the scaling property.

4.2 Asymptotic Studies of (Mth , t≥0),

t) and Mth − Exh(Γ [h(Γt )] , t≥0

h(Γt ) Ex [h(Γt )] , t≥0

243

We make upon h the extra hypothesis : h(x) ≥ α1[0,β] (x)

(α, β > 0)

(4.78)

In fact, this hypothesis is not really needed : it only helps to make our arguments a little simpler. From (4.78), we have : x < β ⇒ h(x) > α

(4.79)

Hence, we obtain : √ √ π √ √ δt ≥ tαE0 h( tS1 )(S1 − X1 ) − 1 tS1 <β 1|S1 −X1 − π |> c√ 2 2 α t √ β/ t b √ 1 (2b − a)2 = tα √ db (2b − a) exp − √ 2 2π 0 (β+ αc ) √1t − π2 π |da |b − a − 2 (with the help of the law of the pair (X1 , S1 ) given by (4.66)) 2 β/√t 2b−(β+ c ) √1 +√ π √ α 2 t π 1 x = |dx tα √ db x exp − |x − b − 2 2 2π 0 b (from the change of variables 2b − a = x) 2 √π/2 αβ π x −→ √ |dx > 0 x exp − |x − t→∞ 2 2 2π 0 4) In the same spirit as for point 1) of Theorem 4.10, we can show that, for two probability density functions h, g : R+ → R+ , one has : ∞ g h |h(y) − g(y)|dy (4.80) lim E0 [|Mt − Mt |] = t→∞ 0 ∞ and lim λ(t)E0 [|h(St ) − g(St )|] = |h(y) − g(y)|dy (4.81) t→∞

0

We now show (4.80) We ﬁrst show that : E0 [|Mth − Mtg |] ∼ E0 [|h(St ) − g(St )|(St − Xt )] t→∞

from (4.62) and the fact that E0

∞ St

h(y)dy −→ 0 (and the same result for t→∞

g, instead of h) ∼ E0 [|h(Lt ) − g(Lt )||Xt |]

t→∞

with the help of L´evy’s equivalence Theorem.

(4.82)

244

4 A General Principle and Some Questions about Penalisations

On the other hand, for every bounded Borel function ϕ, we know that the y process (ϕ(Lt )|Xt | − Φ(Lt ), t ≥ 0) where Φ(y) := 0 ϕ(x)dx, is a martingale. Thus, we obtain : (4.83) E0 [ϕ(Lt )|Xt |] = E0 (Φ(Lt )) Using this result for ϕ = |h − g|, we get : ∞ 2 u 2 u g h E0 [|Mt − Mt |] ∼ du exp − |h(y) − g(y)|dy t→∞ πt 0 2t 0 ∞ −→ |h(y) − g(y)|dy t→∞

0

√ (after making the change of variables u = x t). On the other hand, the proof of (4.81), is completely elementary, and hinges (law) √ t|X1 |. on the fact that St = 5) With the help of L´evy’s equivalence Theorem (see Chapter 1, Item B ) once again, we may write point 1) of Theorem 4.10 in the form : ∞ u h (4.84) lim E0 [|λ(t)h(Lt ) − Mt |] = (exp(−u))1 − 2 du t→∞ π 0 with, this time :

Mth = h(Lt )|Xt | +

∞

h(y)dy

(4.85)

Lt

We then replace our reﬂecting Brownian motion (|Xt |, t ≥ 0) by the Bessel process with dimension d = 2(1 − α), with 0 < α < 1, and we obtain, with : ∞ α h 2α λ(t) = t , Mt := h(Lt )Xt + h(y)dy (4.86) Lt

(α) h lim E0 [|λ(t)h(Lt ) − Mt |] = E 1 −

t→∞

1 α e Γ(1 + α)

(4.87)

where e denotes a standard exponential variable. The proof of (4.87) uses the same arguments as the second proof of point 1) of Theorem 4.10. (law)

It hinges on the scaling property : (Lt , Rt2α ) = (tα L1 , tα R12α ) and upon : (law)

(L1 , R1 ) = (g1α 1 ,

!

√ 1 − g1 2e)

(4.88)

where in (4.88), g1 , 1 and e are assumed to be independent, and where e is a standard exponential, g1 is a beta (α, 1 − α)-variable, and 1 denotes the local time at 0 for the standard Bessel bridge with dimension d = 2(1 − α) (concerning (4.88), see Chapter 3)).

4.2 Asymptotic Studies of (Mth , t≥0),

h(Γt ) Ex [h(Γt )] , t≥0

t) and Mth − Exh(Γ [h(Γt )] , t≥0

245

4.2.5 Asymptotic study of the family (|λ(t)h(Γt ) − Mth |, t ≥ 0) in a general set up In our study of the penalisation by a function h of St , we have just shown that the family (|λ(t)h(St ) − Mth |, t ≥ 0) is not uniformly integrable. Our arguments relied essentially upon the explicit knowledge of the joint law of the pair (Xt , St ). We shall now develop a discussion which, as much as possible, is free from explicit knowledge of distributions. We begin with a remark : Let (Γt , t ≥ 0) denote a process which satisﬁes the penalisation principle. We take x = 0 and h such that : h(y)ν 0 (dy) = 1; therefore : λ(t)E0 [h(Γt )] −→ 1 t→∞

(4.89)

Thus, we have : Qh0

λ(t)h(Γt ) = E0 [λ(t)h(Γt )] −→ 1 t→∞ Mth

Consequently, the family of r.v.’s Qh0

λ(t)h(Γt ) ,t Mth

(4.90)

≥ 0 is tight under Qh0 , since :

λ(t)h(Γt ) 1 1 h λ(t)h(Γt ) > c ≤ Q0 −→ h h t→∞ c c Mt Mt

Thus, there exists a sequence (tn , n ≥ 0), tn −→ ∞ and a r.v. Z such that, n→∞

under Qh0 : λ(tn )h(Γtn ) (law) −→ Z n→∞ Mthn

(4.91)

We now consider the following hypothesis : • H4) the law of Z is not the Dirac measure at 1. This hypothesis H4) is satisﬁed in all our main examples and we shall discuss this point in detail in Section 4.4. Theorem 4.12. Let (Γt , t ≥ 0) be a process which satisﬁes both H1) and H4). Then : 1)

lim E0 (|λ(t)h(Γt ) − Mth |) > 0

t→∞

(4.92)

the 2) If, furthermore, hypothesis H2) of Theorem 4.5, (Γt , t ≥ 0) satisﬁes h(Γt ) h then the family Mt − , t ≥ 0 is not uniformly integrable. E0 (h(Γt ))

246

4 A General Principle and Some Questions about Penalisations

Proof of Theorem 4.12 Point 2) of Theorem 4.12 follows immediately from point 1) and the fact that h(Γt ) −→ 0 in probability. Mth − E0 (h(Γt )) t→∞ We now prove point 1) We have, for 0 < a < 1 (recall that h(y)ν 0 (dy) = 1) : E0 [|λ(t)h(Γt ) − Mth |] ≥ E0 [(λ(t)h(Γt ) − Mth )1Mth 1 ] ≥ a a Mth 1 − a h λ(t)h(Γt ) 1 Q0 = > a a Mth

(4.93)

In the same manner, for 0 < b < 1, we have : E0 [|λ(t)h(Γt ) − Mth |] ≥ E0 [(Mth − λ(t)h(Γt ))1λ(t)h(Γt )
(4.94)

Gathering (4.93) and (4.94), we obtain, for every continuous function ψ : R → [0, 1] such that ψ ≤ 1V (1)c , where V (1)c is the complement of a neighborhood of 1, the existence of a constant C > 0, such that : λ(t)h(Γt ) h h E0 [|λ(t)h(Γt ) − Mt |] ≥ C Q0 ψ Mth hence :

λ(tn )h(Γtn ) lim E0 [|λ(t)h(Γt ) − Mth |] ≥ lim C Qh0 ψ t→∞ n→∞ Mthn = CE[ψ(Z)] > 0

as soon as V (1) is chosen to be small enough, from H4). Remark 4.13. No care was needed to consider the process under Qh0 , since if T0h := inf{t : Mth = 0}, then :

(1/Mth , t

≥ 0)

Qh0 (T0h < ∞) = E0 [1(T0h <∞) MTh0 ] = 0 Thus, Qh0 (Mth > 0 , for every t ≥ 0) = 1.

t) 4.3 Asymptotic study of Ex 1Λ Exh(Γ (h(Γt )) for Λ ∈ F∞

247

t) for Λ ∈ F∞ 4.3 Asymptotic study of Ex 1Λ Exh(Γ (h(Γt ))

4.3.1 We begin by showing that, for some Λ ∈ F∞ , h(Γt ) lim Ex 1Λ may diﬀer from Qhx (Λ) t→∞ Ex (h(Γt ))

(4.95)

Clearly, it suﬃces to ﬁnd Λ ∈ F∞ such that : Px (Λ) = 1 and Qhx (Λ) = 0 h(Γt ) which implies that : for every t, Ex 1Λ = 1. Ex (h(Γt ))

(4.96)

satisﬁes the penalisation Theorem 4.14. Let (Γt , t ≥ 0) be a process which principle, and let h : E → R+ such that : 0 < E h(y)ν x (dy) < ∞. 1) (4.97) If Px {Mth −→ 0} = 1 , then : Qhx {Mth −→ ∞} = 1 t→∞

t→∞

(thus, the set Λ∗ := −→ 0) satisﬁes (4.96)). t→∞ Moreover, under the same hypothesis, we have, with U a uniform variable on [0, 1] : (Mth

1 U

under Px

inf Mth = U

under Qhx

(law)

sup Mth = t≥0

whereas

(law)

t≥0

2) If (Γt ≡ At , t ≥ 0) satisﬁes the hypothesis of Theorem 4.5, then, with the notation of Theorem 4.5 : Px (A∞ < ∞) = 0

and

Qhx (A∞ < ∞) = 1

(4.98)

(thus, the set Λ∗∗ := (A∞ = ∞) satisﬁes (4.96)). Proof of Theorem 4.14 1) For any b > 1, we have, with Tb := inf{t ≥ 0; Mth > b} : h Qhx (Tb < ∞) = Ex [1(Tb <∞) MThb ] = Ex [MThb ] (since M∞ = 0) = 1 , by Doob’s optional Theorem.

Thus : sup Mth = ∞, Qhx a.s. t≥0

Now, (1/Mth , t ≥ 0) is a positive martingale under Qhx ; hence, it converges Qhx a.s., and since sup Mth = ∞, it must be that : Mth −→ ∞, Qhx a.s. t≥0

t→∞

248

4 A General Principle and Some Questions about Penalisations

We also deduce from : 1 = Ex [1(Tb <∞) MThb ] = b Px (Tb < ∞) 1 under Px . U The analogous statement under Qhx is proven in the same manner, or simply by exchanging the roles of Qhx and Px . (law)

the fact that : sup Mth = t≥0

2) We now prove (4.98) By hypothesis, we have Px (A∞ < ∞) = 0. Thus, it remains to show that : Qhx (A∞ < ∞) = 1

(4.99)

∞

We may suppose, without loss of generality, that

h(y)ν x (dy) = 1. 0

For every a > 0, we have :

Qhx (As < a) = lim Ex [1(As
≥ lim Ex [1(At
since (At , t ≥ 0) is an increasing process. Thus : a h Qx (As < a) ≥ h(y)ν x (dy) (from (4.5)) 0

Hence, letting s → ∞ : Qhx (A∞ < a) ≥

a

h(y)ν x (dy) . 0

Finally, letting a → ∞, we obtain : Qhx (A∞

< ∞) =

∞

h(y)ν x (dy) = 1 , 0

which proves (4.99). 4.3.2 We shall now show that, despite Theorem 4.14, there are many sets Λ ∈ F∞ , such that : h(Γt ) (4.100) lim Ex 1Λ = Qhx (Λ) t→∞ Ex (h(Γt )) For ease of writing, we shall now consider r.v.’s Θ rather than indicators 1Λ .

t) 4.3 Asymptotic study of Ex 1Λ Exh(Γ (h(Γt )) for Λ ∈ F∞

249

Theorem 4.15. Let (Γt , t ≥ 0) be a process which satisﬁes the penalisation principle. Then, for every x and every function h such that 0 < E h(y)ν x (dy) < ∞, the set of r.v.’s Θ which satisfy h(Γt ) (4.101) lim Ex Θ = Qhx (Θ) t→∞ Ex (h(Γt )) is dense in L∞ (Ω, F∞ , Px ) and in L∞ (Ω, F∞ , Qhx ). Proof of Theorem 4.15 1) To any process (Zs , s ≥ 0), which is predictable and bounded, we associate, for λ > 0 : ∞ e−λs Zs ds

Z (λ) :=

(4.102)

0

We note that : i) Every r.v. Zu ∈ L∞ (Ω, Fu , Px ) may be written in the form (4.102), since, denoting Zs := Zu 1s≥u , s ≥ 0, we get : ∞ 1 (λ) := Z e−λs Zs ds = e−λu Zu λ 0 ii) The product of two r.v.’s of the form (4.102) is the sum of two r.v.’s of the same form, with the help of integration by parts. We denote by V the set of all linear combinations of elements in L∞ (Ω, F∞ , Px ) which may be written in the form (4.102). This set is stable by multiplication and contains L∞ (Ω, Fu , Px ), for every u ≥ 0. Hence, the “strong” version of the monotone class Theorem (see [M], T20, p.28) asserts that V is dense in L∞ (Ω, F∞ , Px ) and in L∞ (Ω, F∞ , Qhx ). 2) It remains to see that every variable Z in V satisﬁes (4.101). Now, every such Z in V may be “well” approximated as follows : for every s ≥ 0 there )s ∈ L∞ (Ω, Fs , Px ) (= L∞ (Ω, Fs , Qhx )) and ε(s) −→ 0 such that : exists Z s→∞

)s ||L∞ (Ω,F ,Qh ) ≤ ε(s) )s ||L∞ (Ω,F ,P ) + ||Z − Z ||Z − Z ∞ x ∞ x

(4.103)

Indeed, it suﬃces to show (4.103) for Z (λ) given by (4.102); then, we obtain : s ∞ ||Z (λ) − e−λv Zv dv||L∞ (Ω,F∞ ,Px ) ≤ || e−λv |Zv |dv||L∞ (Ω,F∞ ,Px ) 0

s

1 C ≤ e−λs sup ||Zv ||L∞ (Ω,F∞ ,Px ) ≤ e−λs λ λ v (4.104) since the process (Zv , v ≥ 0) is bounded (the same argument applies with Qhx s instead of Px ). Thus, to obtain (4.103), it suﬃces to choose Z)s = 0 e−λv Zv dv.

250

4 A General Principle and Some Questions about Penalisations

3) Let Z be given by (4.102). Then, from (4.103), we get : h(Γt ) h(Γt ) h ) − Qx (Z) ≤ Ex Zs − Qhx (Z)s ) Ex Z Ex (h(Γt )) Ex (h(Γt )) h(Γt ) )s |) ) +Ex |Z − Zs | + Qhx (|Z − Z Ex (h(Γt )) h(Γt ) ) (from (4.103)) ≤ E x Zs − Qhx (Z)s ) + ε(s) Ex (h(Γt )) Hence, from (4.8) : lim Ex Z

t→∞

h(Γt ) Ex (h(Γt ))

− Qhx (Z) ≤ ε(s)

Now, since ε(s) is arbitrary, we obtain : h(Γt ) lim Ex Z = Qhx (Z) t→∞ Ex (h(Γt )) for every Z of the form (4.102), hence for every Z ∈ V.

4.4 Convergence in Law, as t → ∞ of the Family (t) of Processes (Xu ; 0 ≤ u ≤ 1):= X√tut ; 0 ≤ u ≤ 1 Here, we study the convergence of the family of processes X•(t) under the (t) probability Px , resp. : under Qhx . We shall make this study for two examples : the ﬁrst one, in the framework of Feynman-Kac penalisations; the second one for penalisations with respect to a function of St . Unfortunately, we have no result of a general nature. 4.4.1 Study in the framework of a Feynman-Kac penalisation. We recall our notation and our results (see [RVY,I] or Chapter 2) : i) q is a positive Radon measure on R such that : ∞ (1 + |x|)q(dx) < ∞ 0<

(4.105)

−∞

(q) and (Γt , t ≥ 0) = (At := R Lyt q(dy), t ≥ 0). √ (q) ii) tEx exp − 12 At −→ ϕq (x), where ϕq is the unique solution of t→∞

ϕ = qϕ , ϕ (+∞) = −ϕ (−∞) =

2 π

(4.106)

(t)

4.4 Convergence in Law, as t → ∞ of the Processes (Xu ; 0 ≤ u ≤ 1)

251

From these relations, we deduce : ϕq (x) ϕq (x)

∼

x→±∞

1 x

ϕ (x) q sup x − 1 ≤ k ϕq (x) x∈R

and

(4.107)

iii) For every s ≥ 0 and every Λs ∈ Fs : ⎞ ⎛ (q) exp − 12 At ⎠ = Ex [1Λs Ms(q) ] := Q(q) lim Ex ⎝1Λs x (Λs ) (4.108) (q) t→∞ Ex exp − 12 At (q) and Ms :=

ϕq (Xs ) ϕq (x)

(q) exp − 12 As , s ≥ 0 is a positive martingale such

(q)

that M0 = 1, Px a.s. iv) Under Q(q) x , the canonical process (Xt , t ≥ 0) satisﬁes :

t

Xt = x + Bt + 0

ϕq (Xu ) du ϕq (Xu )

(4.109)

where (Bt , t ≥ 0) is a Q(q) x Brownian motion starting from 0. The process (Xt , t ≥ 0) is transient and satisﬁes : ∞ dy ϕ2q (y) dy −∞ ϕ2q (y) x

Q(q) x {Xt −→ +∞} := α(x) = ∞ t→∞

= 1−Q(q) x {Xt −→ −∞} (4.110) t→∞

Theorem 4.16. 1. Asymptotic study of the processes X (t) , as t → ∞ under Q(q) x (t) Under Q(q) converges in law as t → ∞ towards the process x , the process X def

(Ru(q) ; 0 ≤ u ≤ 1) = (εRu(ε) ; 0 ≤ u ≤ 1) where ε, a Bernoulli variable, R(1) and R(−1) are independent, with : P (ε = 1) = α(x) , P (ε = −1) = 1 − α(x) (i)

and where (Ru , 0 ≤ u ≤ 1), i = −1, +1 are two 3-dimensional Bessel processes starting from 0. Consequently, |X (t) | converges in law, as t → ∞, towards a 3-dimensional Bessel process starting from 0. (t)

2. Asymptotic study of the processes X (t) , under Px as t → ∞ (t) Under the probability Px deﬁned by : ⎞ ⎛ (q) exp − 12 At ⎠ Px(t) (Λ) = Ex ⎝1Λ (q) Ex exp − 12 At

(4.111)

252

4 A General Principle and Some Questions about Penalisations (q)

the process X (t) converges in law, as t → ∞, towards the process (mu , 0 ≤ (ε) u ≤ 1) := (ε mu , 0 ≤ u ≤ 1), where ε, m(1) and m(−1) are independent, (i) P (ε = 1) = α(x), P (ε = −1) = 1 − α(x) and where (mu , 0 ≤ u ≤ 1), i = −1, +1 are two Brownian meanders (see, Chapter 1, Item G ). Remark 4.17. Thus, the limits in law of X (t) , as t → ∞, exist under both (t) Q(q) x and Px (see question 4.1.4v) in the introduction to this Chapter) and they are diﬀerent. Moreover, they depend very little on q and they are also very much alike, owing to Imhof’s absolute continuity relationship : 2 (3) (4.112) E0 (F (Ru , 0 ≤ u ≤ 1)) = E[F (mu , 0 ≤ u ≤ 1)•m1 ] π Proof of Theorem 4.16 1) The equation (4.109), becomes, after scaling : √ x = √ + Bu(t) + t t

ϕq (Xvt ) dv (4.113) 0 ϕq (Xvt ) √ √ u ϕq ( tXv(t) ) x (t) = √ + Bu + t (4.114) √ (t) dv t 0 ϕq ( tXv ) (t) where (Bu , 0 ≤ u ≤ 1) = √1t But , 0 ≤ u ≤ 1 is a Brownian motion starting Xu(t)

u

from 0. Now, using (4.107) and the fact that (Xt ) is transient under Q(q) x , we get : √ √ (t) tϕq (Xvt ) 1 ϕq ( tXv ) √ (t) = √ (t) • tXv (t) ϕq (Xvt ) Xv ϕq ( tXv ) 1 ∼ a.s. t→∞ X (t) v

and Xv(t)

√ (t) ϕq ( tXv ) 1 √ (t) − (t) ϕq ( tXv ) Xv

:= ε(v, t)

(4.115) (4.116)

(4.117)

with ε(v, t) −→ 0 a.s. t→∞

and |ε(v, t)| < k (t)

(4.118)

We now apply Itˆ o’s formula to (4.114) to compute (Xu )2 . After introducing : (t) (t) u := sgnXu •dBu(t) , and using (4.118), we obtain : dB u u x2 (t) 2 (t) (t) (Xu ) = +2 |Xs |dBs + 2 (1 + ε(v, t))dv + u (4.119) t 0 0

(t)

4.4 Convergence in Law, as t → ∞ of the Processes (Xu ; 0 ≤ u ≤ 1)

253 (t)

We then deduce from (4.119) that the family of processes ((Xu )2 , 0 ≤ u ≤ 1) is tight and that every limiting process obtained from this family satisﬁes : u! 2 s + 3u Xu = 2 Xs2 dB (0 ≤ u ≤ 1) (4.120) 0

s , s ≤ 1) is a Brownian motion; hence, (Xu2 , u ≤ 1) is a squared 3where (B dimensional Bessel process, starting from 0. In order to conclude, it remains (t) to observe that the sign of Xu = √1t Xut is constant a.s. for all t’s bigger than a certain t0 (ω) and that this sign is positive with probability α(x) and negative with probability 1 − α(x) owing to (4.110). 2) We now prove point 2) of Theorem 4.16 Let F : C([0, 1] → R) → R+ be a continuous bounded functional. Our aim is to study the limit, as t → ∞, of : ρ(t) := Px(t) (F (X (t) )) ⎡

⎤ (q) exp − 12 At ⎦ := Ex ⎣F (X (t) ) (q) Ex exp − 12 At

(4.121)

Thanks to (4.108), we may write ρ(t) in the form : F (X (t) ) ϕq (x) (q) Qx ρ(t) = (q) ϕq (Xt ) Ex exp − 12 At

(4.122)

Now, from (4.106), we have : √

tEx

ϕq (x) −→ 1 (q) t→∞ exp − 12 At

(4.123)

and, from (4.107) and (4.109) : 1 ∼ ϕq (Xt ) t→∞

2 |Xt | π

Q(q) x a.s.

(4.124)

Hence : F (X (t) ) √ ρ(t) ∼ t t→∞ ϕq (Xt ) π (t) ∼ Q(q) (G(X )) • x t→∞ 2

Q(q) x

(4.125) (4.126)

254

4 A General Principle and Some Questions about Penalisations

where the functional G is deﬁned by : G(Xv , 0 ≤ v ≤ 1) = F (Xv , 0 ≤ v ≤ 1)

1 |X1 |

(4.127)

Hence, with the help of point 1) of Theorem 4.16, we obtain : ρ(t) −→

t→∞

% & π 1 (ε) E F (ε Ru , 0 ≤ u ≤ 1)• (ε) 2 |R |

(4.128)

1

which proves point 2) of Theorem 4.16, thanks to Imhof’s relation (see Chapter 1, Item G ) : E[F (mu , 0 ≤ u ≤ 1)] =

π (3) 1 E F (Ru , 0 ≤ u ≤ 1) 2 0 R1

(4.129)

4.4.2 Study in the framework of a penalisation by a function of St ∞ Let h : R+ → R+ be such that : 0 h(y)dy = 1. Recall that we have (see Example 4.2), for every s ≥ 0, and Λs ∈ Fs : h(St ) (4.130) lim E0 1Λs = E0 (1Λs Msh ) := Qh0 (Λs ) t→∞ E0 (h(St )) ∞ where (Msh := h(Ss )(Ss − Xs ) + Ss h(y)dy, s ≥ 0) is a positive martingale such that : P0 a.s. (4.131) M0h = 1 Theorem 4.18. 1. Asymptotic study of X (t) under Qh0 Under Qh0 , the process X (t) converges in law, as t → ∞, towards the opposite of a 3-dimensional Bessel process starting from 0. (t) 2. Asymptotic study of X (t) under P0 (t) Under P0 , X (t) converges in law, as t → ∞, towards the opposite of a Brownian meander. We have stated this Theorem for x = 0, but it remains valid for every x. Proof of Theorem 4.18 We shall provide two proofs. First proof 1. We ﬁrst prove point 1. We shall use the two following results relative to Qh0 (see [RVY,IV]) : i) S∞ < ∞ a.s. (see point 2) of Theorem 4.14)

(4.132)

ii) the process (Rt := 2St − Xt , t ≥ 0) is a 3-dimensional Bessel process, starting from 0, and independent of S∞ (4.133)

(t)

4.4 Convergence in Law, as t → ∞ of the Processes (Xu ; 0 ≤ u ≤ 1)

255

Consequently, we may write : 1 1 Xu(t) = √ Xtu = √ (2Stu − Rtu ) t t

(0 ≤ u ≤ 1)

(4.134)

However, 2Stu √ −→ 0 Qh0 a.s., (from (4.132)) (4.135) t t→∞ and √1t Rtu , 0 ≤ u ≤ 1 is a 3-dimensional Bessel process, starting from 0, using the scaling property satisﬁed by R. Point 1) of Theorem 4.18 follows immediately. 2. We now prove point 2. Let F : C([0, 1] → R) → R+ denote a continuous and bounded functional, and let gε : R+ → R+ denote a continuous function such that : 0 ≤ gε (x) ≤ 1 for x ≥ 0 , gε (x) = 0 for x < ε , gε (x) = 1 for x ≥ 2ε (4.136) (t)

We deduce from the deﬁnition of the probability P0 that : St − Xt (t) √ E0 F (X (t) )gε t St − Xt h(St ) √ = E0 F (X (t) )gε E0 (h(St )) t & % ∞ Mth − St h(y)dy St − Xt (t) √ = E0 F (X )gε (E0 (h(St )))(St − Xt ) t

(4.137)

from ∞ (4.131). We note that, in (4.137), the term which corresponds to St h(y)dy does not contribute asymptotically. Indeed : & ∞ St − Xt St h(y)dy √ lim E0 F (X )gε t→∞ (E0 (h(St )))(St − Xt ) t √ ∞ St − Xt t √ ≤ lim K E0 gε h(y)dy t→∞ St − Xt t St ∞ gε (S1 − X1 ) ≤ lim K E0 h(y)dy =0 √ t→∞ S1 − X1 tS1 %

(t)

since the function y →

gε (y) y

is bounded. Thus, from (4.137), we deduce :

St − Xt h(St ) √ E0 F (X (t) )gε E0 (h(St )) t √ St − Xt 2 t (t) h √ E0 F (X )gε M ∼ t→∞ π St − Xt t t

256

4 A General Principle and Some Questions about Penalisations

√ St − Xt 2 h t √ Q0 F (X (t) )gε by deﬁnition of Qh0 ; π St − Xt t 2 (3) 1 E −→ F (−Ru ; 0 ≤ u ≤ 1)gε (R1 ) t→∞ π 0 R1 (from the preceding point 1), using the notation in (4.133)) = E(F (−mu ; 0 ≤ u ≤ 1)gε (m1 )) (4.138) =

from Imhof’s relation (see Item G , in Chapter 1) and where m denotes the Brownian meander. We note again that we were able to pass to the limit to obtain (4.138) because : y → gεy(y) is continuous and bounded. We shall now show how to get rid of gε Indeed : St − Xt (t) √ lim E0 F (X (t) ) 1 − gε t→∞ t √ St − Xt √ ≤ lim K E0 1 − gε th(St ) t→∞ t √ √ ≤ lim K E0 [(1 − gε (S1 − X1 )) th( tS1 )] t→∞

(4.139)

(law)

Using the identity (S1 , S1 − X1 ) = (U R1 , (1 − U )R1 ), we obtain that the expression in (4.139) is : ∞ dx h(xR1 )1(R1 ≤2ε) ≤KE 0 1 =KE 1(R1 ≤2ε) R1 which goes to 0 as ε → 0 since : E R11 < ∞. Point 2) of Theorem 4.18 now follows easily from (4.138) and (4.139). Second proof of Theorem 4.18 1. We begin with a reminder from enlargement theory (see [MY], Table 2, p.34). We denote by (Gt , t ≤ 1) the ﬁltration (Ft , t ≤ 1) initially enlarged with the r.v. S1 : (4.140) Gt = Ft ∨ σ(S1 ) Let γ : R+ → R+ be deﬁned by : 2 exp − x2 2 γ(x) := x exp − y2 dy 0

(4.141)

(t)

4.4 Convergence in Law, as t → ∞ of the Processes (Xu ; 0 ≤ u ≤ 1)

257

In this case, the enlargement formula may be stated as follows : there exists a (Gt , t ≤ 1) Brownian motion (Bt , 0 ≤ t ≤ 1) independent from S1 , such that : t Ss −Xs S1 − Xs 1 1Ss <S1 ds (0≤t≤1) Xt = Bt + −√ γ √ 1Ss =S1 + 1−s 1−s 1−s 0 (4.142) We deduce from this relation (4.142) that : i) the laws of (Xt , 0 ≤ t ≤ 1), conditionally to S1 = y admit a weakly continuous version with respect to y; ii) taking S1 = 0 in (4.142), which implies Ss = 0 for every s < 1, hence : 1Ss <S1 = 0, then, conditionally on S1 = 0, we obtain : t −Xs 1 √ Xt = Bt − γ √ ds (4.143) 1−s 1−s 0 or equivalently : −Xt = −Bt + 0

t

1 √ γ 1−s

−X √ s 1−s

ds

(0 ≤ t ≤ 1)

(4.144)

On the other hand (see [AY2 ]), the law of the Brownian meander (mu , 0 ≤ u ≤ 1) is characterized by the fact that (mu , 0 ≤ u ≤ 1) solves the SDE : t −Xs 1 t + √ Xt = B γ √ ds (4.145) 1−s 1−s 0 Finally, conditionally on S1 = 0, (Xu , 0 ≤ u ≤ 1) is the opposite of a Brownian meander. 2. We now prove point 2. of Theorem 4.18 (t) By deﬁnition of P0 , there is the identity : h(St ) (t) E0 [F (X (t) )] = E0 F (X (t) ) E0 (h(St )) √ 1 E0 [F (X• )h( tS1 )] = (by scaling) E0 (h(St )) 2 ∞ √ 2 x 1 exp − = E0 [F (X• )|S1 = x]h( tx)dx E0 (h(St )) 0 π 2 (4.146) (conditioning with respect to S1 ) 2 ∞ y πt 2 1 2 y √ exp − ∼ E0 F (X• )S1 = √ h(y)dy t→∞ 2 π t 0 π 2t t √ (after making the change of variables : x t = y)

258

4 A General Principle and Some Questions about Penalisations

−→ E0 [F (X• )|S1 = 0]

t→∞

h(y)dy

= E[F (−mu , 0 ≤ u ≤ 1)] from the points i) and ii) discussed above.

(4.147)

3. We now prove point 1. of Theorem 4.18 Let F : C([0, 1] → R) → R denote a continuous and bounded functional. One has : Qh0 [F (X (t) )] = E0 [F (X (t) )Mth ] = E0 F (X (t) ) h(St )(St − Xt ) +

∞

h(y)dy

(from (4.131))

St

(4.148) ∞ The term St h(y)dy in (4.148) does not contribute in the limit t → ∞. Indeed, we have : ∞ ∞ (t) h(y)dy ≤ K E0 h(y)dy −→ 0 (4.149) E0 F (X ) St

St

t→∞

Thus : Qh0 [F (X (t) )] ∼ E0 [F (X (t) )h(St )(St − Xt )] t→∞ √ √ = E0 [F (X• ) th( tS1 )(S1 − X1 )] (by scaling) 2 ∞ √ √ x 2 exp − = t E0 [F (X• )(x − X1 )|S1 = x]h( tx)dx π 2 0 by conditioning with respect to S1 . √ Then, making the change of variables : x t = y, we obtain : ∞ 2 y 2 y y = exp − E0 F (X• ) √ − X1 S1 = √ h(y)dy π 0 2t t t ∞ 2 E0 [F (X• )(−X1 )|S1 = 0] −→ h(y)dy t→∞ π 0 (from the above point i)) 2 = E[F (−mu , 0 ≤ u ≤ 1)m1 ] π (from the above point ii)) (3)

= E0 [F (−Ru , 0 ≤ u ≤ 1)]

(4.150)

from Imhof’s absolute continuity result (see Item G in Chapter 1.)

4.5 Some Final Remarks

259

Remark 4.19. We have shown Theorem 4.18 in the framework of penalisation by a function of St . With similar technics we should also be able to (n) penalise by a function of Lt , or of Dta,b , or of Vgt , and so on...

4.5 Some Final Remarks Let (Γt , t ≥ 0) denote a process which satisﬁes the penalisation principle. Using the notation of the Generic Theorem, one has, from (4.8) : Qhx|Fs = Msh •Px|Fs

(4.151)

Consider then T , a (Ft ) stopping time which satisﬁes : Px (T < ∞) = 1. Can we replace, in (4.151) Fs by FT , i.e.: is it true that : Qhx|FT = MTh •Px|FT

?

(4.152)

Qhx|FT ∩(T <∞) = MTh •Px|FT

(4.153)

It is a general fact that (4.151) implies :

Thus, the question (4.152) amounts from (4.153) to know whether a given stopping time T , which is a.s. ﬁnite under Px satisﬁes Qhx (T < ∞) = 1

(4.154)

Of course, any (Ft ) stopping time T , which is Px a.s. ﬁnite, does not necessarily satisfy (4.154). To be totally convinced of this, it suﬃces to choose with the hypothesis of Theorem 4.14 : T := Tb = inf{t ≥ 0; At > b}

(b > 0)

(4.155)

Here is a situation where (4.154) is satisﬁed : Proposition 4.20. Let (Xt , t ≥ 0) denote the canonical Brownian motion, starting from 0, and (Γt , t ≥ 0) = (St := sup Xs , t ≥ 0) its one-sided s≤t

supremum. Let furthermore h : R+ → R+ a probability density, and T a (Ft ) stopping time such that (Xt∧T )√ t≥0 is uniformly integrable (this is well-known to be satisﬁed as soon as E( T ) < ∞). Then : 1) Qh0 (T < ∞) = 1 h 2) (Mt∧T , t ≥ 0) is uniformly integrable

(4.156) (4.157)

260

4 A General Principle and Some Questions about Penalisations

We note that the proof which we provide below is still valid if we replace (St , t ≥ 0) by (Lt , t ≥ 0), or by (Dta,b , t ≥ 0), and so on... (we may also replace 0 by x any real number). Proof of Proposition 4.20 For every function k : R+ → R, which is continuous with compact support, using both the martingale (Mtk , t ≥ 0) and the hypothesis that (Xt∧T )t≥0 is uniformly integrable, we obtain : ∞ ∞ dy k(y) = E0 k(ST )(ST − XT ) + dy k(y) (4.158) 0

ST

Next, from the monotone class Theorem, this identity still holds for any k : R+ → R+ , Borel. In particular, if k = h is a probability density on [0, ∞), then the RHS of (4.158) is equal to 1, which proves both 1) and 2). We note that Proposition 4.20 allows - in particular - to compute the law of (ST , XT ) under Qh from the law of (ST , XT ) under P : ∞ h dy h(y) P (ST ∈ ds, XT ∈ dx) Q (ST ∈ ds, XT ∈ dx) = h(s)(s − x) + s

Bibliography [AY2 ] [M]

[MY] [RVY,I]

[RVY,II]

[RVY,IV]

[RVY,V]

[RY,VIII]

J. Az´ema and M. Yor. Etude d’une martingale remarquable. (French) [Study of a remarkable martingale]. S´em. de Prob. XXIII, 88–130, LNM 1372, Springer, Berlin, 1989. P.A. Meyer. Probabilit´es et potentiel. Publications de l’Institut de Math´ematique de l’Universit´e de Strasbourg, N◦ XIV. Actualit´es Scientiﬁques et Industrielles, N◦ 1318. Hermann, Paris, 1966. R. Mansuy and M. Yor. Random times and enlargements of ﬁltrations in a Brownian setting. LNM 1873. Springer, Berlin, 2006. B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I. Studia Sci. Math. Hungar., 43(2):171–246, 2006. B. Roynette, P. Vallois, and M. Yor. Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time. II. Studia Sci. Math. Hungar., 43(3):295–360, 2006. B. Roynette, P. Vallois, and M. Yor. Some extensions of Pitman’s and RayKnight theorems for penalized Brownian motion and their local time. IV. Studia Sci. Math. Hungar., 44(4):469–516, 2007. B. Roynette, P. Vallois, and M. Yor. Penalizing a BES (d) process (0 < d < 2) with a function of its local time. V. Studia Sci. Math. Hungar., 45(1):67–124, 2008. B. Roynette and M. Yor. Ten penalisation results of Brownian motion involving its one-sided supremum until ﬁrst passage time VIII. Journal Funct. Anal., 255(9):2606–2640, 2008.

Appendices

261

local time in a1

≥ 0)

(h(Sgt )1Xt >0 , t ≥ 0)

(h(Sgt ), t ≥ 0)

(h(St ), t ≥ 0)

2 St , one-sided supremum

(f (Lat 1 )g(Lat 2 ), t

Lat 2 local time in a2

Lat 1

+h− (Lt )1Xt <0 , t ≥ 0)

(h (Lt )1Xt >0

+

0

∞ −

0

0

0

∞

∞

∞

h(y)dy = 1

h(y)dy = 1

h(y)dy = 1

=1

(h (y) + h (y)) = 1

+

f (x)g(y)νa0 1 ,a2 (dx, dy) R+ ×R+

1 2

0

[RY,VIII]

[RY,VIII]

Chap. 1

[RVY,II]

Chap. 2

[RVY,II]

Chap. 1

Penalisation functional (Integrability conditions) References 1 Lt , local time at 0 ∞ h(y)dy = 1 [RVY,II] (h(Lt ), t ≥ 0)

A Tables

Lt

∞

h(y)dy

f (Lat 1 R+ ×R+

+ 1 )g(Lat 2 + 2 )

(d 1 , d 2 ) (See Sec. 2.6 for the ex-

St

∞

h(y)dy

St

h(y)dy

+ St

∞

h(y)dy

h(Sgt )Xt++ h(St )(St − Xt+ )

+

1 h(Sgt )|Xt | + h(St )(St − Xt+ ) 2 ∞

h(St )(St − Xt ) +

martingale associated to this penalisation

plicit form of the measure νax1 ,a2 and the

a ,a2

ν X1t

h+ (Lt )Xt+ + h− (Lt )Xt− ∞ + 12 (h+ + h− )(y)dy Lt

h(Lt )|Xt | +

Limiting Martingale

262 Appendices

R

(1 + |x|)q(dx) < ∞

R+

decay at +∞ h(y)ν0 (dy) = 1

Chap. 2

(h(Aqt ), t ≥ 0)

Chap. 1

[RVY,I]

[RVY,I] Chap. 1 h with “subexponential”

exp − λ2 Aqt , t ≥ 0

R

t 4 Aqt = q(Xs )ds 0 y Lt q(dy) =

h(0) = 1,h(+∞) = 0

h(y)dy = 1

[a, b](a < b) Δh(n) := h(n) − h(n + 1) [a,b] (Δh(Dt ), t ≥ 0)

0

∞

h decreasing,

exp

n≥1

− λ2 Aqt ,

0

∞

(continued)

h(Aqt + y)νXt (dy)

ϕλq solution of : ϕ = λqϕ, ϕ (+∞) = −ϕ (−∞) = π2

ϕλq (Xt ) ϕλq (0)

h+ (Sgt )Xt+ + h+ (St )(St − Xt+ ) ∞ +h− (St )(St − Xt ) + (h+ + h− )(y)dy St ∞ [RY,VIII] h(St )(St − Xt ) + h(y)dy St h(n) b − Xt 1+ 1[σ2n ,σ2n+1 [ (t) [RVY,II] 2 b−a n≥0 h(1+n) Xt −a + 2 + 1[σ2n+1 ,σ2n+2 [ (t) b−a h(n+1) b−Xt Xt −a 1 + b−a + h(n) 2 2 b−a σ2n := inf{t ≥ σ2n−1 , Xt < a}, σ2n+1 = inf{t ≥ σ2 n; Xt > b} [a,b] = 1(σ2n ≤t) Dt

(h+ + h− )(y)dy = 1 [RY,VIII]

downcrossings on the interval

0

∞

h : R+ → [0, 1]

3 Dt[a,b] = number of

(h(Sdt ), t ≥ 0)

+h− (Sgt )1Xt <0 , t ≥ 0)

(h+ (Sgt )1Xt >0

Appendices 263

(Xt , t ≥ 0) : reﬂected BM

(h((Vgt )1/2 ), t ≥ 0)

(1)

excursion before gt

5 Vg(1) t = length of the longest

(h(L•t ), t ≥ 0)

(L•t ) = (Lyt , y ∈ R)

Brownian motion gt Aqt := q1 (Xs )ds 0 t dt + q2 (Xs )ds + q3 (Xs )ds gt t exp − 12 Aqt , t ≥ 0

(Xt , t ≥ 0) reﬂected

q = (q1 , q2 , q3 )

(continued)

0

∞

i = 1, 2, 3

(1 + x)qi (dx) < ∞

0

∞

Ω

h( )Λ0 (d ) = 1

zh (z)dz = −1

at inﬁnity

h inﬁnite “subexponential”

→ R+ h : E(R → R+ ) = Ω

Chap. 3

h(L•t + )ΛXt (d )

|Xt | (1) (Vg −At )+ t

x

(1)

|Xt | (Vgt −At )+

2 X2 h1 At + v2t exp − v2 dv 0 ∞ 2 At := t − gt ; φ(x) := π2 exp − u2 du ∞ x h1 (x) := − √ zh (z)dz 2 π

of the measures (Λx , x ∈ R)

See Chap. 2 for the explicit form

Ω

φqi (0) = 1, φi decreasing ψqi (0) = 0, ψq i (0) = 1

φ = qi φ, ψ = qi ψ and :

(1) 1/2 2 )|Xt |+h1 (At )φ π (h((Vgt )

[RVY,VII] +

Chap. 2

Chap. 2

gt

φqi and ψqi solutions, on R+ , of :

−φq1 (0+ )ψq2 (Xt ) 1 1 t exp − q1 (Xu )du − q2 (Xu )du 2 0 2 gt 1 t +φq1 (Xt ) exp − q1 (Xu )du 2 0

264 Appendices

(n)

(n)

t

(1(V (n) ≤x) , t ≥ 0)

excursion before t

: length of the nth longest

(n)

x ﬁxed

x ﬁxed

x ﬁxed

gt

1

gt

n

|Xt | √ n x

2 π

x

∞

2 exp − u2 du

2 π

x

∞

+

2 exp − u2 du

Γxt := {u ≤ t; Au = x} |Xt | |X | 2 1 √ t √ −1 π + n 1At ≤x φ (n−1) x x−At Γx t + 1 − n−1

At := t − gt ; φ(x) :=

gt

1(V (n−1) ≤x)

+

|Xt | −1 + n1 1At ≤x φ √x−A t Γx + 1 − nt

2 π

At := t − gt ; φ(x) :=

gt

1(V (n) ≤x)

x

Γxt := {u ≤ t; Au = x} See Section 3.9 of Chap. 3 for the explicit form of the martingale (Mtn,x1 ,...,xn ) associated to this penalisation

Γxt := {u ≤ t; Au = x} |Xt | |Xt | −1 1(V (n−1) ≤x) (n−1)√x π2 + n1 1At ≤x φ √x−A t gt Γx t Chap. 3 + 1 − n−1 +∞ 2 At := t − gt ; φ(x) := π2 exp − u2 du

Chap. 3

Chap. 3

n ≥ 2, (Xt , t ≥ 0) : reﬂected Brownian motion (1) (n) (Vgt , ..., Vgt ) list of the n ﬁrst excursions before gt , x1 ≥ x2 ≥ ... ≥ xn ﬁxed Chap. 3 ranked in decreasing order (1(V (1) ≤x ) .....1(V (n) ≤x ) , t ≥ 0)

dt

(1(V (n) ≤x) , t ≥ 0)

excursion before dt

Vdt : length of the nth longest

Brownian motion

n ≥ 2, (Xt , t ≥ 0) : reﬂected

Vt

Brownian motion

(Xt , t ≥ 0) : reﬂected

gt

(1(V (n) ≤x) , t ≥ 0)

excursion before gt

Vgt : length of the nth longest

Appendices 265

266

Appendices

B Some Commutative Diagrams The penalisation process has remarkable “continuity” properties. Here are some illustrations via some examples. 1) Convergence of the number of excursions greater than ε towards the local time at 0 (x > 0 ﬁxed, see Chap. 3)

Penalisation of P 1(V (n) ≤xn−2 ) , t ≥ 0 −−−−−−−−−−−−→ t t→∞

−2

⏐ ⏐ ⏐ ⏐n→∞ 3

⏐ ⏐ ⏐ n → ∞ ⏐ a.s. 3

Qx,xn

Penalisation of P 1Lt ≤√ πx , t ≥ 0 −−−−−−−−−−−−→ 2 t→∞

x Q

with : Qx,xn |Ft

−2

−2

:= Mtx,xn •P|Ft

x := M (x •P|F Q t |Ft t and : Mtx,xn

−2

%

& −2 |Xt | |Xt | 2 Λxn t √ = 1(n2 V (n) ≤x) + Φ − 1 + 1 − gt π n(xn−2 )1/2 n x − At ∞ a.s. (x −→ M hx (y)dy t := hx (Lt )|Xt | +

n→∞

Lt

with hx (y) :=

+

2 √ (y). πx 1[0, πx 2 ]

2) Convergence of the number of downcrossings towards the local time at 0 (see [RVY,II]) Let G : R+ → [0, 1], G of class C 1 , decreasing and such that G(0) = 1, G(+∞) = 0. Let (Dtε , t ≥ 0) the number of downcrossings on the interval [0, ε] (ε > 0) : ε ε σ2n := inf{t ≥ σ2n−1 , Xt < 0} ε ε σ2n+1 := inf{t ≥ σ2n , Xt > ε} ε ≤t) 1(σ2n Dtε := n≥0

B Some Commutative Diagrams

267

Let Δε G(n) := G(2εn) − G(2ε(n + 1))

Δε G(Dtε ) ε

Penalisation of P , t ≥ 0 −−−−−−−−−−−−→ t→∞

Qε,G ⏐ ⏐ ⏐ ⏐ε↓0 3

⏐ ⏐ ⏐ ε ↓ 0 ⏐ a.s. 3 Penalisation of P (−G (Lt ) , t ≥ 0) −−−−−−−−−−−−→ t→∞

Q−G

with : ε,G •P|F Qε,G t |Ft = Mt

−G Q−G •P|F t |Ft = Mt

and :

G(2εn) ε − Xt G(2ε(1 + n)) Xt ε ,σ ε (t) 1[σ2n 1 − + 2n+1 [ 2 ε 2 ε n≥0 G(2ε(1 + n)) ε − Xt G(2εn) Xt ε ε +1[σ2n+1 1+ + ,σ2n+2 [ (t) 2 2 2 2 ∞ a.s. −→ Mt−G = −G (Lt )|Xt | + −G (y)dy

Mtε,G =

ε→0

Lt

t 1 1[−ε,ε] (Xs )ds towards the local time at 0 (see Chap. 1, 2ε 0 Item A and Chap. 2, Corollary 2.10) t ε 1 1 i) Let q ε := 1[−ε,ε] , Aqt = 1[−ε,ε] (Xs )ds. 2ε 2ε 0 Let h : R+ → R+ continuous with “subexponential” decay at inﬁnity: 3) Convergence of

ε Penalisation of P (h(Aqt ), t ≥ 0) −−−−−−−−−−−−→ t→∞

Qq

ε,h

⏐ ⏐ ⏐ ⏐ε↓0 3

⏐ ⏐ ⏐ ε ↓ 0 ⏐ a.s. 3 Penalisation of P (h(Lt ) , t ≥ 0) −−−−−−−−−−−−→ t→∞ with : ε,h

Qq|Ft = Mtq

ε,h •

P|Ft

h = M (h •P|F Q t |Ft t

h Q

268

Appendices

and : Mtq

ε,h

∞

ε

t→∞

with ν x (dy) =

ε

h(Lqt + y)ν qXt (dy) 0 ∞ h ( −→ Mt = h(Lt + y)ν Xt (dy) =

0

2 π 1[0,∞[ (y)dy

+

2 π |x|δ0 (dy).

a

ii) Following the same idea, let q = 1[−a,0] and t 1Xs <0 ds := A− t .

a Aqt

t

=

a.s.

1[−a,0] (Xs )ds −→

0

a→∞

0

Let h : R+ → R+ continuous with “subexponential” decay at inﬁnity: a Penalisation of P (h(Aqt ), t ≥ 0) −−−−−−−−−−−−→ t→∞

Qq

a

,h

⏐ ⏐ ⏐ ⏐ a → +∞ 3

⏐ ⏐ ⏐ a → +∞ ⏐ a.s. 3 Penalisation of−→ P −−−−−−−−−−− (h(A− t ) , t ≥ 0) t→∞

Q−,h

with : a,h

Qq|Ft = Mta,h •P|Ft −,h Q−,h •P|F t |Ft = Mt

and :

Mta,h

a.s.

−→

0 ∞

a→∞

with ν − x (dy) := x+ ample 2.4.5.d).

∞

=

2 π δ0 (dy)

0

+

a

a

h(Aqt + y)ν qXt (dy) −,h − h(A− t + y)ν Xt (dy) = Mt

1 π

2 −) 1[0,∞[ (y) √dyy (see Chap. 2, Exexp − (x2y

4) “Projectivity” of probabilities Qn,x1 ,...,xn (see Chap. 3, Remark 9.4) (1) (n) Let n ﬁxed, and Vgt ≥ ... ≥ Vgt the sequence of the n longest excursions before gt , ranked in decreasing order. Let x1 ≥ x2 ≥ ... ≥ xn−1 ≥ xn ﬁxed.

C Index of Main Notations

269

Penalisation of P 1(V (1) ≤x ,...,V (n) ≤x ) , t ≥ 0 −−−−−−−−−−−−→ Q(n,x1 ,...,xn ) 1 n gt gt t→∞

x1 , x2 , ..., xn−1

1(V (n) ≤x ) , t ≥ 0 gt

⏐ ⏐ ⏐ ⏐ x1 , x2 , ..., xn−1 → +∞ 3

⏐ ⏐ ⏐ → +∞ ⏐ a.s. 3

Penalisation of−→ P (n,xn ) −−−−−−−−−−− Q t→∞

n

with : (n,x1 ,...,xn )

= Mt

(n,xn )

:= Mt

Q|Ft

Q|Ft

(n,x1 ,...,xn ) (n,xn )

•

•

P|Ft

P|Ft

(n,x1 ,...,xn )

where Mt

(n,x1 ,...,xn )

Mt

is deﬁned in Section 3.9 of Chap. 3 and where : 2 a.s. |Xt | −→ 1 (n) x1 ,x2 ,...,xn−1 →+∞ (Vgt ≤x) πx + |Xt | 1 Δxt + 1At ≤x Φ √ −1 + 1− n n x − At

(see Section 3.5 of Chap. 3).

C Index of Main Notations • a+ = a ∨ 0, a− = (−a) ∨ 0, |a| = a+ + a− • sgn(x)=1 if x ≥ 0, = −1 if x < 0 • Γ the Gamma function : Γ(t) =

∞ 0

e−u ut−1 du (t ≥ 0)

• Jν the Bessel function of index ν • Kν the MacDonald function of index ν • Iν the modiﬁed Bessel function of index ν • Φ(α, γ; •) the conﬂuent hypergeometric function of index α, γ • K the Markov kernel deﬁned by : Kf (y) = 0

∞

2 z z exp − f (z)dz y 2y

270

Appendices

• Δ the Laplace operator the Laplace Beltrami operator on the unit sphere • Δ • Φ(x) =

x 0

ϕ(y)dy, with ϕ Borel, positive and ϕ ∈ L1 (R+ , dx) **************************

• fZ the density of the r.v. Z • Sβ an exponential r.v. with mean 1/β (law)

• e = S1 a standard exponential r.v. • (T1 , T2 , ..., Tn , ...) the sequence of jumps of a standard Poisson process T1 T2 Tn • (ρ1 , ρ2 , ..., ρn , ...) = , , ..., , ... a sequence of independent T2 T3 Tn+1 r.v. where ρn is beta (n, 1) distributed (for any n ≥ 1) (law)

• γα a gamma (α) r.v. with density : fγα (x) =

1 −x α−1 e x 1[0,∞[ (x) Γ(α)

************************** • C(I → I ) the space of continuous functions from I to I • (C(R+ → R), (Xt , Ft )t≥0 , F∞ , Px (x ∈ R)) or (C(R+ → R), (Xt , Ft )t≥0 , F∞ , Wx (x ∈ R)) the canonical Brownian motion of dimension 1 • Px (or Wx ) the Wiener measure s.t. Px (X0 = x) = 1 • P = P0 , W = W0 • (Xt , t ≥ 0) the coordinate process : Xt (ω) = ω(t), ω ∈ C(R+ → R) • (Ft , t ≥ 0) the natural ﬁltration • (Gt , t ≥ 0) an enlarged ﬁltration of (Ft , t ≥ 0) • F∞ =

> t≥0

Ft

• b(Ft ) the space of bounded real valued Ft -measurable functions • θt the usual time translation operator : Xs ◦ θt = Xs+t (s, t ≥ 0)

C Index of Main Notations

271 (d)

• (C(R+ → Rd ), (Xt , Ft )t≥0 , F∞ , Px (x ∈ Rd )) or (d) (C(R+ → Rd ), (Xt , Ft )t≥0 , F∞ , Wx (x ∈ Rd )) the canonical Brownian motion of dimension d (d)

(d)

(d)

• Px (or Wx ) the d-dimensional Wiener measure s.t. Px (Xt = x) = 1 (1)

(1)

• Px = Px , Wx = Wx (d)

(d)

• P (d) = P0 , W (d) = W0

• (C(R+ → R), |Xt |, t ≥ 0, Px , x ∈ R+ ) the canonical reﬂecting Brownian motion (d)

• (C(R+ → R+ ), (Rt , Ft )t≥0 , F∞ , Px , x ∈ R+ ) the canonical d-dimensional Bessel process (3)

• P0

the law of the 3-dimensional Bessel process started at 0 (α)

• (C(R+ → R+ ), (Rt , Ft )t≥0 , F∞ , Px , x ∈ R+ ) the canonical Bessel process of dimension d = 2(1 − α), with 0 < α < 1 (α)

• Px the law of the d-dimensional Bessel process started at x (x ≥ 0, d = 2(1 − α), 0 < α < 1) • (bu , 0 ≤ u ≤ 1) the standard Brownian bridge (α)

• (ru , 0 ≤ u ≤ 1) or (ru , 0 ≤ u ≤ 1) the standard Bessel bridge of dimension d = 2(1 − α), with 0 < α < 1

1 Rgt +(t−gt )u ; 0 ≤ u ≤ 1 the Bessel met − gt ander, where (Rs , s ≥ 0) is a d-dimensional Bessel process of dimension d = 2(1 − α), with 0 < α < 1 (α)

• (mu , 0 ≤ u ≤ 1) :=

(d)

√

(1/2)

• (mu , 0 ≤ u ≤ 1) = (m u , 0 ≤ u ≤ 1) 1 Xgt +(t−gt )u ; 0 ≤ u ≤ 1 := √ t − gt the Brownian meander, where (Xs , s ≥ 0) is a Brownian motion • (Mt , t ≥ 0) a generic notation for a positive martingale • (St := sup Xs , t ≥ 0) the one-sided supremum process of X s≤t

272

Appendices

• (Lt , t ≥ 0) (or (L0t , t ≥ 0)) the continuous process of local time at level 0 • (τ , ≥ 0) the right continuous inverse of (Lt , t ≥ 0) : τ := inf{t ≥ 0; Lt > } • (Lyt , t ≥ 0, y ∈ R) the bicontinuous family of local times • Ta := inf{t ≥ 0; Xt = a} (a ∈ R) (α)

• Pr (T0 < u) = Φ(α)

r √ u

(r, u ≥ 0) with : ∞ ! 1 e−u uα−1 du Φ(α) (r) := P ( 2γα > r) = Γ(α) r2 /2

• Φ

(1/2)

(r) =

[a,b]

• (Dt

2 π

∞

(α ∈]0, 1[, r ≥ 0)

e−x /2 dx 2

r

, t ≥ 0) the process of downcrossings on the interval [a, b] (a < b)

• (Jt , t ≥ 0) := (inf s≥t Rs , t ≥ 0), the future inﬁmum process of (Rs , s ≥ 0), where (Rs , s ≥ 0) is a d-dimensional Bessel process (d > 2) • q a positive Radon measure on R (or on R2 ) • ϕq a particular solution of Sturm-Liouville equation ϕ = qϕ

(or Δϕ = qϕ)

• I = {q; 0 <

R

(1 + |x|)q(dx) < ∞}

• q+ the restriction of q to R+ • q− the image by the application x → −x of the restriction of q to R− • I+ = {q, positive measure on R+ s.t. 0 < • < q, >:= (q)

• At :=

R

R

R+

(1 + x)q(dx) < ∞}

:= C(R → R+ )) (y)q(dy) (q Radon measure on R, ∈ Ω

t Lyt q(dy) = 0 q(Xs )ds if q is absolutely continuous

C Index of Main Notations

• gt := sup{s ≤ t; Xs = 0}

273

(t ≥ 0)

• dt := inf{s ≥ t; Xs = 0} = t + T0 ◦ θt

(t ≥ 0)

• (At , t ≥ 0) = (t − gt , t ≥ 0) the age (of excursions) process • TxA := inf{t ≥ 0; At > x} (x ≥ 0); TxA ≥ x a.s. (1)

• Vgt (or Σt ) the length of the longest excursion above 0 before gt • A∗t = sup As = Vg(1) ∨ (t − gt ) t s≤t

• dt − gt the length of the excursion straddling t −→ (1) (2) (n) • Vgt = (Vgt , Vgt , ..., Vgt , ...) the sequenceof lengths of excursions above Vg(i) = gt 0, before gt , ranked by decreasing order t i

→ − (1) (2) (n) of lengths of excursions above • Vt = (Vt , Vt , ..., Vt , ...) the sequence 0, before t, ranked by decreasing order (i) sequence and Vt = t

t − gt is an element of this

i

−→ (1) (2) (n) • Vdt = (Vdt , Vdt , ..., Vdt , ...) the sequence of lengths of excursions above (i) 0, before dt , ranked by decreasing order Vdt = dt i

− • → v = (v (1) , v (2) , ..., v (n) , ...) the sequence of lengths of excursions above 0, ranked by decreasing order, of a Bessel bridge of dimension d = 2(1 − α), 0 < α < 1 • S ↓ = {σ = (σ1 , σ2 , ..., σn , ...); σ1 ≥ σ2 ≥ ... ≥ σn ≥ ... ≥ 0} • Pα,β the Poisson Dirichlet distribution with parameter (α, β) (α, β ≥ 0) • BESx (d) : the law of the Bessel process of dimension d, starting from x (3)

=BES0

(α)

=BESx

• P0

• Px

(3)

1− d2

(0 < d < 2, d = 2(1 − α))

274

Appendices

• H(P, Q) the Hellinger distance between the probabilities P and Q • (λ(t), t ≥ 0) a normalisation function ************************** = C(R → R+ ) • (Λx , x ∈ R) a family of positive and σ-ﬁnite measures on Ω • Λ = Λ0 (q)

• (ν x , x ∈ R) a family of positive and σ-ﬁnite measures on R+ , associated with q ∈ I (q) → R+ , iq () =< , q > • ν x is the image of Λx by the application iq : Ω (q)

• ν (q) = ν 0

• (Wx , x ∈ R) a family of positive and σ-ﬁnite measures on (C([0, ∞[, R), F∞ ) • W = W0 “the master measure” deﬁned by Prop. 2.4, Chap. 2 • Λx is the image of Wx by the application Θ (: local time at inﬁnity) : C([0, ∞[, R) → C(R → R+ ) Θ(Xt , t ≥ 0) = (Ly∞ , y ∈ R) • Π a positive and σ-ﬁnite measure on S ↓

D Classiﬁcation of Rigorous Results and Meta-theorems in this Volume For ease of the reader, we have classiﬁed in three categories the results found in this monograph. Category 1: Results which are fully proven, either in this monograph or in one of the references e.g. [RVY,i]; i=I,...X. Category 2: Results which depend on the validity of Conjecture (C), p. 148. Category 3: Meta-results, that is: results which we conjecture, and which necessitate precise hypotheses under which they may be proven.

D Classiﬁcation of Rigorous Results and Meta-theorems in this Volume

275

Here are the precise entries for this classiﬁcation: Category 1: a) Examples 0.1, 0.2, 0.3, 0.4, 0.5, 0.13, 0.15, 0.18, 0.20, 0.21, 0.22, 0.23i), 0.24, 0.25; 4.2, 4.3, 4.4 b) Remarks 2.11, 2.19 c) Propositions 1.7, 1.13, 1.18, 1.20; 2.2, 2.4; 3.2, 3.3, 3.6, 3.7, 3.18, 3.29 d) Corollaries 2.6, 2.7, 2.9 e) Theorems 0.9, 0.10, 0.11, 0.12, 0.16; 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.8, 1.9, 1.10, 1.12, 1.14, 1.19, 1.21; 2.1, 2.4*, 2.5, 2.12, 2.13, 2.18; 3.1, 3.4i), 3.5, 3.10, 3.13, 3.16, 3.19, 3.23, 3.25, 3.28; 4.5, 4.7, 4.9, 4.10, 4.12, 4.14, 4.15, 4.16, 4.18 Category 2: a) Example 0.23i) b) Remark 1.17 c) Propositions 1.15, 3.22 d) Theorems 1.16, 3.4ii), 3.8, 3.9, 3.15 Category 3: •

Meta-theorem of penalisation (Section 0.3)

•

Generic Theorem (Section 4.1)

•

Corollary 2.10

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