Table Of ContentLecture Notes in Mathematics 6251
Editors:
A. Dold, Heidelberg
B. Eckmann, Ztirich
E Takens, Groningen
.J Az6ma R A. Meyer M. Yor ).sdE(
Sdminaire de
Probabilitds XXVI
Springer-Verlag
Berlin Heidelberg NewYork
London Paris Tokyo
Kong Hong Barcelona
Budapest
Editors
Jacques Az6ma
Marc Yor
Laboratoire de Probabilit6s
Universit6 Pierre et Marie Curie
4, place Jussieu, Tour 56
F-75252 Paris Cedex 05, France
Paul Andr6 Meyer
Institut de Recherche Math6matique Avanc6e
Universit6 Louis Pasteur
7, rue Ren6 Descartes
F-67084 Strasbourg, France
Mathematics Subject Classification (1991): 60GXX, 60HXX, 60JXX
ISBN 3-540-56021-1 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-56021-1 Springer-Verlag New York 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, re-use of
illustrations, recitation, broadcasting, reproduction on microfilms 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-Verlag.
Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1992
Printed in Germany
Typesetting: Camera-ready by author/editor
46/3140-543210 - Printed on acid-free paper
Le "S&minaire de Probabilit&s" atteint cette annie l'~ge
inesp6r6 de 25 ans.
La situation des probabilit~s a bien chang~ en un quart
de si&cle. Nous avons essay6 de suivre cette ~volution
alors que les sujets tendaient a se diversifier et le
volume des publications ~ augmenter.
Nous n'avons ni te d~sir, ni les moyens, de nous trans-
former en journal math~matique pourvu d'un vaste comit6
de r~daction.
Cela nous amine, d~s ce volume, mais encore davantage
partir du volume XXVII en preparation, ~ concentrer
nouveau nos efforts sur nos th~mes traditionnels : calcul
et analyse stochastiques, 6rude du mouvement brownien et
sujets connexes.
En outre, nous esp6rons, comme autrefois, accueillir lar-
gement des articles d'exposition.
J.A., P.-A.M., M.Y.
SEMINAIRE DE PROBABtLITES XXVl
TABLE DES MATIERES
R. Bass, D. Khoshnevisan : Stochastic calculus and the continuity of local
times of L~vy processes.
E. Mayer-Wolf, D. Nualart, V. Perez-Abreu : Large deviations for multiple II
Wiener - It6 integral processes.
A. Xia : Weak convergence of jump processes. 32
L. Miclo : Recuit simul6 sans potentiel sur un ensemble fini. 47
L. Pratelli : Une caract6risation de la convergence dans L I. Application aux 61
quasimartingales.
M.T. Barlow, P. Imkeller : On some sample path properties of Skorokhod inte- 70
gral processes.
.K Burdzy, .D Marshall : Hitting a boundary point with reflected Brownian 81
motion.
T.S. Mountford : Quasi-everywhere upper functions. 95
T.S. Mountford : A critical function for the planar Brownian convex hull. 107
IIIV
J.C. Taylor : Skew products, regular conditional probabilities and stochastic 113
differential equations : a technical remark.
M. Pontier, A. Estrade : Rel~vement horizontal d'une semimartingale cadtag. 127
M. Arnaudon : Connexions et martingales dans les groupes de Lie. 146
L. Dubins, M. Smorodinsky : The modified, discrete L~vy transformation is 157
Bernoulli.
H. Boutabia, B. Maisonneuve : Lois conditionnelles des excursions markoviennes 162
D. L6pingle : Orthogonalit6 et 6quiint6grabilit6 des martingales discretes. 167
D. Bakry, D. Michel : Sur les in6galit6s FKG. 170
J.R. Norris : A complete differential formalism for stochastic calculus in 189
manifolds.
M. Baxter : Markov processes on the boundary of the binary tree. 210
Ph. Biane : Fronti~re de Martin du dual de SU(2). 22:5
P. Mc Gill : Generalized transforms, quasi-diffusions, and D6sire Andr6's 234
equation.
J. Az6ma, M. Yor : Sur les z6ros des martingales continues. 248
J. Az6ma, P.A. Meyer, M. Yor : Martingales relatives. 307
Th. Jeulin, M. Yor : Une ddeomposition non-canonique du drap brownien. 322
XI
J. Bertoin : Une famille de diffusions qui s'annulent sur les z6ros d'un 348
mouvement brownien r6fldchi.
P. Vallois : Amplitude et juxtaposition des excursions positives et n6gatives 361
pour le mouvement brownien.
R. Abraham : Un arbre al6atoire infini associ~ A l'excursion brownienne. 374
F. Delbaen : Infinitesimal behavior of a continuous local martingale. 398
F. Utzet : Les processus & accroissements ind6pendants et les ~quations de 405
structuFe.
J.L, Sol~, F. Utzet : Une note sur l'int6grale multiple de Stratonovich pour 410
le processus de Poisson.
T. Kazumi, I. Shigekawa : Measures of finite (r,p) energy and potentials on 415
a separable metric space.
S.E. Kuznetsov : More on existence and uniqueness of decomposition of exces- 445
sive functions and measures into extremes.
S.E. Kuznetsov : On the existence of a dual semigroup. 473
P.J. Fitzsimmons, R.K. Getoor : Some applications of quasi-boundedness for 485
excessive measures.
A. Joffe : Renouvellement : g~n~rateur du processus de l'~ge. 498
R. Rebolledo : Les "principes d'invariance" en probabilit~ sur l'espace de 501
Wiener.
X
O. Bobadilla, R. Rebotledo, E. Saavedra : Sur la convergence d'int6grales 505
anticipatives.
D. Applebaum : An operator theoretic approach to stochastic flows on manifolds 514
M. Kikuchi : A note on the energy inequalities for increasing processes. 533
S. Sato : On the reconstruction of a killed Markov process. 540
D.N. Hoover : Extending probability spaces and adapted distribution. 560
Y.Z. Hu : Une formule d'It6 pour ie mouvement brownien fermionique. 575
Y.Z. ttu : S6rie de Taylor stochastique et formule de Campbell - Hausdorff, 579
d'apr6s Benarous.
Y.Z. Hu : Sur un travail de R. Carmona et D. Nualart. 587
Y.Z. Hu : Une remarque sur l'in6galit6 de HSlder non-commutative. 595
E.I. Trofimov : Sobolev topologies in semimartingale theory. 596
S. Ladouceur, M. Weber : Note & propos d'un r6sultat de Kowada sur les flots 608
analytiques.
S. Attal : Probl~mes d'unicit6 dans les repr6sentations d'op6rateurs sur 619
l'espace de Fock.
Corrections aux volumes ant6rieurs. 633
Note de la R6daction : Les difficult6s de communication avec la Russie nous
ont emp6ch6s d'obtenir une version d6finitive, corrig6e de certaines erreurs
typographiques, des deux articles de S. Kuznetsov contenus dans ce volume,
articles dont la qualit6 nous semble n6anmoins - et c'est aussi l'avis du
referee - justifier une publication sans d61ai suppi6mentaire.
Stochastic Calculus and the Continuity of Local Times of L~vy Processes
Richard Bass* and Davar Khoshnevisan
1. Introduction. Let Zt be a one dimensional L~vy process with characteristic func-
tion
E exp(iuZt) = exp(-t¢(u)),
where
£
)I.I( ¢(u) = -iau + la2u2 - (e i"" - I - iuzl(id<l))u(dz ).
2 oc
Here u satisfies f(1 A z2)u(dz) < oo.
We are interested in those L~vy processes for which 0 is regular for {0} and either
2 a > 0 or u(R- {0}) = cx~. In this case (see [K]) there exists a bounded continuous
function g that is a density for the 1-resolvent:
/ 0/
(1.2) f(x)g(x - y)dx = E y e-tf(Zt)dt, f > ,O y e IR.
(If G(x, y) is the Green function for Zt killed at an independent exponential time with
parameter 1, the relationship between g and G is given by g(x) = G(O, z) = G(a, a + x)
for any a ~ R and a(~, y) = g(~ - x).)
For each x,
(1.3) g(x) = ~ 1 +
For each x, g(x - .) is the 1-potential of an additive functional L~ that is continuous in
t. Moreover, a version of L~(w) may be chosen that is jointly measurable in (x,t,w).
See [GK] for details. L~ is called the local time of Zt at x. L~ is also a density of
occupation time measure: if f _> 0,
'0/ /
(1.4)
f(Zs)ds-= f(x)L~dx, a.s..
A number of people have studied the question of the continuity of L~ in the space
variable (see [Bo], [Me], [GK] and [MT]), culminating in the works [B1], [BH], and [B2],
where a necessary and sufficient condition for the joint continuity of L~ in t and x is
given.
* Research partially supported by NSF grant DMS-8822053
The purpose of this paper is to give a stochastic calculus proof of the following
sufficient condition for joint continuity. Let ~ : R +--- ,0[ co) be defined by
(1.5) ~2(x) = (1 - cos ux)Re + 1 ¢(u)dU-
Let d(a, )b = ~(b - a) and let H(u) be the logarithm of the smallest number of d-balls
of radius less than u that are needed to cover [-1, .]1 Define
0f
( 1
(1.6) =
Theorem I.I. (a)/f F(0+) < %0 then L~ has a jointly continuous version.
(b) For each t,
limsup sup sup ~L[ - [~L < 2(supL~)l/2,
a.S.
8<_t F( (a - b)) -
Theorem 1.1(a) was first proved in [BH], where it was also remarked that the
entropy condition was equivalent to one involving the monotone rearrangement of ~.
Part (b) was also proved in [BH], with, however, the constant 2 replaced by a larger
constant (namely 416). In [B2] it was shown that part (b) holds with the constant 2
under the additional assumption that ~ is regularly varying (but not slowly varying)
and that the constant 2 is sharp. (The principle result of [B2] was that the condition
F(0+) < oo is necessary as well as sufficient for joint continuity.) Marcus and Rosen
[MR] have recently obtained necessary and sufficient conditions for the joint continuity
of local times of certain Markov processes. Theorem 1.1 for symmetric L~vy processes
is a special case of their results.
In Section 2 we prove Theorem 1.1 assuming that ess supz L~ < oo, a.s. We establish
this latter fact in Section 3.
2. Modulus of continuity. Our proof is modeled after that of [McK]. Let us begin
by assuming for this section that esssupxL ~ < co, a.s. Let R be an exponential variable
with parameter ,1 independent of Zt. Since g(x - .) is the 1-potential of L~, we have
(2.1) E"LbR = g(b - a).
