Table Of ContentLecture Notes in Mathematics 1655
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S6minaire
,.ed Probabilit6s XXXI
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Marc Yor
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(Lecture notes in mathematics ; Vol. 1655)
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IIV
SI~MINAIRE DE PROBABILITIES XXXI
TABLE DES MATIERES
J. Warren : Branching processes, the Ray-Knight theorem, and sticky
Brownian motion.
R. L~andre, J. Norris : Integration by parts and Cameron-Martin
formulae for the free path space of a compact Riemannian manifold. 16
A. S. Ustiinel, M. Zakai : The change of variables formula on Wiener
space. 24
O. Mazet : Classification des semi-groupes de diffusion sur R associ6s
une famille de polynSmes orthogonaux. 40
S. Fang, J. Franchi : A differentiable isomorphism between Wiener space
and path group. 54
J. Jacod, V. Perez-Abreu : On martingales which are finite sums of
independent random variables with time dependent coefficients. 62
J.-M. Aza'/s, M. Wschebor : Oscillation presque sfire de martingales
continues. 69
F. Gao : A note on Cramer's theorem. 77
S. He, J. Wang : The hypercontractivity of Ornstein-Uhlenbeck semi-
groups with drift, revisited. 80
B. Cadre : Une preuve "standard" au principe d'invariance de Stoll. 85
J.-F. Le Gall : Marches al~atoires auto-~vitantes et mesures de polym~res. 103
K.D. Elworthy, X.M. Li, M. Yor : On the tails of the supremum and
the quadratic variation of strictly local martingales. 113
L. I. Galtchouk, A. A. Novikov : On Wald's equation. Discrete time
126
case.
L. Miclo : Remarques sur l'hypercontractivit~ et l'~volution de l'entropie
pour des chaines de Markov finies. 136
VIII
M. Deaconu, S. Wantz : Comportement des temps d'atteinte d'une
diffusion fortement rentrante. 168
M. l~mery : Closed sets supporting a continuous divergent martingale. 176
D. Khoshnevisan : Some polar sets for the Brownian sheet. 190
P. Majer, M.E. Mancino : A counter-example concerning a condition
of Ogawa integrability. 198
Y. Chiu : The multiplicity of stochastic processes. 207
N. Eisenbaum : ThSor~mes limites pour les temps locaux d'un processus
stable sym~trique. 216
P. Gosselin, T. Wurzbacher : An It5 type isometry for loops in (cid:127)d via
the Brownian bridge. 225
J. Jacod : On continuous conditional Gaussian martingales and stable
convergence in law. 232
J. Feldman, M. Smorodinsky : Simple examples of non-generating
Girsanov processes. 247
P. A. Meyer : Formule d'It8 g~nSralis~e pour le mouvement brownien
lin~aire, d'apr~s FSllmer, Protter, Shyriaev. 252
K. Takaoka • On the martingales obtained by an extension due to Saisho,
Tanemura and Yor of Pitman's theorem. 256
B. Rauseher • Some remarks on Pitman's theorem. 266
J. Pitman, M. Yor • On the lengths of excursions of some Markov
processes. 272
J. Pitman, M. Yor • On the relative lengths of excursions derived from
a stable subordinator. 287
M. Yor : Some remarks about the joint law of Brownian motion and its
supremum. 306
A. Estrade : A characterization of Markov solutions for stochastic
differential equations with jumps. 315
R. L~andre : Diffeomorphism of the circle and the based loop space. 322
F. Coquet, J. M6min : Correction `5 : Vitesse de convergence en loi pour
des solutions d'Squations diff~rentielles stochastiques vers une diffusion
(volume XXVIII). 327
C. Rainer : Correction .5 : Projection d'une diffusion sur sa filtration lente
(volume XXX). 329
Branching processes, the Ray-Knight
theorem, and sticky Brownian motion
JONATHAN WARREN 1
University of Bath, U.K.
1 Introduction
Diffusions with boundary conditions were studied by Ikeda and Watanabe 5 by means
of associated stochastic differential equations. Here we are interested in a fundamental
example. Let 0 and x be real constants satisfying 0 < 0 < r and 0 < x < c~. Suppose
(~t, (~t)t>0, P) is a filtered probability space satisfying the usual conditions, and that
(Xt; t > 0) is a continuous, adapted process taking values in ,0 (x)) which satisfies the
stochastic differential equation
0
(1.1) Xt = x + 0 Ilx,=o}ds + I{xs>o}dWs,
where (Wt; t _> 0) is a real valued (Ut)-Brownian motion. We say that Xt is sticky
Brownian motion with parameter 0, started from x. Sticky Brownian motion has a
long history. Arising in the work of Feller 3 on the general strong Markov process on
,0 c~) that behaves like Brownian motion away from 0, it has been considered more
recently by several authors, see Yamada 12 and Harrison and Lemoine 4, as the
limit of storage processes, and by Amir 1 as the limit of random walks.
Ikeda and Watanabe show that (1.1) admits a weak solution and enjoys the
uniqueness-in-law property. In 2, Chitashvili shows that, indeed, the joint law of
X and W is unique (modulo the initial value of W), and that X is not measurable
with respect to W, so verifying a conjecture of Skorokhod that (1.1) does not have a
strong solution. The filtration (her) cannot be the (augmented) natural filtration of W
and the process X contains some 'extra randomness'. It is our purpose to identify this
extra randomness in terms of killing in a branching process. To this end we will study
the squared Bessel process, which can be thought of as a continuous-state branching
process, and a simple decomposition of it induced by introducing a killing term. We
will then be able to realise this decomposition in terms of the local-time processes of
X and W. Finally we will prove the following result which essentially determines the
conditional law of sticky Brownian motion given the driving Wiener process.
Theorem 1. Suppose that X is sticky Brownian motion starting from zero, and that
W is the driving Wiener process, also starting from zero. Letting Lt = sups<t(-W~),
the conditional law of X given W satisfies
e(Xt < zl,,(W)) = exp ( - 20(W, + L, - x)) a.s.
for x C ,0 Wt + Lt.
ljwl @maths.bath.ac.uk
Note in particular that Xt E 0, Wt + Lt a.s.. The proof of this result is given in
Section 4, and depends on the construction of the pair (X, W) discussed in Section
3. Section 2 is essentially independent, but helps provide us with the intuitive reason
for believing Theorem 1.
We begin with a simple but illuminating lemma on sticky Brownian motion, and
fix some notation we will need in the sequel.
We denote:
(1.2)
A + = I{x,>o}ds; ~+ = inf{u : A + > t};
(1.3)
A~ = I{x~=oids; at ~ = inf{u : A ~ > t}.
Then we have
Lemma 2. If we time change both sides o (1.1) with a +, the right-continuous inverse
of A +, we find that (X +, t _> )O solves Skorokhod's reflection equation
x~ t = w~ + + ~L +
erehw FW = x+f;' +o ~Wd~o>.x~I is a B~ow.ian motion, andLt = sup<, .)OV)2W-((
Proof. On time changing we have
xo t = w~ + + CA~
Observe that W + is a Brownian motion by L~vy's characterization. Now, A ~ is a
continuous and increasing function of t, A + is a continuous and strictly increasing
function of t, and so L + = ~A~ is also a continuous and increasing function of t.
Furthermore it is constant on the set {t : Xa+ > 0}. The criteria of Skorokhod's
lemma, see 9, are thus satisfied and L + = SUPs<t(-W +) as claimed.
This lemma shows us that sticky Brownian motion is just the time change of a
reflecting Brownian motion so that the process is held momentarily each time it visits
the origin. In this way it spends a real amount of time at the origin, proportional to
the amount of local time the reflecting Brownian motion has spent there, in fact,
(1.4) 0A~ = L +.
The laws of t+A ,A~ and other quantities can be obtained directly from this, as has
been accomplished by Chitashvili and Yor 13.
2 A decomposition of the squared Bessel process
We consider two processes (Rt, t > 0) and (Yt, t > 0) satisfying
(2.1a) dRt = 2v~tdBt - 2~Rt dt, Ro = x,
(2.1b) dYt = 2v/~td~t + 20nt dt, oY = ,O
where B and/~ are independent Brownian motions.
