Probability Abstracts 19

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303. MULTIVARIATE RECORDS AND ORDERED RANDOM SCATTERING

Charles M. Goldie and Sidney I. Resnick

Suppose $({\bf X}_n)_{n=1,2,\dots}$ is a sequence of iid random vectors in
$R^2$. Say ${\bf X}_n$ is a record if there is a record simultaneously in both
coordinates at index $n$. The total number of records of a sequence is
frequently finite so we consider the behaviour of the records in a fixed
rectangle  $A$ conditional on there being a large number of records $N_A$ in
the rectangle. We join the records up to form a random parametrized curve, and
prove that, under suitable conditions, as the number of records in the
rectangle goes to infinity the random curves converge conditionally in
probability to a non-random limit parametrized curve, which we characterize.
We also prove a large-deviation result for the random curves.

Finally, if $A<B$ are two regions of $R^2$ such that every point of $A$ is
south-west of every point of $B$, then $N_A$ and $N_B$ are independent. We
call this property ordered independent scattering and discuss which other
notions of record give rise to it.

C.M.Goldie@qmw.ac.uk

304. A SOBOLEV INEQUALITY AND NEUMANN HEAT KERNEL ESTIMATE FOR UNBOUNDED DOMAINS

Z. Q. Chen, R. J. Williams and Z. Zhao

Suppose $D$ is an unbounded domain in $R^d$ ($d \geq 2$) with
compact boundary and that $D$ satisfies a uniform interior
cone property. We show that for $1\leq p < d$, there exists
a constant $c=c(D, p)$ such that for each
$f\in W^{1,p}(D)$ the following Sobolev inequality holds:
$$    \| f\|_q \leq c  \| \nabla f\|_p,                 $$
where $1/q=1/p -1/d$ and for $r=p, q$, $\| \cdot \|_r$
denotes the norm in $L^r(D)$.
As an application of this Sobolev inequality, assuming 
in addition that $D$ is a Lipschitz domain in $R^d$ with
$d\geq 3$, we obtain a Gaussian upper bound estimate
for the heat kernel on $D$ with zero Neumann boundary condition.

zchen@euclid.ucsd.edu (LaTeX file available)

305. ON THE EXISTENCE OF POSITIVE SOLUTIONS FOR SEMILINEAR ELLIPTIC EQUATIONS WITH NEUMANN BOUNDARY CONDITIONS

Z. Q. Chen, R. J. Williams and Z. Zhao 

We give sufficient conditions for the existence of positive solutions  
to some semilinear elliptic equations in unbounded Lipschitz domains 
$D \subset R^d$ $(d\geq 3)$, having compact boundary,
with nonlinear Neumann boundary conditions on the boundary of $D$.
For this we use an implicit probabilistic representation, 
Schauder's fixed point theorem, and a recently proved Sobolev
inequality for $W^{1,2} (D)$. 
Special cases include equations arising from the study of pattern 
formation in various models in mathematical biology and from
problems in geometry concerning the conformal deformation of metrics.

zchen@euclid.ucsd.edu (LaTeX file available)

306. SUR LA FORME DES COMPOSANTES CONNEXES DU COMPLEMENTAIRE DE LA COURBE BROWNIENNE PLANE

Wendelin Werner

We study the shape of the small connected components of the complement of a 2-dimensional Brownian path.
We show the existence of an asymptotic law for this shape. Moreover, we prove
a limit theorem that shows that the family of all the connected components
of the complement of a single path contains all the information about this
law.

W.J.J.Werner@statslab.cam.ac.uk (LateX file available)

307. SUR LES POINTS AUTOUR DESQUELS LE MOUVEMENT BROWNIEN PLAN TOURNE BEAUCOUP

Wendelin Werner

We study the asymptotic behaviour of the area $A_n$ of the set of points
around which the planar Brownian motion winds about $n$ times on a given
time-intervall $[0,t]$. We prove that $A_n$ is equivalent in the $L^2$
sense to $t/(2 \pi n^2)$ as n tends to infinity

W.J.J.Werner@statslab.cam.ac.uk (LaTeX file available)

308. FORMULE DE GREEN, LACET BROWNIEN PLAN ET AIRE DE LEVY

Wendelin Werner

We study some functionals related to the windings of the planar
Brownian loop. We derive an analog to Green's formula in the case
where the boundary is given by a planar Brownian loop. This leads
to approximations of Levy's stochastic area.

W.J.J.Werner@statslab.cam.ac.uk (LaTeX file available)

309. ESTIMATION ASYMPTOTIQUE DE RAYON DU PLUS GRAND DISQUE RECOUVERT PAR LA SAUCISSE DE WIENER PLANE

Thierry Meyre, Wendelin Werner

Let $S_t$ be the Wiener sausage of radius 1 associated with a planar
Brownian motion started at 0, on the time-interval $[0,t]$. Denote by $R(t)$
the radius of the largest disc centered at the origin covered by $S_t$. We 
obtain asymptotic results for $R(t)$ as $t \rightarrow \infty$. In particular,
we prove the existence of $c \in [1/8,1]$ such that almost surely,
$$\limsup_{t \rightarrow \infty} (\log R(t))^2/(\log t \log \log \log t)=c$$

W.J.J.Werner@statslab.cam.ac.uk (TeX-file available)

310. COMPORTEMENT ASYMPTOTIQUE DU NOMBRE DE TOURS EFFECTUES PAR LA TRAJECTOIRE BROWNIENNE PLANE

Jean Bertoin, Wendelin Werner

If $\theta_t$ denotes a continuous version of the argument
of a two-dimensional Brownian motion started at 1, we prove that
a.s., $$\limsup_{t \rightarrow \infty} {\theta_t \over f(t) \log t}
=0 or \infty}$$
according as the integral
$$\int^\infty {dt \over t f(t) \log t}$$
converges or diverges (f is a positive increasing function).

W.J.J.Werner@statslab.cam.ac.uk (LaTeX file available)

311. A WEAK QUASI-LINDELOF PROPERTY AND QUASI-FINE SUPPORTS OF MEASURES

P. J. Fitzsimmons and R. K. Getoor

We show that the fine topology of a right Markov process always enjoys a weak
form of Doob's quasi-Lindel\"of property. Using this property, we
construct $\lambda$-quasi-fine supports for measures not charging
$\lambda$-semipolar sets, where $\lambda$ is a fixed initial distribution.
We provide several other applications of the weak quasi-Lindel\"of property,
including a representation theorem for measures not charging $\lambda$-semipolar
sets.

pfitzsim@ucsd.edu  (plaintex file available)

312. ABSOLUTE CONTINUITY OF SYMMETRIC DIFFUSIONS

P. J. Fitzsimmons

Let $X$ and $Y$ be symmetric diffusion processes with a common state space, and
let $P^m$ (resp. $Q^\mu$) be the law of $X$ (resp. $Y$) with its symmetry
measure $m$ (resp. $\mu$) as initial distribution.  We study the consequences of
the  condition: $Q^\mu$ locally absolutely continuous with respect to
P^m$.  We show that under this condition there is a ``smooth'' version
$\rho$ of
the Radon-Nikodym derivative $d\mu/dm$ such that
${1\over 2}[\log\rho(X_t)-\log\rho(X_0)]=M_t+N_t$,$t<\sigma$,
where $M$ is a continuous local martingale additive functional,
$N$ is a zero-energy continuous additive functional, and $\sigma$ is an
explosion time.  The Girsanov density $L_t:=dQ^\mu|_{\FF_t}/dP^m|_{\FF_t}$ then
admits the representaion
$L_t=\exp(M_t-{1\over 2}\langle M\rangle_t)1_{\{t < \sigma\}}$.
The density $\rho$ also serves to link the Dirichlet forms of $X$ and $Y$
in a simple way.  Our identification of $L$ relies on new notions of {\it
even} and {\it odd\/} for additive functionals.  These notions complement
Fukushima's decomposition and the ``forward-backward martingale''
decomposition of Lyons and Zheng.

pfitzsim@ucsd.edu  (plaintex file available)

313. CONDITIONAL PROPAGATION OF CHAOS AND A CLASS OF QUASI-LINEAR PDE

Weian Zheng
	
      We consider in this paper conditional propagation of chaos 
and use it to solve a class of quasi-linear equations of parabolic type. 
Our method imposes less smoothness conditions on the coefficients and allow 
a degenerate non-linear weight before a divergence form operator. As its 
application, we gives solution to Stefan's problem under more general 
assumptions. We wish this probabilistic approach gives a better microscopic 
picture to explain related phase transition problems.

wzheng@math.uci.edu (LaTeX file available)

314. ASYMPTOTIC WINDINGS OF PLANAR BROWNIAN MOTION REVISITED VIA THE ORNSTEIN-UHLENBECK PROCESS

Jean Bertoin and Wendelin Werner

A celebrated Theorem of Spitzer suggests that the number of windings
made by a planar Brownian motion $Z$ around the origin and taken in the 
logarithmic time-scale , is asymptotically close to a Cauchy process.
The purpose of this paper is to show that this informal consideration
can be made precise by introducing the Ornstein-Uhlenbeck process
$X(t)= e^{-t/2} Z(e^t)$. This yields short proofs of known results
as well as some new features on the asymptotic behaviour of the 
winding number (in distribution and pathwise).

jbe@ccr.jussieu.fr or W.J.J.Werner@statslab.cam.ac.uk (Tex-file available)

315. THE INTERMEDIATE PHASE FOR THE CONTACT PROCESS ON A TREE

Rick Durrett and Rinaldo Schinazi

 
We show that in between the two
critical values for the contact process on trees, there are infinitely
many extremal nontranslation invariant stationary distributions. 
This conclusion is reminiscent of results of
Grimmett and Newman (1990) for percolation on the product of
a tree with the integers where there is an intermediate phase with
infinitely many infinite clusters.

schinazi@vision.uccs.edu      (plain Tex file available)

316. A RENORMALIZATION RESULT FOR THE DISCRETE INTERSECTION LOCAL TIME IN HIGH DIMENSIONS

Sergio Albeverio and  Xian Yin Zhou


  In this paper we derive some general conditions on stable random walks in
$Z^d$, under which the central limit theorem holds for the normalized
intersection local time of them. In particular, we prove that the process,
given by the normalized intersection local time of the simple random walk
in $Z^d$, with $d\ge 3$, is weakly convergent to the standard Brownian
motion.

Xian.Zhou@ruba.rz.ruhr-uni-Bochum.dbp.de

317. ADAPTED SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS

Xuerong Mao

Backward stochastic differential equations have been recently studied
by Pardoux and Peng with applications to quasilinear parabolic partial
differential equations.  However, so far the existence and uniqueness
of the adapted solution to the backward stochastic differntial equation
have only been established under the uniform Lipschitz condition. It
is still open whether the local Lipschitz condition works or not. The
difficulty here is that the techniques of stopping time and localization
seem not work for backward stochastic differential equations. On the
other hand it is somehow too strong to require the uniform Lipschitz
continuity in various applications e.g. in dealing with quasilinear
partial differential equations. The question is: Is there any weaker
condition than the Lipschitz one under which the backward stochastic
differential equation has a unique adapted solution? In this paper we
shall give a positive answer.

xuerong@stams.strath.ac.uk