Probability Abstracts 16

Click here to see the list of all abstract titles.

257. STRATONOVICH CALCULUS WITH SPATIAL PARAMETERS AND ANTICIPATIVE PROBLEMS IN MULTIPLICATIVE ERGODIC THEORY

Ludwig Arnold and Peter Imkeller

Let $u(t,x), t\in {\bf R},$ be an adapted process parametrized by a variable
$x$ in some metric space $X$, $\mu(\omega, dx)$ a probability kernel on the
product of the probability space $\Omega$ and the Borel sets of $X$. We deal
with the question whether the Stratonovich integral of $u(.,x)$ with respect
to a Wiener process on $\Omega$ and the integral of $u(t,.)$ with respect to
the random measure $\mu(., dx)$ can be interchanged. This question arises for
example in the context of stochastic differential equations. Here $\mu(., dx)$
may be a random Dirac measure $\delta_{\eta}(dx)$, where $\eta$ appears as an
anticipative initial condition.
We give this random Fubini type theorem a treatment which is mainly based on
ample applications of the real variable continuity lemma of Garsia, Rodemich
and Rumsey. As an application of the resulting ''uniform Stratonovich
calculus'' we give a rigorous verification of the diagonalization algorithm
of a linear system of stochastic differential equations.

258. ON THE EXISTENCE OF POSITIVE SOLUTIONS OF SEMILINEAR ELLIPTIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

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

We use an implicit probabilistic representation and
Schauder's fixed point theorem to obtain sufficient conditions for
the existence of positive continuous solutions to the
following semilinear elliptic equation with Dirichlet boundary conditions:
$$ {1\over 2}\Delta u  + F ( x , u ) = -g(x )  \quad  \hbox{for } x \in   D, $$
$$u(x) = \phi (x) \quad \hbox{for } x\in  \partial D, $$
$$\lim_{{|x| \to \infty} \atop {x \in D} } u(x)  = \alpha \quad \hbox{when
} \ D \ \ \hbox{ is unbounded,} $$
where $ D$ is a regular domain in $R ^d$ ($d\geq 3$), $\phi  $ is
a non-negative  continuous function on $ \partial D$, 
$\alpha$ is a non-negative constant  such that
when $\partial D $ is unbounded,
$\lim _{|x|\to\infty,\, x\in \partial D} \phi (x)=\alpha $,  
$ g$ is a  non-negative Green-tight function on  $D$, and 
$F$ is a real-valued Borel measurable function  defined on $ D\times
(0,b) $ for some $ b \in (0, \infty ]$ such that 
$F (x, \,\cdot \, )  $ is  continuous on $(0, b)$ for each $x \in D$  
and $ -U(x) u \leq F(x, u) \leq V(x) f(u) $ for all $( x, u) \in D \times
 (0, b)$, where $U$ and $ V$ are  
non-negative  Green-tight functions
on $D$ and $f $ is a non-negative Borel measurable  
function defined on $ (0, b)$.
Special cases of  equations of this type 
arise in connection with  superprocesses and   
problems in geometry concerning conformal deformation
of metrics.

zchen@euclid.ucsd.edu

259. ERGODIC THEORY ON GALTON-WATSON TREES, I: SPEED OF RANDOM WALK AND DIMENSION OF HARMONIC MEASURE

Russell Lyons, Robin Pemantle and Yuval Peres.
                    
We consider simple random walk on the family tree $T$ of a
nondegenerate supercritical Galton-Watson branching process and show
that the resulting harmonic measure has a.s.\ strictly smaller
Hausdorff dimension than that of the whole boundary of $T$.
Concretely, this implies that an exponentially small fraction of the
$n$th level of $T$ carries most of the harmonic measure.  First
order asymptotics for the rate of escape, Green function and the
Avez entropy of the random walk are also determined.  Ergodic theory
of the shift on the space of random walk paths on trees is the main
tool; the key observation is that iterating the transformation
induced from this shift to the subset of ``exit points'' yields a
nonintersecting path sampled from harmonic measure.

TeXfile or hardcopy available from  rdlyons@silver.ucs.indiana.edu
  pemantle@math.wisc.edu   or    peres@stat.berkeley.edu

260. ON THE FUTURE INFIMA OF SOME TRANSIENT PROCESSES

Davar Khoshnevisan, Thomas M. Lewis and Wenbo V. Li

Let $(X(t), ~t \in S)$ be a real-valued
transient stochastic process starting at zero.
In this paper we study a class of such processes
in terms of their reluctance to tend to infinity.
This includes determining the rate of escape of the
associated process $( \inf_{s \ge t} X(s), ~t \in S),$ the
so-called future infima process.

lewis@frmnvax1.bitnet

261. QUASI-HOMEOMORPHISMS OF DIRICHLET FORMS

Zhen-Qing Chen, Zhi-Ming Ma and Michael Rockner

We prove that an arbitrary (non-symmetric) Dirichlet form on a general
topological state space is quasi-regular if and only if it is
quasi-homeomorphic to a regular DIrichlet form on a locally compact
separable metric space. Several ways for implementing this 
``regularization" are described..

zchen@euclid.ucsd.edu,   unm30b@uni-bonn.de

262. SINGULARITY OF SUPER-BROWNIAN LOCAL TIME AT A POINT CATALYST

D.A. Dawson, K. Fleischmann, Y. Li, and C. Mueller

	The catalytic process, which was recently introduced by 
Dawson and Fleischmann, is an extension of the super-Brownian 
motion.  Instead of birth and death taking place everywhere, it 
can only take place on the support of the catalytic measure, 
which is a preassigned measure.  The larger the measure, the 
faster the birth and death.  One should imagine that the birth 
and death represents a chemical reaction, which can only take 
place in the presence of catalysts, which are represented by the 
catalytic measure. 
	In this work, we study the catalytic process with the 
catalyst at a single point $c$ of $\bold R$.  We expect the 
rocess to be particularly irregular at the point $c$, since this 
is the location of all the birth and death.  We define an 
occupation time measure $\lambda^{c}$ for particles at the point 
$c$, and study its properties.  Our main result is that this 
measure is not absolutely continuous with respect to Lebesgue 
measure on the time axis.
	The proof is based on a probabilistic characterization 
of the law of the Palm canonical clusters $\chi$appearing in the 
L\'{e}vy-Khinchin representation of $\lambda^{\frak c}$ in a 
historical process setting and the fact that these $\chi$ have 
infinite left upper density (with respect to Lebesgue measure) 
at the Palm time point.

cmlr@troi.cc.rochester.edu 	(latex file available)

263. AN ANALYTIC APPROACH TO FLEMING-VIOT PROCESSES WITH INTERACTIVE SELECTION

Ludger Overbeck, Michael Roeckner, Byron Schmuland

We study a class of (non-symmetric) Dirichlet forms (E,D(E)) having a
space of measures as state space S and derive some general results about them.
We show that under certain conditions they "generate" diffusion processes M. 
In particular, if M is ergodic and (E,D(E)) symmetric, w.r.t. every starting 
point the large deviations of the empirical distribution of M are governed by E.
We apply all of this to construct Fleming-Viot processes with interactive selec
tion and prove some results on their behaviour. Among other things we show some
support properties for these processes using capacitary methods.

UNM30A@IBM.rhrz.uni-bonn.de

264. MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

Zhi-Ming Ma, Ludger Overbeck, Michael Roeckner

We prove that ( as for a Dirichlet form) quasi-regularity of a Semi-
Dirichlet form is necessary and sufficient for the existence of an associated
special standard Markov process. Such a form by definition has the contraction
property only w.r.t. one of its arguments. Because of lack of duality some new
results on the analytic potential theory of Semi-Dirichlet forms on measurable
spaces are required and are proved in this paper.

UNM30A@IBM.rhrz.uni-bonn.de

265. HOW MANY DIFFUSIONS EXIST ON THE VICSEK SNOWFLAKE?

Volker Metz

We consider Kusuoka's construction of Dirichlet forms on
the Vicsek snowflake, a nested fractal. His method is
generalized and all irreducible, local Dirichlet forms
which can be constructed in this way are characterised.
We end up with a one-parameter family of different possible
forms. This proves that "Brownian motion" on nonnested
fractals is not unique if the isometry group of the fractal 
is too small.

metz@mathematik.uni-bielefeld.de    (AMStex file available)

266. A SKOROHOD-TYPE LEMMA AND A DECOMPOSITION OF REFLECTED BROWNIAN MOTION

Krzysztof Burdzy and Ellen Toby

We consider 2-dimensional reflected Brownian motions in sharp thorns 
pointed downward with horizontal vectors of reflection.  We present 
a decomposition into a Brownian motion and a process which has bounded 
variation away from the tip of the thorn.  The construction is based
on a new Skorohod-type lemma.

toby@math.tamu.edu

267. THE HEAT EQUATION WITH LEVY NOISE

Carl Mueller

SPDE with white noise often fail to have function solutions in 2 or
more spatial dimensions.  This makes it hard to give meaning to 
nonlinear terms.  One alternative is to consider Levy noise
$L(dyds)$.  This noise is defined in the same way as the Brownian 
sheet, but with marginal distributions given by a Levy process.
We consider Levy noise which is nonnegative and stable with index
$p\in (0,1)$.  We are able to show short time existence for the 
equation
        u_t=\Delta_\alpha u+u^\gamma\dot L(t,x)
	u(t,x)=0  for x\in D^c         
	u(0,x)=u_0(x)
provided
	d<\frac{(1-p)\alpha}{\gamma p-(1-p)}.
Here, $\Delta_\alpha$ is the $\alpha$ power of the Laplacian.  We
assume that $x\in D$, where $D$ is a smooth bounded domain in $R^d$.
$\dot L$ is the mixed partial derivative of the sheet $L$.  $dot L$
has many of the same properties as white noise.  One motivation for
studying equations of this form is given by the stable measure-valued
branching processes studied by Watanabe, Dawson, and others.  If 
such processes have a density $u(t,x)$, it is thought that it would
satisfy the above equation, with $\gamma=1/p$.  Unfortunately, the 
case most often studied is for $p>1$, which we cannot deal with.  
We only give some tenous connections between our equation and such 
measure-valued processes.  For the proof, we look at how close atoms
of the Levy measure $L$ can be.  If they are too close, the process 
may blow up immediately, but otherwise we can show existence of a 
solution.  We cannot show uniqueness, but our solution is minimal.
The solution is rather singular, since it has infinities at a countable
dense set of points.

cmlr@troi.cc.rochester.edu       (AMSLatex file available)

268. DIRICHLET FORM METHODS FOR UNIQUENESS OF MARTINGALE PROBLEMS AND APPLICATIONS

Sergio Albeverio and Michael Roeckner

Some new results on Markov uniqueness of generators (L,D(L)) of (non-symmetric)
Dirichlet forms are proved.More precisely,we show that in certain cases L is
already completely determined on some subdomain D in D(L) of "smooth" functions
(e.g.,D=compactly supported infinitely differentiable functions in the case of
finite dimensional state spaces or D=smooth bounded cylinder functions in
infinite dimensional cases) as the only operator having D in its domain and
generating a strongly continuous sub-Markovian contraction semi-group.We also
discuss applications to prove uniqueness of the martingale problems given by
(L,D).In particular,we prove that the Fleming-Viot processes with interactive
selection recently constructed by L.Overbeck,B.Schmuland,and M.Roeckner
are the unique solutions of certain martingale problems.We also describe
special cases where D is a core for (L,D(L)) (i.e., in the symmetric case
(L,D) is essentially self-adjoint) reviewing recent results by J.Kondratiev
and the two authors.

UNM30B@IBM.rhrz.uni-bonn.de

269. PLANAR FIRST PASSAGE PERCOLATION TIMES ARE NOT TIGHT

Robin Pemantle and Yuval Peres

Let the edges of Z^2 be assigned IID passage times that are 
exponentials of mean one, and let T(0,n) denote the resulting 
first-passage time from the origin to the point (0,n).  We show
that T(0,n) is not tight around its median.  A fractional power 
lower bound for the dispersion of T(0,n) may be obtained by combining 
this method with that of Newman and Piza.

pemantle@math.wisc.edu (latex file available)

270. GALTON-WATSON TREES WITH THE SAME MEAN HAVE THE SAME POLAR SETS

Robin Pemantle and Yuval Peres

Steve Evans defines a notion of what it means for a set B to
be polar for a process indexed by a tree, T.  The main result
herein is that a tree picked from a Galton-Watson measure whose 
offspring distribution has mean m and finite variance will 
almost surely have precisely the same polar sets as a deterministic 
tree of the same growth rate.  
	This implies that deterministic and nondeterministic trees 
behave identically in a variety of probability models. Mapping subsets 
of Euclidean space to trees and polar sets to capacity criteria, 
it also follows that certain random Cantor sets are capacity-equivalent 
to each other and to deterministic Cantor sets.   

pemantle@math.wisc.edu (latex file available)

271. EXISTENCE AND UNIQUENESS OF SEMIMARTINGALE REFLECTING BROWNIAN MOTIONS IN CONVEX POLYHEDRONS

J. G. Dai and R. J. Williams

We consider the problem of existence and uniqueness of
semimartingale reflecting Brownian motions (SRBM's) in convex
polyhedrons. Loosely speaking, such a process has a semimartingale
decomposition such that in the interior of the polyhedron the process
behaves like a Brownian motion with a constant drift and covariance
matrix, and at each of the $(d-1)$-dimensional faces that form the
boundary of the polyhedron, the bounded variation part of the process
increases in a given direction (constant for any particular face), so
as to confine the process to the polyhedron.  For historical reasons,
this ``pushing'' at the boundary is called instantaneous reflection.
For simple convex polyhedrons, we give a necessary and sufficient
condition on the geometric data for the existence and uniqueness of an
SRBM. For non-simple convex polyhedrons, our condition is shown to be
sufficient. It is an open question as to whether our condition is also
necessary in the non-simple case. From the uniqueness, it follows that
an SRBM defines a strong Markov process.  Our results have application
to the study of diffusions arising as heavy traffic limits of
multiclass queueing networks and in particular, the non-simple case
has application to multiclass fork and join networks.  Our proof of
weak existence uses a patchwork martingale problem introduced by T. G.
Kurtz, whereas uniqueness hinges on an ergodic argument similar to
that used by L. M. Taylor and R. J. Williams to prove uniqueness for
SRBM's in an orthant.

dai@isye.gatech.edu