Probability Abstracts 12

Click here to see the list of all abstract titles.

197. CONDITIONED SUPERPROCESSES AND THEIR WEIGHTED OCCUPATION TIMES

Stephen Krone

  A (Dawson-Watanabe) superprocess which starts at a finite measure will
die in finite time.  We consider a class of measure-valued Markov
processes obtained by conditioning such a process to stay alive forever.
More specifically, we study the asymptotic behavior of the weighted
occupation times for these "conditioned superprocesses".  This behavior
is determined by that of the underlying motion for the superprocess.

krone@math.utah.edu  (Latex file available)

198. COUPLING AND HARNACK INEQUALITIES FOR SIERPINSKI CARPETS

M.T. Barlow and R.F. Bass 

Uniform Harnack inequalities for harmonic functions on the pre-- and graphical 
Sierpinski carpets are proved using a probabilistic coupling argument. Various 
results follow from this, including the construction of Brownian motion on 
Sierpinski carpets embedded in $\R^d$, $d\geq 3$, estimates on the fundamental 
solution of the heat equation, and Sobolev and Poincar\'e inequalities. 
(This short paper is an announcement of results which will be published
in detail later.)

bass@math.washington.edu or barlow@unixg.ubc.ca  (AMSTeX file available)

199. SUPERPROCESS LOCAL AND INTERSECTION LOCAL TIMES AND THEIR CORRESPONDING PARTICLE PICTURES

Robert J. Adler

The super Brownian motion has a local time up to three dimensions, an
intersection (over disjoint time sets) local time up to dimension seven,
and a renormalised self-intersection local time up to dimension five.
The branching Brownian motions that, in the infinite density limit, provide
a particle picture for the super Brownian motion have a local time in only
one dimension, an intersection local time up to three dimensions, and a
renormalised self-intersection local time only in dimensions one and two.
This paper is primarily  concerned with the ``dimension gap" between the
particle picture and the superprocess, and shows how to explain this gap
in terms of weak convergence of functionals of the finite system to
functionals on the superprocess.
The first section gives a detailed review of the recent literature on local
and intersection local times for super Brownian and super stable processes.

ierhe01@technion.technion.ac.il

200. SOME MIXING CONDITIONS FOR STATIONARY SYMMETRIC STABLE STOCHASTIC PROCESSES

Aaron Gross

We derive some necessary and sufficient conditions for mixing
and weak mixing of stationary symmetric stable processes in terms of their
spectral representations.  The results apply to those stationary processes
which have a spectral representation $(\int{f_t} \, dM)_{t \in \R}$ where 
$M$ is an independently scattered symmetric stable random measure and the
process $(f_t)_{t \in \R}$ is also stationary.

gross@sunrise.huji.ac.il

201. APPROXIMATE SOLUTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH PATHWISE UNIQUENESS

Xuerong Mao

In this paper we study the Caratheodory arrpoximation for the solution
of a stochastic differential equation.  It is shown that the pathwise
uniqueness of the solution implies the convergence in the strong
sense of the Caratheodory approximation to the solution. This general
result together with the existing results on the pathwise uniqueness
of the solution enable us to obtain the approximate solutions for 
stochastic differential equations with Holder continuous coefficients.
Perhaps, it should be pointed out that the Picard approximation
problem for stochastic differential equations with Holder continuous
coefficients is still open.

xuerong@uk.ac.strath.stams

202. YULE PROCESS APPROXIMATION FOR THE SKELETON OF A BRANCHING PROCESS

Neil O'Connell

The skeleton (sub-tree of individuals with infinite line of
descent) of a supercritical Galton-Watson process
with offspring mean $1+r$, $r\ge 0$, and finite offspring
variance, is considered.  When $r=0$ it is trivial.  If 
$r>0$ is small and the time unit is taken as $\alpha /r$
generations ($\alpha > 0$) then the skeleton can be approximated
by a Yule (linear pure birth) process of rate $\alpha$.
This approximation can be used to study the evolution of
genetic types over a long period of time in an exponentially
growing population.

neil@stat.berkeley.edu (LaTeX file available)

203. THE GENEALOGY OF BRANCHING PROCESSES AND THE AGE OF OUR MOST RECENT COMMON ANCESTOR

Neil O'Connell

We obtain an stickbreaking branching process
approximation for the reduced family tree
in a near-critical Markov branching process
when the time interval considered is long,
and apply it to the problem of estimating
the age of our most recent common female
ancestor.

neil@stat.berkeley.edu (LaTeX file available)

204. WEIGHTED OCCUPATION TIMES FOR BRANCHING PARTICLE SYSTEMS AND A REPRESENTATION FOR THE SUPERCRITICAL SUPERPROCESS

Steven N. Evans and Neil O'Connell

We obtain a representation for the supercritical
Dawson-Watanabe superprocess in terms of a subcritical
superprocess with immigration, where the immigration
at a given time is governed by the state of an underlying
branching particle system.  The proof requires a new 
result on the laws of weighted occupation times for 
branching particle systems.

neil@stat.berkeley.edu (LaTeX file available)

205. MEASURE-VALUED BRANCHING DIFFUSIONS WITH SINGULAR INTERACTIONS

Steven N. Evans and Edwin A. Perkins

The usual critical super-Brownian motion is 
a measure-valued process that arises as a high density limit 
of a system of branching Brownian particles in which the 
branching mechanism is critical.  In this work we consider 
analogous processes that model the evolution of a system of 
two such populations
in which there is inter-species competition or predation.

We first consider a competition model in which inter-species collisions
may result in casualties on both sides.  Using a Girsanov approach, we obtain
existence and uniqueness of the appropriate martingale problem in
one dimension.  In two and three dimensions we establish existence only.
Although the supports of two independent super-Brownian motions collide in
dimensions four and five, we show that there is no solution to the martingale
problem in these cases.

We next study a predation model in which collisions only affect
the ``prey'' species.  Here we can show both existence and uniqueness in
one, two and three dimensions.  
Again, there is no solution in four and
five dimensions.  As a tool for proving uniqueness, we
obtain a representation of martingales for a super-process as
stochastic integrals with respect to the related orthogonal martingale measure.
  
We also obtain existence and uniqueness for a related single population
model in one dimension in which particles are killed at a rate 
proportional to the local density.  This model appears as a limit of a
rescaled contact process as the range of interaction goes to infinity.

evans@gandalf.Berkeley.EDU (Latex file available)
 

206. A CHARACTERIZATION OF STOPPING TIMES (AN EXTENSION OF A RESULT OF A.O. PITTENGER)

Frank Knight

A result of Pittenger (1981) asserts that if $X_t$ is a right 
process with natural augmented filtration $\Cal F_t$, a random 
time $0\leq R \in \Cal F_\infty$ is a stopping time if (and 
only if), given $\Cal F_R,\, \, X_{R+(\cdot)}$ is Markovian with the 
transition function of $X$.  We generalize this as follows.  Let 
$X_t$ be any measurable process with natural augumented pseudo-
path filtration $\Cal F_t$, and let $Z_t$ be the process of 
wide-sense conditional futures of $X_t$ given $\Cal F_{t+}$, where 
$Z_t$ is right-continuous in a weak topology of measures.  Then 
$0 \leq R \in \Cal F_{\infty}$ is a stopping time if (and only 
if) $Z_R$ defines the conditional distribution of $X_{R+(\cdot)}$ given 
$\Cal F_{R+}$.  An analogous characterization is provided for 
previsible stopping times.  The proofs illustrate the use of the 
author's ``prediction process.''

britt@symcom.math.uiuc.edu
 

207. EXISTENCE OF QUASI STATIONARY DISTRIBUTIONS. A RENEWAL DYNAMIC APPROACH.

P. A. Ferrari, H. Kesten, S. Martinez and P. Picco

We consider Markov chains on the positive integers for which the origin is an
absorbing state.  Quasi stationary distributions (qsd) are described as fixed
points of a transformation $\Phi$ in the space of probability measures. Under
the assumption that the absorption time at the origin, $R$, of the chain
starting from state $x$ goes to infinity in probability as $x\to\infty$, we
show that the existence of a qsd is equivalent to $E_xe^{\lambda R}<\infty$
for some positive $\lambda$ and $x$.  We also prove that a subsequence of
$\Phi^n\de_x$ converges to a minimal qsd. For a birth and death chain we prove
that $\Phi^n\de_x$ converges along the full sequence to the minimal qsd.  The
method is based on the study of the renewal process with interarrival times
distributed as the absorption time of the chain with a given initial measure
$\mu$. The key tool is the fact that the residual time in that renewal process
has as stationary distribution the distribution of the absorption time of
$\Phi\mu$.

pablo@ime.usp.br (plain tex file available)
 

208. ON THE FILTRATION OF HISTORIAL BROWNIAN MOTION

 M. T. Barlow and Edwin A. Perkins 

We show that the historical Brownian motion may be recovered from 
ordinary super-Brownian motion when the dimension $d$ of the underlying 
Brownian motion is greater than 4.  We outline a proof, involving
a pair of interacting measure-valued diffusions, which shows that this 
conclusion is false if $d\leq 3$. We also give in full a much simpler 
argument for the case $d=1$. The state of affairs in the critical 
dimension $d=4$ is left unresolved. Some extensions are given for $1+\beta$ 
stable branching mechanisms where $\beta \in(0,1]$. 

barlow@unixg.ubc.ca

209. FRACTALS AND DIFFUSION-LIMITED AGGREGATION

M.T. Barlow

This is a short survey paper, which gives an introduction to DLA. It 
contains a description of the model, a brief account of Kesten's lower 
bound on the cluster growth rate. Various questions about the asymptotic 
shape of the cluster are raised, as well as some possibly more tractable 
problems. This paper will appear in Bull. Math. Sci. in 1993, in the 
proceedings of the 1992 St. Cheron 'Round Tables'. 

barlow@unixg.ubc.ca

210. ASYMPTOTIC CRITICAL VALUE FOR A COMPETITION MODEL

Richard Durrett and Rinaldo Schinazi

In this paper we study a model of the competition of annual
and perennial plants proposed by Crawley and May (1987).
Specifically, we calculate the asymptotic behavior of the 
critical value for coexistence in the biologically reasonnable 
limit in which annual seeds are dispersed over a large number
of sites. Our results are related to earlier work of Durrett and
Swindle (1991), and prove a conjecture made in that paper.

schinazi@vision.uccs.edu

211. THE SELF-AFFINE CARPETS OF MCMULLEN AND BEDFORD HAVE INFINITE HAUSDORFF MEASURE

Yuval Peres

We show that the self-affine sets considered by
McMullen(1984) and Bedford(1984) have infinite Hausdorff
measure in their dimension,except in the (rare) cases where
their Hausdorff dimension coincides with their Minkowski (=Box)
dimension.This resolves a question of McMullen,which was
partially answered by Lalley and Gatzouras(1992) who showed
The Hausdorff measure is EITHER zero or infinite.
 Almost Precise estimates on gauge functions
are obtained(for ANY gauge function,the Carpets have
either zero or infinite Hausdorff measure).The proof
of the upper bound relies on the Rogers-Taylor density
Theorem to control "typical" points,and large deviation
estimates to handle nontypical points.

peres@math.yale.edu (TeX file available)

212. VARIATION OF ITERATED BROWNIAN MOTION

Krzysztof Burdzy

Suppose that $X^1,X^2$ and $Y$ are independent standard Brownian
motions starting from 0 and let
$$X(t) = \cases X^1(t)  & \text{if } t\geq 0,\\
                X^2(-t) & \text{if } t < 0. \endcases$$
The process $\{Z(t) = X(Y(t)), t\geq 0\}$ will be called
``iterated Brownian motion'' or IBM.
 The 4-th variation of IBM is equal to $3t$. The 3-rd
variation is equal to 0.
 Let $t_0 \leq t_1 \leq \dots \leq t_n$ be a partition
of $[0,t]$ and let
$$V_n(t_m) = \sum _{k=1}^m (Z(t_k) - Z(t_{k-1}))^2
sign(Z(t_k) - Z(t_{k-1})).$$
Extend $V_n$ continuously to all positive reals.
When the mesh of the partition goes to 0,
the processes $V_n$ converge in distribution 
to a Brownian motion with variance $3t$. In other words,
the signed quadratic variation of IBM is equal
in a weak sense to a Brownian motion.

burdzy@math.washington.edu (AmSTeX file available)

213. A SUPER-BROWNIAN MOTION WITH A SINGLE POINT CATALYST

Don Dawson, Klaus Fleischmann 

     A one-dimensional continuous measure-valued branching process is 
discussed, where branching occurs only at a single point, i.e. at a point 
catalyst. If we exclude the location  c  of the catalyst, then a jointly 
continuous mass density field exists, which stochastically degenerates to 0 
approaching the catalyst at a fixed time. Roughly speaking, normally mass 
will be killed by the catalyst. On the other hand, the related occupation 
time process has a density field which is everywhere jointly continuous. The 
occupation density measure even at  c  has carrying Hausdorff dimension 1, 
that is, there is production of mass even at a set of time points of full 
dimension. - The main analytical tool is a class of non-linear reaction 
diffusion equation in which delta-functions enter in three different ways.

fleischmann@iaas-berlin.dbp.de

214. ENHANCING THE MODEL OF PHYSICAL BROWNIAN MOTION

Joseph C. Watkins

ABSTRACT. Consider a bath of molecules, each having mass epsilon evolving in
equilibrium in a gravitational and electric field. Molecules can be introduced
or created and intermolecular collisions are simulated by replacing actual
collisions with collisions based on a statistical choice under local 
equilibrium conditions. Introduce a massive Brownian particle and allow it to
interact with the molecules via elastic collisions. We investigate the motion
of the Brownian particle and describe its evolution in the limit as epsilon 
tends to zero keeping the mean kinetic energy of the molecules fixed. This
yields a diffusion which reduces, in a homogeneous environment to the Ornstein
Uhlenbeck process. The physcial parameters in the model are typically quite
large, thus we rescale to give apprpriate analogies to Einstein's model of
physical Brownian motion.

jwatkins@math.arizona.edu