Probability Abstracts 5

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78. EXISTENCE OF QUASI STATIONARY DISTRIBUTIONS. A RENEWAL DYNAMICAL APPROACH

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

Quasi stationary distributions (qsd) are described as fix points of a
transformation $T$ in the space of probability measures. For a given
probability measure $\mu$ we give sufficient conditions for the
existence of $\lim_{n\to\infty} T^n\mu$ and for this limit to be a
qsd.  For the birth and death chain, calling $R(\delta_1)$ the
absorbing time of the chain with starting measure $\delta_1$, that
concentrates mass on the state $1$, we show that the existence of qsd
is equivalent to $Ee^{\theta R(\delta_1)}<\infty$ for some positive
$\theta$. Moreover we prove that $T^n\de_1$ converges to the minimal
qsd.  The method is based on the study of the renewal process with
interarrival times distributed as the absorbing time of the chain with
initial measure $\mu$. The key tool is the fact that the residual time
of that renewal process has the same distribution as the absorbing
time of $T\mu$.

pablo@ime.usp.br, smartine@uchcecvm.bitnet, picco@acf3.nyu.edu
     plain tex file available

79. LOCAL TIMES FOR SUPERDIFFUSIONS

Stephen M. Krone

In this work we study local times for a class of measure-valued 
Markov  processes known as superprocesses.
We begin by deriving analogues of well-known properties of ordinary
local times.  Then, restricting our attention to a class of 
superprocesses (which includes the important case of super-Brownian
motion), we prove more detailed properties of the
local times, such as joint continuity and a global H\"older
condition.  These are then used to obtain path properties of
the superprocesses themselves.  For example, we compute
 the Hausdorff dimension of the ``level
sets'' of  super-Brownian motion. 

krone@math.utah.edu

80. ERGODIC THEOREMS FOR INFINITE SYSTEMS OF LOCALLY INTERACTING DIFFUSIONS

Ted Cox and Andreas Greven

Let $x(t) = \{ x_i(t) , i \in \Bbb Z^d\}$ be the solution of the
system of stochastic differential equations 
$$ 
dx_i(t) = \left(\sum_{j\in\Bbb Z^d} a(i,j)x_j(t) - x_i(t)\right)dt +
\sqrt{2g(x_i(t))} dw_i(t), \quad i \in \Bbb Z^d .  
$$ 

Here $g:[0,1] \to \Bbb R^+$ satisfies $g>0 \text{ on } (0,1)$,
$g(0)=g(1)=0$, $g$ is Lipschitz, $a(i,j)$ is an irreducible random
walk kernel on $\Bbb Z^d$, and $\{w_i(t), i \in \Bbb Z^d\}$ is a
family of standard, independent Brownian motions on $\Bbb R$ .  $x(t)$
is a Markov process on $X=[0,1]^{\Bbb Z^d}$.  The special case
$g(v)=v(1-v)$ has been studied extensively, especially by Shiga.

We show that the long term behavior of $x(t)$ depends only on 
$\widehat a(i,j)= ((a(i,j)+a(j,i))/2$ and
is universal for the entire class of $g$ considered.
If $\widehat a(i,j)$ is transient, then there 
exists a family $\{\nu_{\theta}, \theta \in [0,1]\}$ of extremal, 
translation invariant equilibria.  Each $\nu_{\theta}$ is mixing,
and has density $\theta=\int x_0 d\nu_{\theta}$.
If $\widehat a(i,j)$ is recurrent, then the
set of extremal translation invariant equilibria consists of the
point masses $\{\delta _{\bk 0}, \delta_{\bk 1}\}$.  The
process starting in a translation invariant, shift ergodic measure
$\mu$ on $X$ with $\int x_0d\mu=\theta$ converges weakly as 
$t \to \infty$ to $\nu_{\theta}$ if $\widehat a(i,j)$ is transient, and to 
$(1-\theta) \delta_{\bk 0} + \theta \delta_{\bk 1}$ if
$\widehat a(i,j)$ is recurrent. For the case $\widehat a(i,j)$ 
transient we use methods developed for infinite particle systems
by Liggett and Spitzer (see \cite{13}). For the case 
$\widehat a(i,j)$ recurrent we use a duality comparison argument.

jtcox@mailbox.syr.edu

81. RANDOM WALKS ON A TREE AND CAPACITY IN THE INTERVAL.

Itai Benjamini and Yuval Peres

Every compact set K in the unit interval corresponds to a tree
via base b expansions .For instance,in base 3 the classical
Cantor set corresponds to a binary tree ;	In base 5
the same set corresponds to a very complicated tree .
Nevertheless,in general transience of the simple random walk
on this kind of tree depends only on the compact set K
and not on the base (in fact transience is equivalent to
K being nonpolar for planar brownian motion).
Other results concerning the harmonic measures for the random 
walk and their Hausdorff dimension are obtained.We note the
following criterion,which is a consequence of the work
of Terry Lyons and Russell Lyons:
"Simple RW on a tree is transient if for some real C
there are arbitrarily many vertices with average meeting height
(above the root) less than C."
   
peres@lom1.math.yale.edu (Hardcopy available from  Yuval Peres.)

82. MARKOV CHAINS INDEXED BY TREES.

Itai Benjamini and Yuval Peres
    
Performing a random walk on a graph can often enlighten us on
its structure.However if the graph is very large(e.g.  a 
high-dimensional lattice or a Cayley graph of a nonamenable
group) the random walk will be transient and will visit 
only a tiny part of G. Consequently,many results on such
random walks involve estimating probabilities rather than
sample path properties.Tree indexed chains provide a solution.
An ordinary Markov chain is a random mapping from the integers
to the state space G ;In this paper we study random mappings from
a fixed tree to G with the Markov property.
(in the special case where the tree arises from
a Galton-Watson process,branching random walk is recovered).
The transience / recurrence questions for these chains are
considerably richer than for ordinary Markov chains and are the
main focus of this paper.Dimension notions for trees
(e.g. Hausdorff dimension, Tricot packing dimension)
play a vital role.

peres@lom1.math.yale.edu (Hardcopy available from  Yuval Peres.)

83. A TOPOLOGICAL CRITERION FOR HYPOTHESIS TESTING

Amir Dembo and Yuval Peres

Suppose you are given ,sequentially,independent observations X1,X2,...
from an unknown distribution with finite variance on the real line,
with the goal of deciding whether the mean is in A or in B  
(with A,B given disjoint sets).For each n,after viewing 
X1,...,Xn   you guess A or B  ;
When can you ensure ,with probability 1,that you will guess correctly
from some point on?
Extending previous work of Cover and Kulkarni&Zeitouni,
this note shows this is possible iff A and B are seperated
by F-sigma sets  (e.g. the Cantor set and its complement are O.K
                  but the rationals versus the irrationals are not).

peres@lom1.math.yale.edu (Hardcopy available from Y. Peres.)
amir@playfair.stanford.edu (TeXfile available from A. Dembo.)  

84. TRACES OF RANDOM VARIABLES ON WIENER SPACE AND THE ONSAGER MACHLUP FUNCTIONAL

Rene A. Carmona and David Nualart

Let F be a square integrable random variable on the classical Wiener space and 
let us denote by f_n the  symmetric kernels appearing in the chaos expansion 
of F. We give sufficient conditions for the unambiguous definition of F at the
points of the Cameron Martin space. We also give conditions for the existence 
of approximate limits (in the sense of Denjoy). These results are first 
proved for multiple integrals I_n(f_n) before they are extended to general 
random variables F. The conditions are given in terms of the L_p norms of the 
kernels f_n. The last section of the paper is devoted to an original 
application of our estimates to the proof of the existence of the 
Onsager-Machlup functional for a nonanticipative stochastic differential 
equation of a special type.

rcarmona@orion.oac.uci.edu

85. REFLECTING BROWNIAN MOTIONS AND COMPARISON THEOREMS FOR NEUMANN HEAT KERNELS

Rene A. Carmona and Weian Zheng

We consider the challenging  problem of the Chavel's conjecture on the
domain monotonicity of the fundamental solution of the Neuman problem.
It says that if D  contained in  D' are open convex domains and if 
p(t,x,y) and p'(t,x,y) denote the corresponding parabolic heat
kernels for the respective Neuman problems, then: 

p(t,x,y) >= p'(t,x,y) for all x and y in D.

 The quantities p(t,x,y) and p'(t,x,y) can be interpreted as the 
transition densities of the reflecting Brownian motions  in D and  D'
respectively. The conjecture can be restated as a comparison problem
for Brownian motions with reflecting boundary conditions. We
give a detailed analysis of the small time asymptotics of the Brownian
paths reflected on the boundary of a polyhedron and we give a
probabilistic proof of Chavel's conjecture for small time t and x and
y away from some parts of the boundary. An interesting byproduct of 
our proof is that it does not require the large domain to be convex. 

rcarmona@orion.oac.uci.edu

86. ITERATING VON-NEUMANN'S PROCEDURE FOR EXTRACTING RANDOM BITS

Yuval Peres

Given a sequence of independent biased random bits,
von-Neumann's simple procedure extracts independent unbiased bits.
We show that iterating this procedure on the information it discards
yields an algorithm which extracts unbiased bits at a rate arbitrarily
close to the entropy bound.

peres@lom1.math.yale.edu

87. ANALYTIC DEPENDENCE OF LYAPUNOV EXPONENTS ON TRANSITION PROBABILITIES

Yuval Peres

Consider the product of i.i.d random matrices,
with each factor chosen from the k nonsingular matrices
A1,A2,...,Ak
with corresponding probabilities
p1,p2,...,pk.
Theorem:If the top Lyapunov exponent(i.e., the rate of exponential
         growth) for this random product is different from the next 
         exponent,then it is locally a real analytic function
         of the probabilities p1,p2,...pk.
         
This contrasts with examples due to Henkin,Simon and Taylor
which show that the top Lyapunov exponent might not depend smoothly
on the entries of A1,...,Ak.

When the matrices A1,A2,... are positive ,
we exhibit explicit domains of analytic continuation
for the top Lyapunov exponent as a function of the probabilities
p1,p2,...,pk.

peres@lom1.math.yale.edu

88. AN INTRODUCTION TO THE THEORY OF (NON-SYMMETRIC) DIRICHLET FORMS

Zhi-Ming Ma and Michael Roeckner

This book is intended to give an (as economic as possible) introduction
to the theory of (non-symmetric) Dirichlet forms on general state spaces
inluding both the analytic and probabilistic part of the theory.Topological
assumptions on the state space (though of a very weak kind) are only necessary
in the latter part,whereas in the first it only has to be a measuarble space.
The specific topics covered by the book are presented in the "Contents" below.
In addition, we emphasize the following results which are included.
a)We give a new proof for M. Fukushima's result that" Markov property" and
"unit contraction property" are equivalent for closed symmetric forms which
also works in the non-symmetric case.
b)We present a proof of the equivalence of the sector condition of a Dirichlet
form with the analyticity of the corresponding semigroup.
c)We identify an analytic property of a Dirichlet form called "quasi-
regularity" and prove that it is equivalent with the existence of an associated
standard Markov process respectively of an associated pair of right processes.
d)All results known about the analytic (and probabilistic) potential theory
of regular Dirichlet forms on locally compact state spaces are extended to
quasi-regular Dirichlet forms on arbitrary topological state spaces.
e)We prove a one to one correspondence between all (equivalence classes of)
pairs of "m-sectorial" right processes and all quasi-regular Dirichlet forms.
f)We characterize those quasi-regular Dirichlet forms for which the associated
process is a diffusion and those for which it is a Hunt process.
g)We describe a general compactification method for quasi-regular Dirichlet
forms which make all results from the classical "locally compact regular"
theory applicable to this much wider class.
h)Several examples where the state space is infinite dimensional are discussed.

The book is based on various joint papers of S. Albeverio and the two authors,
and a lecture course held by the second-named author at the University of Bonn
in winter-term 1990/91.All new results have been presented in detail at the
"Spring School on Dirichlet forms"at Paseky,CSFR,April 1991,at the "EC-Twinning
Project" meeting at Bielefeld,March 1991,and at the "International Conference
on Potential Theory" at Amersfoort,The Netherlands,August 1991.

Contents
  0.Introduction
I.Functional analytic background
  1.Resolvents,semigroups,generators
  2.Coercive bilinear forms
  3.Closability
  4.Contraction properties
  5.Notes/References
II.Examples
  1.Input: operator
  2.Input: bilinear form - finite dimensional case
  3.Input: bilinear form - infinite dimensional case
  4.Input: semigroup of kernels
  5.Input: resolvent of kernels
  6.Notes/References
III.Analytic potential theory of Dirichlet forms
  1.Excessive functions and balayage
  2.Exceptional sets and capacities
  3.Quasi-continuity
  4.Notes/References
IV.Markov processes associated with Dirichlet forms
  1.Basics on Markov processes
  2.Association of right processes and Dirichlet forms
  3.Construction of the process
  4.Examples of quasi-regular Dirichlet forms
  5.Necessity of quasi-regularity and some probabilistic potential theory
  6.One to one correspondences
  7.Notes/References
V.Characterization of Hunt processes and diffusions
  1.Local property and continuity of sample paths
  2.A new capacity and Hunt processes
  3.Notes/References
VI.Reduction to the locally compact regular case
  1.Compactification
  2.Consequences
  3.Notes/References
VII.Appendix
  1.Some basics from functional analysis
  2.The Banach-Alaoglu and Banach-Saks theorems
  3.Supplements on Markov processe and Ray resolvents
Terminology index
List of symbols
References

UNM30B@DBNRHRZ1.bitnet (M. Roeckner)

89. QUASI-REGULAR DIRICHLET FORMS AND MARKOV PROCESSES

S. Albeverio,Z.M.Ma and M. Roeckner

We identify an analytic property of a (non-symmetric) Dirichlet form on
general (topological) state space called "quasi-regularity" which we then
prove to be equivalent with the existence of an associated standard Markov
process,respectively the existence of an associated pair of right processes.
In addition, we prove a one to one correspondence between all quasi-regular
Dirichlet forms and all (equivalence classes of) pairs of "m-sectorial"
right processes.

UNM30B@DBNRHRZ1.bitnet (M. Roeckner)

90. CHARACTERIZATION OF HUNT PROCESSES AND DIFFUSIONS ASSOCIATED WITH DIRICHLET FORMS

S.Albeverio,Z.M.Ma and M.Roeckner

Given a quasi-regular Dirichlet form E ( or equivalently a Dirichlet form
having associated to it a standard Markov process) we prove a necessary
and sufficient analytic condition on E for the process to be a Hunt process.
The condition is in terms of a new capacity associated with E.We also extend
the result that sample path continuity (up to lifetime) for the process is
equivalent to the local property of E (proved by M.Fukushima for regular
Dirichlet forms in the locally compact case),to quasi-regular Dirichlet
forms on a large class of (not necessarily locally compact) topological state
spaces.

UNM30B@DBNRHRZ1.bitnet (M. Roeckner)

91. A REMARK ON THE SUPPORT OF CADLAG PROCESSES

S.Albeverio,Z.M.Ma and M.Roeckner

Modifying a method in a joint paper of T.Lyons and the last-named author
we prove that every right continuous stochastic process with left limits
up to lifetime on a metrizable co-Souslin space is supported (before life-
time) by a union of compacts.Some applications are described.

UNM30B@DBNRHRZ1.bitnet (M. Roeckner)

92. NON-SYMMETRIC DIRICHLET FORMS AND MARKOV PROCESSES ON GENERAL STATE SPACE

S.Albeverio,Z.M.Ma and M.Roeckner

This paper is a summary of the main results on Dirichlet forms in recent
joint work.

UNM30B@DBNRHRZ1.bitnet (M. Roeckner)

93. ON THE LIMITING DISTRIBUTION OF A SUPERCRITICAL BRANCHING PROCESS IN A RANDOM ENVIRONMENT

B.M. Hambly

We consider an increasing supercritical branching process in a random
environment and obtain bounds on the Laplace transform and
distribution function of the limiting random variable. There are two
possibilities that can be distinguished depending on the nature of the
component distributions of the environment. If the minimum family size
of each is one, the growth will be as a power depending on a parameter
$\alpha$. If the minimum family sizes of some are greater than one, it
will be exponential, depending on a parameter $\gamma$. We obtain
bounds on the distribution function analogous to those found for the
simple Galton-Watson case. It is not possible to obtain exact estimates
and we are only able to obtain bounds to within $\epsilon$ of the
parameters.

b.hambly@statslab.cam.ac.uk

94. BROWNIAN MOTION ON A HOMOGENEOUS RANDOM FRACTAL

B.M. Hambly

We introduce a simple random fractal based on the Sierpinski gasket and
construct a Brownian motion upon the fractal. The properties of the process
on the Sierpinski gasket are modified by the random environment. A sample
path construction of the process via time truncation is used, which is
a direct construction of the process on the fractal from
the associated Dirichlet forms. We obtain estimates on the
resolvent and transition density for the process and hence a value for
the spectral dimension which satisfies $d_{s} =2 d_{f}/d_{w}$. A
branching process in a random environment can be used to deduce some of
the sample path properties of the process.


b.hambly@statslab.cam.ac.uk

95. LARGE DEVIATIONS FOR CONDITIONED AND NONSTATIONARY LATTICE SYSTEMS

Timo Seppalainen

This paper studies large deviations and the associated rate
functions of conditioned  and nonstationary random  fields. 
The emphasis is on large deviations at levels 2
and 3, with occasional remarks on level 1 results. We start from 
questions related to ergodicity and relative entropy on configuration 
spaces. The first large deviation results involve conditioned fields 
in different settings, but with a common theme of conditional independence. 
Here the rate functions turn out to have
entropy expressions under suitable assumptions on the conditioning.
Next, the classical i.i.d large deviation theory is
generalized to independent but nonstationary fields. Now the rate functions
are no longer given by entropy functionals. The last section  introduces 
dependence through an interaction potential, presents large deviation 
theory for conditioned and nonstationary Gibbs measures of this potential, 
and investigates variational principles.

timosepp@mps.function.ohio-state.edu

96. REFLECTED BROWNIAN MOTION; HUNT PROCESS AND SEMIMARTINGALE REPRESENTATION

R. J. Williams.

The Hunt process associated with a regular Dirichlet form for
reflected Brownian motion on a bounded domain is considered. It is
shown that a  necessary condition for this process 
to be a semimartingale whose bounded variation part 
has an associated smooth measure with finite energy 
integral is that the domain be a Caccioppoli set. 

rjwilliams@ucsd.bitnet

97. FLUCTUATING TIME CHANGES OF REAL DIFFUSIONS AS A RIEMANN-HILBERT PROBLEM

Paul McGill

60J65

David Williams has proposed studying the behaviour of a Markov
process under a fluctuating time change. The object of interest is a
transition kernel P (x, dy) which describes how the new process
jumps to positive points. For real diffusions, where the
time change has one boundary point, computing P (x, dy ) is
related to the first passage problem for L\'evy processes.
The kernel can be studied by using the equation which connects the killed
resolvent with the law of the overshoot. We propose interpreting
this equation as a Riemann-Hilbert problem. Our claim --- that this
is a general method --- is justified in detail for a wide class of examples.
Our results are most complete in cases when the positive mass
is finite with bounded support, but we consider other cases and
especially those having some group symmetry. One interesting consequence 
is the appearance of a simple new convolution formula for the kernel.

pmcgill@uci.edu

98. GENERALISED TRANSFORMS, QUASI-DIFFUSIONS, AND D\'ESIR\'E ANDR\'E'S EQUATION

Paul McGill

60J65

We examine two uniqueness results related to Wiener-Hopf
factorisation of infinitesimal generators, and give a proof when
the `positive' string is short. Our method uses excursion
theory, a Paley-Wiener `type' argument, and some standard spectral
theory for Krein's strings. The results are applied to study a
singular convolution equation.

pmcgill@uci.edu

99. REMARK ON THE INTRINSIC LOCAL TIME

Paul McGill

60J65

The intrinsic local time is a semimartingale in the excursion
filtration. We indicate, inter alia, a new proof of the fact that
its martingale part generates an orthogonal martingale measure in
the sense of Walsh. The calculations avoid explicit use of excursion theory,
relying instead on stochastic calculus in the space variable.

pmcgill@uci.edu