Probability Abstracts 39

839. ENTROPY OF CONVOLUTIONS ON THE CIRCLE

Elon Lindenstrauss, David Meiri and Yuval Peres

Given a sequence of ergodic $p$-invariant measures ${mu_i}$ on the 
1-torus $T=R/Z$,  we prove that if their entropies are bounded away 
from zero, then the entropy of the convolution $\mu_1*...*\mu_n$
converges to $log p$. We show how a refinement of this result can be 
combined with a method of B. Host to prove the following:  
For every $p$-invariant ergodic $\mu$ with positive entropy, 
the Cesaro average of the images of $\mu$ under multiplication by
$c_1,...,c_n$ converges weakly to Lebesgue measure, under a certain 
mild combinatorial condition on the sequence ${c_k}$. This extends a 
result of Johnson and Rudolph (1995), who considered the sequence 
$c_k = q^k$ when $p$ and $q$ are multiplicatively independent.

We also obtain a corollary concerning Hausdorff dimension of sumsets:
For any sequence ${S_i}$ of $p$-invariant closed subsets of $T$, if 
their Hausdorff dimensions are bounded away from zero, then the 
Hausdorff dimension of $S_1+...+S_n$ tends to 1$.

elon@math.huji.ac.il  dafid@math.huji.ac.il  peres@math.huji.ac.il

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840. MARKOV PROCESSES IN RANDOM ENVIRONMENT: LYAPUNOV FUNCTIONS APPROACH

Francis Comets, Mikhail Menshikov, Serguei Popov

We study three typical examples of countable Markov chains in
random environment using the Lyapunov functions method: random 
walk, random string and branching random walk in random environment.
In each case we construct an explicit Lyapunov function. Investigating 
the behavior of this function we get the classification for recurrence, 
transience, ergodicity.

comets@math.jussieu.fr, menshikov@llrs.math.msu.su, popov@llrs.math.msu.su

841. SHARP ASYMPTOTICS FOR THE KPP'S EQUATION WITH GEOMETRIC AND IRREGULAR INITIAL CONDITION

Serge Cohen and St\'ephane Rossignol

In this article sharp asymptotics for the solution of non homogeneous
Kolmogorov Petrovskii Piskunov equation depending on a small parameter
are considered when the initial condition is the characteristic
function of a set $ A \in \RR^d. $ We show how to extend the    Ben Arous
and Rouault's result   that deal with  $d=1$ and initial condition $ A
=\{x \le 0 \}. $ The dependance of the asymptotics on the geometry of 
the boundary of $ A $ is precisely described for the problem with constraint.

cohen@math.uvsq.fr rossignol@math.uvsq.fr or 

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842. FLUCTUATIONS OF A SURFACE SUBMITTED TO A RANDOM AVERAGE PROCESS

P.A. Ferrari and  L. R. G. Fontes

We consider a hyper surface of dimension $d$ imbeded in a $d+1$ space.  For
each $x\in\z^d$, let $\eta_t(x)\in \R$ be the height of the surface at site
$x$ at time $t$. At rate $1$ the $x$-th height is updated to a random convex
combinations of the heights of the `neighbors' of $x$. The distribution of
the convex combination is translation invariant and does not depend on the
heights. This motion, named random average process (RAP), is one of the
linear processes introduced by Liggett (1985). Special cases of RAP are a
type of smoothing process (when the convex combination is deterministic) and
the voter model (when the convex combination concentrates on one of the
neighbors chosen at random). We start the heights located on a
hyperplane passing through the origin but different from the trivial one
$\eta(x)\equiv 0$.  We show that when the convex combination is neither
deterministic nor concentrating on one of the neighbors the variance of the
height at the origin at time $t$ is proportional to the number of returns to
the origin of a symmetric random walk of dimension $d$.  Under mild
conditions on the distribution of the random convex combination, this gives
variance of the order of $t^{1/2}$ in dimension $d=1$, $\log t$ in dimension
$d=2$ and uniformly bounded in $t$ in dimensions $d\ge 3$. We also show that
for each initial hyperplane the process as seen from the height at the
origin converges to an invariant measure on the hyper surfaces conserving
the initial asymptotic slope.  The height at the origin satisfies a weak law
of large numbers and a central limit theorem. To obtain the results we use a
corresponding probabilistic cellular automaton, for which similar results
are derived. This automaton corresponds to the product of (infinitely
dimensional) independent random matrices whose lines are independent.

pablo@ime.usp.br lrenato@ime.usp.br

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843. A LAW OF THE ITERATED LOGARITHM FOR STABLE PROCESSES IN RANDOM SCENERY

D. Khoshnevisan and T.M. Lewis

We prove a law of the iterated logarithm for
stable processes in a random scenery. The proofs rely
on the analysis of a new class of stochastic processes
which exhibit long-range dependence.

davar@math.utah.edu (Postscript file avaliable)

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844. ENTROPY FOR TRANSLATION-INVARIANT RANDOM-CLUSTER MEASURES

Timo  Seppalainen

We study translation-invariant random-cluster measures 
with techniques from large deviation theory and convex 
analysis. In particular, we prove a large deviation 
principle with rate function given by a specific entropy, 
and a DLR variational principle that characterizes
translation-invariant random-cluster measures as solutions 
of the variational equation for free energy. Consequences
of these theorems include results about edge and cluster 
densities of translation-invariant random-cluster measures. 

seppalai@iastate.edu (AMS-TeX file, ps-file and hardcopy available)

845. FLUCTUATION BOUNDS BASED ON THE TOTAL VARIATION OF RANDOM FUNCTIONS

David Steinsaltz

Using standard methods from empirical-process theory, in particular the
symmetrization variant introduced by L. Devroye (1982), we derive
exponential bounds on the fluctuations of an interesting class of
stochastic processes, which may be represented as the averages of many
small functions.  We eliminate some of the very large constants or
polynomial bounds which have appeared in the more general and
asymptotically oriented results of K. Alexander (1984).

While these bounds are in some respects inferior to those of M. Talagrand
(1994) in some comparable cases, they have a different range of
application and generalization.  In particular, we substitute a
dependence on the total variation of the component functions for the
covering-number bounds.  While this restricts our results to the
one-dimensional context, it allows the bounds to be applied to cases in
which the component functions are not jump-functions and not of a uniform
shape.  We also do not need to assume that the functions are identically
distributed.

dstein@math.tu-berlin.de

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846. DEVIATION BOUNDS AND LIMIT THEOREMS FOR THE MAXIMA OF SOME STOCHASTIC PROCESSES

David Steinsaltz

We consider stochastic processes which are defined as averages
$F_n=\rec{n}\sum_{i=1}^n f_i$ of n small, slowly varying, independent (or
almost independent) random cadlag functions $f_i$.  Such processes arise
in many contexts, including queueing and storage problems.  For example,
$F_n(t)$ may be the fraction of n customers present in a self-service
queueing system; that is, a system in which the customers come and go at
random times, independent of one another.  Using slight modifications of
standard empirical-process methods, we derive general inequalities for
the maximum fluctuations of such processes from their means.  This allows
us, for instance, to rederive a well-known functional central limit
theorem for these processes due to E. Gin\'e and J. Zinn:
$n^{1/2}(F_n(t)-E[F_n(t)])$ converges weakly to a Gaussian process if the
$f_i$ are uniformly bounded and stochastically Lipschitz, i.e.,
$E[|f_i(t)-f_i(s)|] \leq k |s-t|$.  And advantage of the present methods
is that they also yield usable conditions in many cases in which the
$f_i$ are not uniformly bounded, and we do not need to assume that the
$f_i$ are identically distributed.

When the expectation $E[F_n(t)]$ itself has a unique maximum, at a point
$t_0$, the maximum of $F_n(t)$ is expected to be near $F_n(t_0)$.  We
then derive second-order bounds for the difference between $\max F_n(t)$
and $F_n(t_0)$.  While $|\max F_n(t) - E[F_n(t_0)|$ is stochastically on
the order of $n^{-1/2}$ for large $n$, $\max F_n(t)-F_n(t_0)$ is strictly
smaller, being only on the order of $n^{-q/(2q-1)}$, where q is the order
of the maximum $E[F_n(t_0)]$, i.e., the smallest q such that the q-th
derivative of $E[F_n(t)]$ is nonzero at $t_0$.

Further, each of these quantities converges in law to a nontrivial
distribution, when divided by the appropriate power of $n$.  Generally
this limit is of the form $\sup_{t\in R} \tau B_t -d t^q$ for positive
constants $\tau$ and $d$, and $B_t$ a two-sided Brownian motion.

dstein@math.tu-berlin.de

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847. APPLICATIONS OF RANDOM TIME CHANGES TO SOME STOCHASTIC-PROCESS LIMIT THEOREMS

David Steinsaltz

A common technique in the theory of stochastic process is to replace a
discrete time-coordinate by a continuous randomized time, defined by an
independent Poisson or other process.  Once the analysis is complete on
this new randomized process, there remains some difficulty in translating
the results back to the context of the original process.  We show here
that under fairly general conditions, if the process $S_n$ and the time
change $\phi_n$ both converge, when normalized by the same constant
$\alpha_n$, to limit processes $\tilde{S}$ and $\tilde{\Phi}$, then the
combined normalized process $\alpha_n (S_n(\phi_n)-E[S_n(\phi_n)])$
converges to
$$
\tilde{S}+\frac{d E[S(t)]}{dt} \tilde{\Phi} \, .
$$

If $S_n$ is defined as the average of $n$ independent random functions,
whose expectation has its maximum at $t^*$, earlier results present
conditions under which $n^{1/3}(max S_n -S_n(t^*))$ converges to the
maximum of $B(t)-a t^2$, where $B(t)$ is two-sided Brownian motion and
$a$ is a positive constant; such limit laws are shown to be unaffected by
these time changes.  As examples, we apply the theorems to processes
which arise in a natural way from sorting procedures, and from random
allocations.

dstein@math.tu-berlin.de

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848. STOPPING SETS: GAMMA-TYPE RESULTS AND HITTING PROPERTIES

Sergei Zuyev

Recently in the paper:  J.M{\o}ller and S.Zuyev. Gamma-type results and 
other related properties of Poisson processes,  Adv. in Appl. Probab., 
28:662-673, 1996, there was established the following Gamma-type
result. Given the number $N$ of a homogeneous Poisson process' points defining
a random figure, its volume is $\Gamma(N,\lambda)$ distributed, where 
$\lambda$ is the intensity of the process. The goal of this paper is to give 
an alternative description of the class of the random sets for which 
the Gamma-type results hold. We show that it corresponds to the 
class of stopping sets with respect to the natural filtration 
of the point process with certain scaling properties. The proof is very short
and uses the martingale technique for directed processes, in particular, the
analog of the Doob's optional sampling theorem. This approach provides a new
inside into the nature of geometrical objects constructed with respect to a
point process. We show, in particular, that in the Poisson case the 
probability of a point to be covered by a stopping set does not depend on
whether it is point of the Poisson process or not.

Sergei.Zuyev@sophia.inria.fr

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• Or here.

849. COMPLETE MARKETS WITH DISCONTINUOUS SECURITY PRICE

Michael Dritschel  and  Philip Protter

A parameterized family of financial market models is presented. These models
have jumps intrinsic to the price processes yet have strict completeness,
equivalent martingale measures, and no arbitrage. For each value of the
parameter $\beta(-2\leq\beta<0)$ the model is just as rich as the standard 
model using white noise (Brownian motion) and a drift; moreover as $\beta$ 
increases to zero the model converges weakly to the standard model. A hedging 
result, analogous to the Karatzas-Ocone-Li theorem, is also presented.

protter@MATH.Purdue.EDU

850. THE TRUE SELF-REPELLING MOTION

B\'alint T\'oth and Wendelin Werner

In this long paper, we construct and study a continuous real-valued
random process, which is of a new type:
It is self-interacting (self-repelling) but only in
a local sense: it
only feels the self-repellance due to its occupation-time
measure density at `immediate neighbourhood'
of the point it is just visiting.
We focus on the most natural process with these
properties that we call  `true self-repelling motion'.
This is the continuous counterpart to the integer-valued
`true' self-avoiding walk, which had been studied among others
by the first author.
One of the striking properties of true self-repelling motion is
that, although the couple $(X_t,$
occupation-time measure of $X$ at time $t)$ is a
continuous Markov process, $X$ is
not driven by a stochastic
differential equation and is not a semi-martingale.
It turns out, for instance,
that it has a finite variation of order 3/2,
which contrasts with the finite quadratic variation
of semi-martingales.
One of the key-tools in the construction of
$X$ is a continuous system of
coalescing Brownian motions
similar to those that have been introduced and constructed by Arratia
(1979a, 1979b). We derive various properties of $X$ (existence and 
properties
of the occupation time densities $L_t(x)$, local variation, etc.) and
an identity that shows that
the dynamics of $X$ can be very loosely speaking described
as follows:  $-dX_t$ is equal to the gradient (in space)
of $L_t(x)$, in a generalized sense,  even if
$x \mapsto L_t(x)$ is not differentiable.

balint@math-inst.hu  wwerner@dmi.ens.fr (hardcopies available)

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• Or here.
• Or here.

851. TRANSITION DENSITY ESTIMATES FOR DIFFUSION PROCESSES ON P.C.F. SELF-SIMILAR FRACTALS

Ben M. Hambly and Takashi Kumagai

We consider the heat kernel for Laplacians on p.c.f. self-similar sets, a
class of
exactly self-similar, finitely ramified fractals without spatial symmetry.
By considering each Laplacian with respect to a natural measure and working
in the effective resistance metric we obtain estimates for the associated heat
kernel which are best possible up to constants.
By considering a range of examples we show that uniform Aronson type estimates
do not hold in general on fractals.
 
kumagai@math.nagoya-u.ac.jp  (LaTex-file available)

852. ASYMPTOTICS FOR THE HEAT CONTENT OF A PLANAR REGION WITH A FRACTAL POLYGONAL BOUNDARY

M. van den Berg and F. den Hollander

Let $k \geq 3$ be an integer. For $0<s<1$, let $D_s \subset \R^2$
be the set that is constructed iteratively as follows. Take a
regular open $k$-gon with sides of unit length, attach regular
open $k$-gons with sides of length $s$ to the middles of the
edges, and so on. At each stage of the iteration the $k$-gons that
are added are a factor $s$ smaller than the previous generation
and are attached to the outer edges of the family grown so far.
The set $D_s$ is defined to be the interior of the closure of the
union of all the $k$-gons. It is easy to see that there must exist
some $s_k>0$ such that no $k$-gons overlap if and only if $0<s\leq s_k$.
We derive an explicit formula for $s_k$.
 
The set $D_s$ is open, bounded, connected and has a fractal polygonal
boundary. Let $E_{D_s}(t)$ denote the heat content of $D_s$ at time $t$
when $D_s$ initially has temperature 0 and $\partial D_s$ is kept at
temperature 1. We derive the complete short-time expansion of $E_{D_s}(t)$
up to terms that are exponentially small in $1/t$. It turns out that there
are three regimes, corresponding to $0<s<1/(k-1)$, $s=1/(k-1)$ respectively
$1/(k-1)<s\leq s_k$. For $s \neq 1/(k-1)$ the expansion has the form
$E_{D_s}(t)= p_s(\log t)t^{1-\frac{d_s}{2}}+A_s t^{\frac{1}{2}}+Bt+O(e^{-\frac{
r_s}{t}})$,
where $p_s$ is a $\log(1/s^2)$-periodic function, $d_s=\log(k-1)/\log(1/s)$
is a similarity dimension, $A_s$ and $B$ are constants related to the edges
respectively vertices of $D_s$, and $r_s$ is an error exponent. For
$s=1/(k-1)$
the  $t^{\frac{1}{2}}$-term carries an additional $\log t$.

denholla@sci.kun.nl

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853. CORRELATION STRUCTURE OF INTERMITTENCY IN THE PARABOLIC ANDERSON MODEL

J. G\"artner  and F. den Hollander

Consider the Cauchy problem $\partial u(x,t)/\partial t = {\cal H} u(x,t)$ $(x
\in \Z^d,t \ge 0)$ with initial condition $u(x,0)\equiv1$ and with ${\cal H}$
the Anderson Hamiltonian ${\cal H} = \kappa \Delta + \xi$. Here $\Delta$ is
the discrete Laplacian, $\kappa \in (0,\infty)$ is a diffusion constant, and
$\xi = \{\xi(x) \colon x \in \Z^d\}$ is an i.i.d.\ random field taking values
in
$\R$. G\"artner and Molchanov (1990) have shown that if the law of $\xi(0)$ is
nondegenerate, then the solution $u$ is asymptotically intermittent. This
means that
$\lim_{t \rightarrow \infty} \langle u^2(0,t) \rangle / \langle u(0,t)
\rangle^2 = \infty$, where $\langle \cdot \rangle$ denotes expectation w.r.t.\
$\xi$, and similarly for the higher moments. Qualitatively their result says
that, as
$t$ increases, the random field $\{u(x,t) \colon x \in \Z^d\}$ develops
sparsely
distributed high peaks, which give the dominant contribution to the moments as
they become sparser and higher.

In the present paper we study the structure of the intermittent peaks for the
special case where the law of $\xi(0)$ is (in the vicinity of) the double
exponential ${\rm Prob}(\xi(0)>s)=\exp[-e^{s/\theta}]$ $(s \in \R)$. Here
$\theta \in (0,\infty)$ is a parameter that can be thought of as measuring
the degree of disorder in the $\xi$-field. Our main result is that, for fixed
$x,y \in \Z^d$ and $t \rightarrow \infty$, the correlation coefficient of
$u(x,t)$
and $u(y,t)$ converges to $\|w_{\rho}\|_{\ell^2}^{-2} \sum_{z \in \Z^d}
w_{\rho}(x+z)w_{\rho}(y+z)$. In this expression, $\rho = \theta/\kappa$ while
$w_{\rho} \colon \Z^d \rightarrow \R^+$ is given by
$w_{\rho}=(v_{\rho})^{\otimes d}$
with $v_{\rho} \colon \Z \rightarrow \R^+$ the unique centered ground state of
the
1-dimensional nonlinear equation $\Delta v + 2\rho v \log v = 0$ (ground state
means the solution in $\ell^2(\Z)$ with minimal $l^2$-norm). Qualitatively our
result says that the high peaks of $u$ have a shape that is a multiple of
$w_{\rho}$
relative to the center of the peak.
 
It turns out that if the right tail of the law of $\xi(0)$ is thicker (or
thinner) than the double exponential, then the correlation coefficient of
$u(x,t)$
and $u(y,t)$ converges to $\delta_{x,y}$ (resp.\ the constant function 1).
Thus,
the double exponential family is the critical class exhibiting a nondegenerate
correlation structure.

denholla@sci.kun.nl

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854. ON SUMS OF IID RANDOM VARIABLES INDEXED BY N PARAMETERS

Davar Khoshnevisan

In this note, I prove maximal inequalities
for sums of iid random variables indexed by N parameters.
These inequalities are close in spirit to Cairoli's
maximal inequality for martingales and Doob's inequality
for reverse martingales. Two immediate consequences are
the condition for L^p boundedness of the "averages" and
the strong law of R.T. Smythe.

davar@math.utah.edu

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855. PERCOLATION DIMENSION OF BROWNIAN MOTION IN R^3

Chad Fargason

Let B(t) be a Brownian motion in $R^3$.  A {\it subpath} of the Brownian 
path $B[0,1]$ is a continuous curve $\gamma(t)$, where $\gamma[0,1] 
\subseteq B[0,1]$, $\gamma(0) = B(0)$, and $\gamma(1) = B(1)$.  It is 
well-known that any subset $S$ of a Brownian path must have Hausdorff 
dimension $dim_h(S) \leq 2$.  This paper defines a subpath of $B[0,1]$ 
with Hausdorff dimension strictly less than 2.  Thus, the percolation 
dimension of Brownian motion in $R^3$ is strictly less than 2.

fargason@math.duke.edu 

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856. THE METASTABILITY OF THE BIASED MAJORITY VOTE PROCESS ON A TORUS

Jun Xie and Dayue Chen

We study the biased majority vote process on a 2-dimensional torus.
The flipping rates are $\delta$ or $\delta^\alpha$ according to the
opinion at the site.  The consensus time is the first time that all
voters hold the same opinion.  We identify the exponent that the mean
of consensus time is asymptotically (1/$\delta)$ to that exponent as
$\delta$ goes to 0.  The proof is by a formula of mean exit time and
by the metastable theory of Markov chains developed in the study of
the stochastic Ising model.
 
dayue@sxx0.math.pku.edu.cn  (TeX file or hardcopy available)

857. POSITIVITY OF BROWNIAN TRANSITION DENSITIES

M. Barlow, R.Bass and K. Burdzy

Let $B$ be a Borel subset of $R^d$ and let
$p(t,x,y)$ be the transition densities 
of Brownian motion killed on leaving $B$. 
Fix $x$ and $y$ in $B$. If $p(t,x,y)$ is 
positive for one $t$, it is positive for 
every value of $t$. Some related results are given.

bass@math.washington.edu, burdzy@math.washington.edu
               (Plain Tex file or hardcopy available)

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