Probability Abstracts 38

819. UNPREDICTABLE PATHS AND PERCOLATION

Itai Benjamini, Robin Pemantle and Yuval Peres

We construct a nearest-neighbor process $S_n$ on the integers that is less 
predictable than simple random walk,  in the sense that given the process
until time $n$, the conditional probability that $S_{n+k}=x$ is uniformly 
bounded by a  constant times $k^{-a}$ for some $a>1/2$.     From this process, 
we obtain a probability measure $\mu$ on oriented paths in $Z^3$ such that the 
number of intersections of two paths chosen independently according to $\mu$, 
has an exponential tail. (For $d > 3$, the uniform measure on oriented paths 
from the origin in $Z^d$ has this property.) We show that on  any graph where 
such a measure on paths exists, oriented percolation clusters are transient if
the retention parameter $p$ is close enough to 1. This yields an extension of 
a recent theorem of Grimmett, Kesten and Zhang, who proved that supercritical
percolation clusters in $Z^d$ are transient for all $d>2$.

itai@wisdom.weizmann.ac.il   pemantle@math.wisc.edu  peres@math.huji.ac.il

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820. BI-INVARIANT SETS AND MEASURES HAVE INTEGER HAUSDORFF DIMENSION

David Meiri and Yuval Peres    

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus
with corresponding eigenvalues relatively prime. We show that
the Hausdorff dimension of any probability measure invariant under both $A$
and  $B$, is an integer. The same holds for any closed bi-invariant set.
Related equidistribution results are also obtained. The proof is based
on the method developed by B. Host for the one-dimensional case,
and combines ideas from harmonic analysis, ergodic theory and probability.

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

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821. BROWNIAN MOTION AND HARMONIC ANALYSIS ON SIERPINSKI CARPETS

Martin T. Barlow and Richard F. Bass 

We consider a class of fractal subsets of $\R^d$ formed in a 
manner analogous to the construction of the Sierpinski carpet. 
We prove a uniform Harnack inequality for positive harmonic
functions; study the heat equation, and obtain upper and lower 
bounds on the heat kernel which are, up to constants, the
best possible; construct a locally isotropic diffusion $X$  
and determine its basic properties; and  extend some classical 
Sobolev and Poincar\'e inequalities to this setting.

bass@math.washington.edu

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822. DENSITY IN SMALL TIME AT ACCESSIBLE POINTS FOR JUMP PROCESSES

Jean Picard

We consider a process $Y_t$ which is the solution of a stochastic
differential equation driven by a L\'evy process with an initial
condition $Y_0=y_0$. We assume conditions under which $Y_t$ has a smooth
density for any $t>0$. We consider a point $y$ that the process can
reach with a finite number of jumps from $y_0$, and prove that, as $t$
tends to 0, the density at this point is of order $t^\Gamma$ for some
$\Gamma=\Gamma(y_0,y)$. Some applications to the potential analysis of
the process are given.

picard@ucfma.univ-bpclermont.fr

823. TREE-VALUED MARKOV CHAINS DERIVED FROM GALTON-WATSON PROCESSES.

David Aldous and Jim Pitman

Let $G$ be a Galton-Watson tree,  i.e. the family tree of a Galton-Watson
branching process.  For $0 \leq u \leq 1$ let $G_u$ be the subtree of $G$
obtained by retaining each edge with probability $u$ and deleting other edges.
We study the tree-valued Markov process $(G_u, 0 \leq u \leq 1)$ and an
analogous process $(G^*_u, 0 \leq u \leq 1)$ in which $G^*_1$ is a critical 
or subcritical Galton-Watson tree conditioned to be infinite.  Results 
simplify and are further developed in the special case of Poisson$(\mu)$
offspring distribution.

Compressed Postscript version available at url below.

aldous@stat.berkeley.edu

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824. DIFFERENT TYPES OF SPDES IN THE EYES OF GIRSANOV'S THEOREM

Hassan Allouba

We prove Girsanov's theorem for continuous orthogonal martingale measures.  
We then define space-time SDEs, and use Girsanov's theorem to establish a 
one-to-one correspondence between solutions of two space-time SDEs 
differing only by a drift coefficient.  For such stochastic equations, 
we give necessary conditions under which the laws of their solutions 
are absolutely continuous with respect to each other.  Using Girsanov's 
theorem again, we prove additional existence and uniqueness results 
for space-time SDEs.  The same one-to-one correspondence and absolute 
continuity theorems are also proved for the stochastic heat and wave 
equations.

allouba@math.duke.edu

825. SUPER-TREE RANDOM MEASURES

Hassan Allouba, Rick Durrett, John Hawkes, and Edwin Perkins

We use supercritical branching processes with random walk steps of 
geometrically decreasing size to construct random measures.  Special 
cases of our construction give close relatives of the 
super-(spherically symmetric stable) processes.  However, other cases 
can produce measures with very smooth densities in any dimension.

allouba@math.duke.edu

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826. SOME LARGE DEVIATION RESULTS FOR SPARSE RANDOM GRAPHS

Neil O'Connell

We obtain a large deviation principle (LDP) for the relative size of the
largest connected component in a random graph with small edge probability.
The rate function, which is not convex in general, is determined explicitly
using a new technique.  As a corollary we present an asymptotic formula
for the probability that the random graph is connected.

We also present an LDP and related result for the number of isolated vertices.
Here we make use of a simple but apparently unknown characterisation, which is
obtained by embedding the random graph in a random directed graph.  The results
demonstrate that, at this scaling, the properties `connected' and `contains no
isolated vertices' are not asymptotically equivalent. (At the threshold
probability they are asymptotically equivalent.)

noc@hplb.hpl.hp.com

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• Or here.

827. BROWNIAN MOTION IN A BROWNIAN CRACK

Krzysztof Burdzy and Davar Khoshnevisan

Let $D$ be the Wiener sausage of width $\eps$ around
two-sided Brownian motion. The components of
2-dimensional reflected Brownian motion in $D$ converge
to 1-dimensional Brownian motion and iterated Brownian 
motion, resp., as $\eps$ goes to $0$.

burdzy@math.washington.edu  (TeX file and hardcopy available)

828. BROWNIAN SHEET AND CAPACITY

Davar Khoshnevisan and Zhan Shi

The main goal of this paper is to present an explicit 
capacity estimate for hitting probabilities of the 
Brownian sheet. As applications, we determine the escape
rates of the Brownian sheet, and also obtain a local 
intersection equivalence between the Brownian sheet and
the additive Brownian motion. Other applications concern
quasi-sure properties in Wiener space.

davar@math.utah.edu

829. BROWNIAN SHEET HAS NO POINTS OF DOUBLE INCREASE

Davar Khoshnevisan and Zhan Shi

We show that points of vertical increase of Brownian sheet
(if any) cannot also be of horizontal increase and vice versa.
The methods extend to show that many "doubly exceptional points"
in Wiener space do not exist.

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

830. STABLE IMAGES AND CARTESIAN PRODUCTS

Davar Khoshnevisan

Let X^1 , ... , X^N denote N independent
isotropic stable Levy processes of index
alpha and dimension d. We show that for
any N dimensional time set T, the stable
image X^1 oplus ... oplus X^N (T) has 
positive d-dimensional Lebesgue measure 
if and only if T has positive d/alpha
dimensional capacity. This solves a problem
of J.-P. Kahane.

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

831. INFINITELY-MANY-SPECIES LOTKA-VOLTERRA EQUATIONS ARISING FROM SYSTEMS OF COALESCING MASSES

Steven N. Evans

We consider  nonlinear, probability measure--valued dynamical systems
that generalise those classical Lotka--Volterra equations in which,
to use ecological terminology, the
total size of a finite number of populations of interacting species
is conserved.  In our generalisation there
is a ``different species'' at each point of an arbitrary measurable space.

Such infinitely--many--species analogues of the
classical Lotka--Volterra equations  appear as hydrodynamic--type limits of
stochastic systems of randomly
coalescing masses related to those that have been used
to model physical and chemical processes of agglomeration, coagulation
and condensation.

One natural instance of our generalisation has closed form solutions,
including a family of solutions that exhibit soliton--like behaviour.
The large time asymptotics of other classes of examples
can be completely described using analogues
of Lyapunov  function techniques.  Moreover, there are
conserved quantities in the form of relative
entropies that generalise those found by Volterra
in the classical case.  Finally, each solution has
a series expansion as a time-varying, geometric mixture of a
fixed sequence of probability measures.  The existence of
this expansion is related to the fact that the system is
in martingale problem duality  with a function--valued Markov process.

evans@stat.Berkeley.EDU

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832. A WEAK CONVERGENCE APPROACH TO THE THEORY OF LARGE DEVIATIONS

Paul Dupuis and Richard S. Ellis

This book presents a comprehensive new approach to the theory of 
large deviations based on the theory of weak convergence of 
probability measures.  The first step in the approach is to replace 
the analysis of the asymptotic behavior of the normalized logarithms 
of probabilities by the analysis of normalized logarithms of 
expectations involving continuous functions.  We refer to the latter 
as the Laplace principle, which is equivalent to a large deviation 
principle with the same rate function.  The second step is to 
represent the normalized logarithms of the expectations as 
variational formulas that are then interpreted, in a natural way, as 
the minimal cost functions of associated stochastic optimal control 
problems.  These minimal cost functions have a form to which the 
theory of weak convergence of probability measures can be applied.  
We consider the introduction of representation formulas to be one 
of our main contributions.  They allow us to replace the exponential 
estimates of standard large deviation approaches by law of large 
numbers-type estimates, which are obtained by the theory of weak 
convergence.  

This book is easily accessible to anyone who has taken courses in 
measure theory and measure-theoretic probability.  No background 
in control theory is assumed or required.  Indeed, control theory 
is used mainly to provide intuition and a convenient terminology.
Much of the material in the book is being published here for the 
first time.

Chapter 1 of the book introduces the Laplace principle and then 
proves a number of general results in large deviation theory from 
the Laplace principle perspective.  This chapter introduces
a main theme that runs through the entire work: namely, the
central role played by relative entropy in large deviation theory.  
The remainder of the book studies two classes of applications: 
empirical measures of Markov chains and random walk models.  
Important special cases of these two classes are treated in 
Chapters 2 and 3.  Chapter 4 develops the representation formulas 
that will be used in subsequent chapters to prove the Laplace
principle.  Chapters 5 through 7 and Chapter 10 are devoted to 
random walk models and their continuous-time analogues, while 
Chapters 8 and 9 treat the empirical measure problem under 
various assumptions.  In all cases the weak convergence approach 
enables us not only to reproduce known results, often under weaker 
conditions, but also to treat new problems.      

The power of the weak convergence approach can be seen in the 
study of Markov processes with discontinuous statistics, which 
arise when modeling queueing systems, in the study of computer 
and communication networks, in the analysis of controlled 
diffusions with discontinuous feedback controls, and in other areas.  
Markov processes with discontinuous statistics are characterized by 
the fact that the components of their generators do not depend 
smoothly on the spatial parameter.  As a result of this nonsmooth 
dependence, classical methods for studying these processes are not 
applicable.  Two examples in which the weak convergence approach is 
particularly useful are given in Chapters 7 and 9 of the book, where we 
treat a random walk model with discontinuous statistics and 
the empirical measures of a Markov chain with discontinuous
statistics.  Markov processes with continuous statistics are 
analyzed in Chapters 6, 9, and 10. 

CONTENTS
Chapter 1.  Formulation of Large Deviation Theory in Terms of 
                the Laplace Principle
Chapter 2.  First Example: Sanov's Theorem
Chapter 3.  Second Example: Mogulskii's Theorem
Chapter 4.  Representation Formulas for Other Stochastic Processes
Chapter 5.  Compactness and Limit Properties for the Random Walk Model
Chapter 6.  Laplace Principle for the Random Walk Model 
                with Continuous Statistics
Chapter 7.  Laplace Principle for the Random Walk Model 
                with Discontinuous Statistics
Chapter 8.  Laplace Principle for the Empirical Measures 
                of a Markov Chain
Chapter 9.  Extensions of the Laplace Principle for the 
                Empirical Measures of a Markov Chain
Chapter 10. Laplace Principle for Continuous--Time Markov Processes 
                with Continuous Statistics
Appendix A. Background Material
Appendix B. Deriving the Representation Formulas via Measure Theory
Appendix C. Proofs of a Number of Results
Appendix D. Convex Functions
Appendix E. Proof of Theorem 5.3.5 When Condition 5.4.1 Replaces
                Condition 5.3.1

rsellis@math.umass.edu

833. EXCEPTIONAL PLANES OF PERCOLATION

Itai Benjamini and Oded Schramm

Consider the standard continuous percolation in $\R^4$, and choose
the parameters so that the induced percolation on a fixed two
dimensional linear subspace is critical. Although two dimensional
critical percolation dies, we show that there are exceptional two
dimensional linear subspaces, in which percolation occurs.  
The setup considered provides another natural dynamic percolation model.

itai@wisdom.weizmann.ac.il     schramm@wisdom.weizmann.ac.il

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834. ANALYSIS AND GEOMETRY ON CONFIGURATION SPACES

Sergio.Albeverio, Yuri G. Kondratiev and Michal R\"ockner

In this paper foundations are presented to a new systematic approach to
analysis and geometry for an important class of infinite dimensional
manifolds, namely, configuration spaces. More precisely, a differential
geometry is introduced on the configuration space $\Gamma_X$ over
a Riemannian manifold $X$. This geometry is ``non-flat'' even if
$X=\Bbb R^d$. It is obtained as a natural lifting of the Riemannian
structure on $X$. In particular, a corresponding gradient
$\nabla^\Gamma$, divergence $\mbox{\rm div}^\Gamma$,
and Laplace-Beltrami operator $H^{\Gamma} =-\mbox{\rm div}^{\Gamma}
\nabla^{\Gamma}$ are constructed. The associated volume elements,
i.e., all measures $\mu$ on $Gamma_X$ w.r.t.\ which $\nabla^{\Gamma}$
and $\mbox{\rm div}^\Gamma$ become dual operators on
$L^{2}(\Gamma_X;\mu)$, are identified as exactly the mixed Poisson
measures with mean measure equal to a multiple of the volume element
$dx$ on $X$. In particular, all these measures obey an integration
by parts formula w.r.t.\ vector fields on $\Gamma_X$. The corresponding
Dirichlet forms $\cal E_\mu^\Gamma$ on $L^2(\Gamma_X;\mu)$ are,
therefore, defined. Each is shown to be associated with a diffusion
process which is thus the Brownian motion on $\Gamma_X$  and which
is subsequently identified as the usual independent infinite particle
process on $X$. The associated heat semigroup  $(T_\mu^\Gamma(t))_{t>0}$
is calculated explicitly. It is also proved  that the diffusion process,
when started with $\mu$, is time-ergodic (or equivalently
$\cal E_\mu^\Gamma$ is irreducible or equivalently
$(T_\mu^\Gamma(t))_{t>0}$ is ergodic) if and only if $\mu$ is Poisson
measure $\pi_{z\,dx}$ with intensity $z\, dx$ for some $z\ge 0$.
Furthermore, it is shown that the Laplace--Beltrami operator
$H^\Gamma =-\mbox{\rm div}^\Gamma\nabla^\Gamma$ on
$L^2(\Gamma_X; \pi_{z\,dx})$ is unitary equivalent to the second
quantization of the Laplacian $-\Delta^X$ on $X$ on the corresponding
Fock space $\bigoplus_{n\ge 0}L^2(X;z\, dx)^{\hat\otimes n}$.
As another direct consequence of our results we obtain a representation
of the Lie-algebra of compactly supported vector fields on $X$ on
Poisson space. Finally, generalizations to the case where $dx$ is
replaced by an absolutely continuous measure and also to interacting
particle systems on $X$ are described, in particular, the case where
the mixed Poisson measures $\mu$ are replaced by Gibbs measures of
Ruelle-type on $\Gamma_X$.

sergio.albeverio@rz.ruhr-uni-bochum.de,kondratiev@physf.uni-bielefeld.de,
      roeckner@mathematik.uni-bielefeld.de

835. APPROXIMATION IN A PARTICLE SYSTEM RUN BY NON-LOCAL GENERAL BRANCHING DYNAMICS

Ingemar Kaj and Serik Sagitov

We present an alternative particle picture for super-stable motion.
It is based on a non-local branching mechanism in discrete time and
only trivial space motion.

ikaj@math.uu.se

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836. THE NUMBER OF INFINITE CLUSTERS IN DYNAMICAL PERCOLATION

Yuval Peres and Jeffrey E. Steif

Dynamical percolation is a Markov process on the space of subgraphs of a given 
graph, that has the usual percolation measure as its stationary distribution.
In previous work with O. Haggstrom, we found conditions for existence of 
infinite clusters at exceptional times. Here we show that for $\Z^d$, with 
$p>p_c$, a.s. simultaneously for ALL times there is a unique infinite cluster,
and the density of this cluster is $\theta(p)$. For dynamical percolation on 
a general tree, we show that for $p>p_c$, a.s. there are infinitely many 
infinite clusters at all times. At the critical value $p=p_c$, the number
of infinite clusters may vary, and exhibits surprisingly rich behaviour: 
  For spherically symmetric trees, we find the Hausdorff dimension of the set 
$T^k$ of times where the number of infinite clusters is $k$, and obtain sharp 
capacity criteria for a given time set to intersect $T^k$. The proof of these 
criteria is based on a new kernel truncation technique.

peres@math.huji.ac.il              steif@math.chalmers.se

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837. THE MARTINGALE PROBLEM FOR DEGENERATE ELLIPTIC OPERATORS

Richard F. Bass

We consider the existence and uniqueness of the martingale problem for
operators of the form 
$${1 \over 2} \sum_{i,j=1}^d a_{ij}(x){{\partial^2 f}\over
{\partial x_i \partial x_j}}(x),$$ where the diffusion coefficients are
continuous but are allowed to degenerate.

bass@math.washington.edu

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838. ASYMPTOTIC BEHAVIOR OF A MULTIPLEXER FED BY A LONG-RANGE DEPENDENT PROCESS

Z. Liu, P. Nain, D. Towsley, and Z.-L. Zhang

In this paper we study the asymptotics of the queue length in a buffer 
fed by an $M/G/\infty$ input process where $G$ is a subexponential 
distribution. More precisely, we derive asymptotic upper and lower 
bounds on the queue length distribution. We find the bounds to be 
tight in some instances, e.g.,$G$ corresponding to either the 
Pareto or lognormal distribution and $c-\rho<1$ where $c$ is the 
rate at which the buffer receives service and $\rho$ is the arrival 
rate to the buffer. 

nain@sophia.inria.fr

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