Probability Abstracts 40
This document contains abstracts 858-872.
They have been mailed on August 6, 1997.
858. GROUP-INVARIANT PERCOLATION ON GRAPHS
Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm
Consider a (possibly dependent) percolation process on a graph $X$, that is
invariant under a closed group $G$ of automorphisms of $X$. Some relations
between properties of the percolation (such as the number of infinite
components, the average degree, and the topology of the components),
and properties of $G$ and $X$, such as amenability and unimodularity,
are presented. It turns out that $G$ is amenable iff for all $\alpha<1$,
there is a $G$-invariant site percolation on $X$ with marginal probability
greater than $\alpha$ for individual sites $x$, and with no infinite
components. When $G$ is not amenable, a threshold $\alpha<1$ appears.
We obtain an inequality for the threshold that is sharp for regular trees.
(The case of regular trees was already treated by O. Haggstrom).
Another interesting fact is: assuming $G$ acts transitively on $X$,
it is nonunimodular iff there is $G$-invariant bond percolation on $X$ where
all components are infinite and the expected degree of any vertex is $<2$.
The investigation of dependent percolation yields results on automorphism
groups of graphs that do not involve percolation, and new insights concerning
Bernoulli percolation. For example, if $X$ is a nonamenable transitive graph
with a unimodular automorphism group and $p$-Bernoulli percolation produces a
unique infinite component a.s., then $p>p_c(X)$.
itai@wisdom.weizmann.ac.il rdlyons@indiana.edu
peres@math.huji.ac.il schramm@wisdom.weizmann.ac.il
859. TWO EXAMPLES FOR INFINITE LIMIT POINTS OF NORMALIZED RANDOM WALK
K. Bruce Erickson
The law of averages is at the heart of classical probability theory and
it is always of some interest to determine what is possible when the
averages are unbounded. Suppose a closed subset $C$ of the unit sphere
$S^{d-1}$, $d\ge 2$, satisfies one of the following: (i) The cone with a
vertex at the origin generated by $C$ is convex; or (ii) $C$ contains a
pair of antipodal points $\pm u$ (but is otherwise arbitrary). Given
such a set the main work of the present paper demonstrates how to
construct a random walk, $W_t = X_1 + \cdots + X_t$, $t \ge 1$, the
$X_i$ being independent and identically distributed, with the property
that the infinite limit points of the sequence of averages $\{W_t/t\}$
coincides exactly with $C$. The infinite limit points of $\{W_t/t\}$
is the set of $z$ such that for some sequence $\{T_n\}$ of random times,
defined for almost all sample paths, $W_{T_n}/|W_{T_n}\to z $
\emph{and} $|W_{T_n}|/T_n \to \infty$. These examples provide counterexamples
to some conjectures made about these sets.
erickson@math.washington.edu
860. RECOVERING TWO CORRELATED BROWNIAN MOTIONS WITH DRIFT FROM
THE MINIMUM PROCESS
Irene Hueter
Suppose that $X_t$ and $Y_t$ are two independent real-valued
Brownian motions started from the origin with drifts $\mu_1$
and $\mu_2$ and scaling parameters $\sigma_1$ and $\sigma_2.$
If at every time $t > 0$ the first and second moment of the
minimum process $\min(X_t,Y_t)$ are known, are the parameters
$\mu_1,$ $\mu_2,$ $\sigma_1$ and $\sigma_2$ uniquely determined ?
We answer this question in the affirmative and show how to
identify the parameters. Interestingly, the It\^{o} formula
relates the minimum process to the local time of $Y_t-X_t$ at
zero. More generally, we consider two correlated Brownian motions
and prove that the knowledge of the first two moments of the
minimum process does not suffice to uniquely determine
$\sigma_1,$ $\sigma_2$ and the correlation. However, if one
parameter is fixed, there is uniqueness in many cases.
This problem is related to the Anderson--Ghurye problem
and was studied in the context of normal random variables.
hueter@math.ufl.edu
861. STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS IN MANIFOLDS
Marc Arnaudon and Anton Thalmaier
We extend the so-called topology of semimartingales to continuous semimartingales
with values in a manifold and with lifetime, and prove that if the manifold
is endowed with a connection $\nabla$ then this topology and the topology
of compact convergence in probability coincide on the set of continuous
$\nabla$-martingales.
For the topology of manifold-valued semimartingales, we give results
on differentiation with respect to a parameter for second order, Stratonovich
and Itô stochastic differential equations and identify the equation
solved by the derivative processes.
In particular, we prove that both Stratonovich and Itô equations
differentiate like equations involving smooth paths
(for the Itô equation the tangent bundles must be endowed with
the complete lifts of the connections on the manifolds).
As applications, we prove that differentiation and antidevelopment of C^1
families of semimartingales commute, and that a semimartingale with values
in a tangent bundle is a martingale for the complete lift of a connection
if and only if it is the derivative of a family of martingales in the manifold.
arnaudon@math.u-strasbg.fr anton.thalmaier@mathematik.uni-regensburg.de
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information provided by the author
click here.
862. COMPLETE LIFTS OF CONNECTIONS AND STOCHASTIC JACOBI FIELDS
Marc Arnaudon and Anton Thalmaier
Differentiable families of $\nabla$-martingales on manifolds are investigated:
their infinitesimal variation provides a notion of stochastic Jacobi fields.
Such objects are known to be martingales taking values in the tangent bundle
when the latter is equipped with the complete lift of the connection $\nabla$.
We discuss various characterizations of TM-valued martingales and apply our
results to establish formulas in connection with the stochastic representation
of the heat flow for harmonic maps between Riemannian manifolds.
As an application, we give local and global gradient estimates for harmonic maps
of bounded dilatation.
arnaudon@math.u-strasbg.fr anton.thalmaier@mathematik.uni-regensburg.de</pre>
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information provided by the author
click here.
863. ON UNIQUENESS OF INVARIANT MEASURES FOR FINITE AND INFINITE
DIMENSIONAL DIFFUSIONS
Sergio Albeverio, Vladimir Bogachev, Michael R\"ockner
We prove uniqueness of ''invariant measures'', i.e., solutions to the
equation $L^*\mu =0$ where $L=\Delta +B\cdot\nabla$ on $\bbbrn$
with $B$ satisfying some mild integrability conditions and $\mu$ is
a probability measure on $\bbbr^n$. This solves an open problem
posed by S.R.S. Varadhan in 1980. The same conditions are shown
to imply that the closure of $L$ on $L^1(\mu)$ generates
a strongly continuous semigroup having $\mu$ as its unique
invariant measure. The question whether an extension of $L$
generates a (unique) strongly continuous semigroup on $L^1(\mu)$
and whether such an extension is unique is addressed
separately and answered positively under even weaker local integrability
conditions on $B$. The special case when $B$ is a gradient of a function
(i.e., the ''symmetric case'') is in particular studied and conditions are
identified ensuring that $L^*\mu =0$ implies that $L$ is symmetric on
$L^2(\mu)$ resp. $L^*\mu=0$ has a unique solution. We also prove
infinite dimensional analogues of the latter two results and a new
elliptic regularity theorem for invariant measures in infinite dimensions.
sergio.albeverio@rz.ruhr-uni-bochum.de
vbogach@mech.math.msu.su
roeckner@mathematik.uni-bielefeld.de
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information provided by the author
click here.
864. EXISTENCE AND UNIQUENESS OF INVARIANT MEASURES: AN APPROACH VIA SECTORIAL FORMS
V. Bogachev, M. R\"ockner, T. S. Zhang
We prove existence and uniqueness for invariant measures of strongly continuous
semigroups on $L^2(X;\mu )$, where $X$ is a (possibly infinite--dimensional)
space. Our approach is purely analytic based on the theory of sectorial
forms. The generators covered are e.g. small perturbations (in the sense of
sectorial forms) of operators generating hypercontractive semigroups. An
essential ingredient of the proofs is a new result on compact embeddings of
weighted Sobolev spaces $H^{1,2}(\rho\cdot dx)$ on $\bbbr^d$ (resp. a Riemannian
manifold) into $L^2(\rho\, dx)$. Probabilistic consequences are also briefly
discussed.
roeckner@mathematik.uni-bielefeld.de
865. WIENER SOCCER AND ITS GENERALIZATION
Yu. Baryshnikov
The trajectory of the ball in a soccer game is modeled by the Brownian
motion on a cylinder, subject to elastic reflections at the boundary
points (as proposed in [KPY]). The score is then the number of windings of
the trajectory around the cylinder. We consider a generalization of this
model to higher genus, prove asymptotic normality of the score and derive
the covariance matrix. Further, we investigate the inverse problem: to
what extent the underlying geometry can be reconstructed from the
asymptotic score. It turns out that, given the area of the surface, the
conformal equivalence class can be recovered unambiguously form the
asymptotic score.
yuliy@mathematik.uni-osnabrueck.de
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information provided by the author
click here.
866. SUPPORTING POINTS PROCESSES AND SOME OF THEIR APPLICATIONS
Yu. Baryshnikov
We introduce the stochastic point process of S-supporting points and
prove that upon rescaling it converges to a Gaussian field. The notion of
S-supporting points specializes to Pareto (or, more generally, cone)
extremal points or to vertices of convex hulls or to centers of
generalized Voronoi tesselations in the models of large scale structure of
the Universe based on Burgers equation for adequately chosen S. The
central limit theorems proven here imply i.a. the asymptotic normality for
the number of convex hull vertices in large Poisson sample from a simple
polyhedra or for the number of Pareto (vector extremal) points in Poisson
samples with independent coordinates.
yuliy@mathematik.uni-osnabrueck.de
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information provided by the author
click here.
867. THE SHERRINGTON-KIRKPATRICK MODEL
WITH SHORT RANGE FERROMAGNETIC INTERACTIONS
F. Comets, G. Giacomin and J.L. Lebowitz.
We consider an Ising model with both random long range
Sherrington-Kirkpatrick and short range ferromagnetic
pair interactions, i.e., with Hamiltonian (divided by temperature)
$T^{-1} H = \beta H^{SK} + \kappa H^{Ising}$.
We generalize known results for the standard high temperature
SK model, $\kappa =0$ and $ \beta<1$, to a region defined by
$\kappa < \kappa_c$ and $ \beta \theta_{\kappa} < 1$,
where $\kappa_c$ and $\theta_\kappa^2$ are respectively
the critical inverse temperature and the sum of squares of the
pair correlation in the pure ferromagnetic Ising model ($\beta =0$).
Our main result is the Central Limit Theorem for the pressure in
this region, which implies that the quenched and the annealed
pressures agree. It holds also at $\kappa=\kappa_c$ for various
Hamiltonians $H^{Ising}$ in dimension $d>4$.
In particular ferromagnetic criticality persists in presence
of small disorder.
We use the stochastic analysis technique and the main technical
estimates arise in studying the overlaps of independent
copies of the Ising model.
comets@math.jussieu.fr gbg@amath.unizh.ch lebowitz@math.rutgers.edu
868. LARGE DEVIATIONS FOR INCREASING SEQUENCES ON THE PLANE
Timo Seppalainen
We prove a large deviation principle with explicit rate
functions for the length of the longest increasing sequence
among Poisson points on the plane. The rate function for
lower tail deviations is derived from a 1977 result of Logan
and Shepp about Young diagrams of random permutations. For
the upper tail we use a coupling with Hammersley's particle
process and convex-analytic techniques. Along the way we obtain
the rate function for the lower tail of a tagged particle in a
totally asymmetric Hammersley's process.
seppalai@iastate.edu (hardcopy, AMS-TeX file and ps-file available)
869. A RANDOM ENERGY MODEL FOR SIZE-DEPENDENCE:
RECURRENCE VS. TRANSIENCE
Christof Kuelske
We investigate the size dependence of disordered mean field models
having an infinite number of Gibbs measures in the framework of a
simplified `random energy model for size dependence'. We introduce
two versions (involving either independent random walks or branching
processes), that can be seen as generalizations of Derrida's random
energy model. Our model is shown to exhibit a recurrence/transience
transition for Gibbs measures (meaning: a.s. existence/nonexistence
of subsequences of volumes converging to a given random infinite volume
state), depending on the growthrate of a function $M_N$ describing the
`number of extremal Gibbs states that can be observed in a finite
volume $N$'. We investigate the model in detail in the `critical regime'
$M_N\sim \left(\log N\right)^p$ (with critical point $p=1$).
We obtain the a.s. large volume asymptotics of the relative weights for
finding a particular state and we compute the set of a.s. cluster points
of the corresponding occupation times (corresponding to the `empirical
metastate'). In the course of the proof we obtain Laws of the Iterated
Logarithm for the pair $X_N^{\mu}, Y_N^{\nu} :=\sup_{{\mu=1,2,\dots,M_N};
{\mu\neq \nu}}X_N^{\mu}$ where $\left(X_N^{\mu}\right)_{\mu=1,\dots,M_N;
N=1,2,\dots}$ is either a branching random walk or a collection
of random walks, independent over $\mu$.
kuelske@cims.nyu.edu kuelske@wias-berlin.de
870. LONG-TIME EXISTENCE FOR SIGNED SOLUTIONS OF THE HEAT
EQUATION WITH A NOISE TERM
Carl Mueller
We consider the following SPDE: $u_t=u_{xx}+g(u)W$. Here,
$x$ takes values on the unit circle, and $W=W(t,x)$ is
2-parameter white noise. We assume that $g(u)<C(1+|u|)^b$
where $b<3/2$. In a previous paper, the author showed that
if $g(u)=u^b$, with $b$ as before, and if the initial
function $u(0,x)$ is nonnegative, then there is no blow-up
in finite time. The proof relied heavily on the fact that
solutions $u(t,x)$ remain nonnegative. In the current
work, we drop that assumption. We add a drift to the
equation which forces solutions to remain nonnegative, and
then revert to the previous proof. As a byproduct, we
show the following. Let $g(u)$ be as before, assume that
$u(0,x)>0$, and let $a>3$. Let $u(t,x)$ be a solution of
$u_t=u_{xx}+u^{-a}+g(u)W$. Then, with probability 1,
$u(t,x)$ remains strictly positive for all $(t,x)$.
cmlr@troi.cc.rochester.edu (AMS-Latex file available)
871. CONVERGENCE OF RANDOM WALKS ON THE CIRCLE
GENERATED BY AN IRRATIONAL ROTATION
Francis Edward Su
Fix \alpha in [0,1). Consider the random walk on the circle S^1 which
proceeds by repeatedly rotating points forward or backward, with
probability 1/2, by an angle 2\pi\alpha. This paper analyzes the rate of
convergence of this walk to the uniform distribution under ``discrepancy''
distance. The rate depends on the continued fraction properties of the
number \xi=2\alpha. We obtain bounds for rates when \xi is any
irrational, and a sharp rate when \xi is a quadratic irrational. In that
case the discrepancy falls as k^{-1/2} (up to constant factors), where k
is the number of steps in the walk. This is the first example of a sharp
rate for a discrete walk on a continuous state space. It is obtained by
establishing an interesting recurrence relation for the distribution of
multiples of \xi which allows for tighter bounds on terms which appear in
the Erdos-Turan inequality. Probabilistic considerations give somewhat
less sharp bounds for arbitrary irrationals.
su@math.hmc.edu
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information provided by the author
click here.
872. AN UPPER BOUND FOR THE RELATIVE ENTROPY OF MARKOV CHAINS
Francis Edward Su
A new inequality is derived for the relative entropy of a Markov chain.
This leads to a very short proof of convergence of reversible Markov chains
in relative entropy. For finite chains, the upper bound yields bounds
using the second largest eigenvalue of the chain (or an associated chain
in the non-reversible case) and Fourier bounds for random walks on groups.
These are closely related to total variation bounds already available.
However, an example shows that convergence rates in total variation and
relative entropy can differ.
su@math.hmc.edu
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information provided by the author
click here.
