Probability Abstracts 41
This document contains abstracts 873-907.
They have been mailed on September 30, 1997.
873. STATIONARY MARKOV PROCESSES RELATED TO STABLE
ORNSTEIN-UHLENBECK PROCESSES AND THE ADDITIVE COALESCENT
Steven N. Evans and Jim Pitman
We consider some classes of stationary,
counting-measure-valued Markov processes
and their companions under time-reversal. Examples arise
in the L\'evy-It\^o decomposition of stable Ornstein-Uhlenbeck
processes and in the large-time asymptotics of the
standard additive coalescent. These processes share the common feature
that points in the support of the evolving counting-measure
are born or die randomly, but each point follows
a deterministic flow during its life-time.
Such processes fit within the framework of
jumping Markov processes considered by Jacod and Skorokhod
following on from work of Davis on piecewise deterministic Markov
processes.
evans@stat.Berkeley.EDU
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information provided by the author
click here.
874. METASTATES IN THE HOPFIELD MODEL IN THE REPLICA SYMMETRIC REGIME
Anton Bovier and Veronique Gayrard
We study the finite dimensional marginals of the Gibbs measure
in the Hopfield model at low temperature when the number of patterns, $M$,
is proportional to the volume with a sufficiently small proportionality
constant $\a>0$. It is shown that even when a single pattern is selected
(by a magnetic field or by conditioning), the marginals do not converge
almost surely, but only in law. The corresponding limiting law is constructed
explicitly. We fit our result in the recently proposed language of
``metastates'' which we discuss in some length. As a byproduct, in a certain
regime of the parameters $\a$ and $\b$ (the inverse temperature),
we also give a simple proof of Talagrand's recent result that the
replica symmetric solution found by Amit, Gutfreund, and Sompolinsky
can be rigorously justified.
bovier@wias-berlin.de
gayrard@cpt.univ-mrs.fr
875. THE KAC VERSION OF THE SHERRINGTON-KIRKPATRICK MODEL
AT HIGH TEMPERATURES
Anton Bovier
We study the Kac version of the Sherrington-Kirkpatrick (SK) model of a
spin glass, i.e. a spin glass with long but finite range interaction
on $\Z^d$ and gaussian mean zero couplings. We prove that for all
$\b<1$, the free energy of this model converges to that of the SK model
as the range of the interaction tends to infinity. Moreover, we prove that
for all temperatures for which the infinite volume Gibbs state is unique,
the free energy scaled by the square root of the volume converges to a gaussian
with variance $c_{\g,\b}$, where $\g^{-1}$ is the range of the interaction.
Moreover, at least for almost all values of
$\b$, this variance tends to zero as $\g$ goes to zero, the value in the
SK model. We interpret our finding as a weak indication that at least at high
temperatures, the SK model can be seen as a reasonable asymptotic model
for lattice spin glasses.
bovier@wias-berlin.de
876. IMMIGRATION PROCESSES ASSOCIATED WITH BRANCHING PARTICLE SYSTEMS
Zeng-Hu Li
Let $N(E)$ be the totality of finite integer-valued measures on the space
$E$. Suppose that $X$ is a branching particle system with state space $N(E)$
and semigroup $Q_t$. Let $\{N_t: t\ge0\}$ be a family of probability measures
on $N(E)$. We call $N_t$ a {\it skew convolution semigroup} associated with
$Q_t$ provided $N_{r+t} = (N_rQ_t)*N_t$ for all $r$ and $t\ge0$. This
condition is necessary and sufficient to ensure that $Q^N_t(\sigma,.) :=
Q_t(\sigma,.)*N_t$ defines another transition semigroup $Q^N_t$. If $Y$ is a
Markov process having $Q^N_t$ as its transition semigroup, we call it an {\it
immigration process} associated with $X$. The intuitive meaning of the
immigration process is clear by the formulation.
The skew convolution semigroup associated with a Dawson-Watanabe superprocess
can be introduced in a similar way. In the earlier work Li (1996a; Chin. Sci.
Bull. 41, 276-280), the author showed that a skew convolution semigroup
associated with the superprocess corresponds uniquely to an infinitely
divisible probability entrance law for the superprocess. The set of infinitely
divisible probability entrance laws and some path properties of the
corresponding immigration processes were studied in Li (1996b; Stoch. Proc.
Appl. 62, 73-86).
In this paper, we show that a skew convolution semigroup associated with the
branching particle system can also be characterized in terms of an infinitely
divisible probability entrance law. The major difference between the
superprocess and the branching particle system is that, started with any
deterministic state, the former is infinitely divisible and the later, which
can only be started with an integer-valued measure, is not. This causes some
difficulties for the study of the immigration particle system -- a minimal
probability entrance law for the superprocess is always infinitely divisible,
but that for the branching particle system is usually not. For this reason,
the characterization of all infinitely divisible probability entrance laws for
a general branching particle system still remains OPEN. But in the special
case where the underlying process is a minimal Brownian motion in a bounded
smooth domain, we give a general representation for all locally integrable
entrance laws for the branching particle system. The convergence of immigration
particle systems to measure-valued immigration processes is also studied.
lizh@ns.bnu.edu.cn (AMS-TeX file and hardcopy available)
877. PERFECT SIMULATION OF SOME POINT PROCESSES FOR THE IMPATIENT USER
Elke Th\"{o}nnes
Recently Propp and Wilson have proposed an algorithm, called Coupling from
the Past (CFTP), which allows not only an approximate but perfect (i.e.
exact simulation of the stationary distribution of certain finite state
space Markov chains. Perfect Sampling using CFTP has been successfully
extended to the context of point processes, amongst other authors, by
H\"{a}ggstr\"{o}m et al.. In H\"{a}ggstr\"{o}m et al. Gibbs Sampling is
applied to a bivariate point process, the pentrable spheres mixture
model.
Fill also introduced an exact sampling algorithm, which, in contrast to
CFTP, is unbiased for user impatience. Fill's algorithm is a form of
rejection sampling and similar to CFTP requires sufficient monotonicity
properties of the transition kernel used. We show how Fill's version of
rejection sampling can be extended to produce perfect samples of the
penetrable spheres mixture process and related models. Following
H\"{a}ggstr\"{o}m et al. we use Gibbs sampling and make use of the partial
order of the mixture model state space. Thus we construct an algorithm
which protects agains bias caused by user impatience and which delivers
samples not only of the mixture model but also of the attractive
area-interaction and the continuum random-cluster process.
elke@stats.warwick.ac.uk
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information provided by the author
click here.
878. RESISTANCE BOUNDS FOR FIRST-PASSAGE PERCOLATION AND MAXIMUM FLOW
Russell Lyons, Robin Pemantle and Yuval Peres
Suppose that each edge $e$ of a network is assigned a random exponential
passage time with mean $r_e$. We show that the expected first-passage time
between two vertices is at least the effective resistance between them for
the edge resistances ${r_e}$. Similarly, suppose each edge is assigned a
random exponential edge capacity with mean $c_e$. Then the expected maximum
flow between two vertices is at least the effective conductance between them
for the edge conductances ${c_e}$. These inequalities are dual to each other
for planar graphs. The second is tight up to a factor of 2 for trees;
this has implications for a herd of gnu crossing a river delta.
rdlyons@indiana.edu pemantle@math.wisc.edu peres@math.huji.ac.il
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information provided by the author
click here.
- Or
here.
879. THE ASYMMETRIC RANDOM CLUSTER MODEL AND COMPARISON OF
ISING AND POTTS MODELS
Kenneth S. Alexander
We introduce the asymmetric random cluster (or ARC) model, which is a
graphical representation of the Potts lattice gas, and establish its basic
properties. The ARC model allows a rich variety of comparisons (in the FKG
sense) between models with different parameter values; we give, for example,
values (beta,h) for which the 0's configuration in the Potts lattice gas is
dominated by the "+" configuration of the (beta,h) Ising model. The Potts
model, with possibly an external field applied to one of the spins, is a
special case of the Potts lattice gas, which allows our comparisons to yield
rigorous bounds on the critical temperaures of Potts models. For example, we
obtain .571 \leq 1 - exp(-beta_c) \leq .600 for the 9-state Potts model on
the triangular lattice, and .926 \leq 1 - exp(-beta_c) \leq .928 for the
30-state Potts model on the hexagonal lattice. Another comparison bounds the
movement of the critical line when a small Potts interaction is added to a
lattice gas which otherwise has only interparticle attraction. ARC models
can also be compared to related models such as the partial FK model, obtained
by deleting a fraction of the nonsingleton clusters from a realization of the
Fortuin-Kasteleyn random cluster model. This comparison leads to bounds on
the effects of small annealed site dilution on the critical temperature of
the Potts model.
alexandr@math.usc.edu
880. A HETEROPOLYMER NEAR A LINEAR INTERFACE
Marek Biskup and Frank den Hollander
We consider a quenched-disordered heteropolymer in the vicinity of
an interface between two solvents. We show that the free-energy
localization concept introduced in [Bolthausen and den Hollander,
1996] is equivalent to pathwise localization. In particular, we
prove that positivity of the excess free energy implies exponential
tightness of the polymer excursions away from the interface, positive
density of intersections with the interface, and convergence of ergodic
averages along the polymer. We include an argument due to G. Giacomin
showing that if the excess free energy is zero then there is pathwise
delocalization in a certain weak sense.
denholla@sci.kun
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information provided by the author
click here.
881. ON MODIFIED LOGARITHMIC SOBOLEV INEQUALITIES
FOR BERNOULLI AND POISSON MEASURES
S. G. Bobkov and M. Ledoux
We show that for any positive function $f$ on the discrete cube
$\{0,1\} ^n$,
$$ {\rm Ent}_{\mu _p^n}(f)\leq { {pq} \over 2} \, {\rm E}_{\mu _p^n}
\Bigl ( {1\over f} \, |Df|^2 \Bigr ) $$
where $\mu _p^n$ is the product measure of the Bernoulli measure with
probability of success $p$. This logarithmic Sobolev inequality may be
shown to imply in the limit the classic Gaussian logarithmic Sobolev inequality
as well as a logarithmic Sobolev inequality for Poisson measure. We further
investigate modified logarithmic Sobolev inequalites to analyze
integrability properties of Lipschitz functions on discrete spaces. In
particular, we obtain, under modified logarithmic Sobolev inequalities,
some concentration results for product measures that extend the classic
exponential inequalities for sums of independent random variables.
ledoux@cict.fr (Plain TeX file available)
882. METASTATES IN DISORDERED MEAN FIELD MODELS II:
THE SUPERSTATES
Christof Kuelske
We continue to investigate the size dependence of disordered mean
field models with finite local spin space in more detail, illustrating
the concept of `superstates', as recently proposed by Bovier and
Gayrard. We discuss various notions of convergence for the behavior
of the paths $\left(t\rightarrow \mu_{[t N]}(\eta)\right)_{t\in (0,1]}$
in the thermodynamic limit $N\uparrow\infty$. Here $\mu_n(\eta)$ is
the Gibbs measure in the finite volume $\{1,\dots,n\}$ and $\eta$
is the disorder variable. In particular we prove refined convergence
statements in our concrete examples, the Hopfield model with finitely
many patterns (having continuous paths) and the Curie Weiss Random
Field Ising model (having singular paths).
kuelske@wias-berlin.de
883. HAMILTONIANS ON RANDOM WALK TRAJECTORIES
Pablo A. Ferrari and Servet Mart\'{\i}nez
We consider Gibbs measures on the set of paths of nearest neighbors
random walks on $Z_+$. The basic measure is the uniform measure on the set
of paths of the simple random walk on $Z_+$ and the Hamiltonian awards each
visit to site $x\in Z_+$ by an amount $\alpha_x\in R$, $x\in Z_+$. We
give conditions on $(\alpha_x)$ that guarantee the existence of the (infinite
volume) Gibbs measure. When comparing the measures in $Z_+$ with the
corresponding measures in $Z$, the so called entropic repulsion appears as a
counting effect.
pablo@ime.usp.br
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information provided by the author
click here.
884. ANALYSIS AND GEOMETRY ON CONFIGURATION SPACES: THE GIBBSIAN CASE
S. Albeverio, Yu. G. Kondratiev, M. R\"{o}ckner
Using a natural ``Riemannian--geometrie--like'' structure on the configuration
space $\G$ over $\R^d$, we prove that for a large class of potentials
$\phi$ the corresponding canonical Gibbs measures on $\G$ can be
completely characterized by an integration by parts formula. That is,
if $\grg$ is the gradient of the Riemannian structure on $\G$ one
can define a corresponding divergence $\div_\phi$ such that the canonical
Gibbs measures are exactly those measures
$\mu$ for which $\grg$ and $\div_\phi$ are dual operators on $L^2(\G ,\mu)$.
One consequence is that for such $\mu$ the corresponding Dirichlet forms
$\EC_\mu^\G$ are defined. In addition, each of them is shown to be
associated with a conservative diffusion process on $\G$ with invariant
measure $\mu$. The corresponding generators are extensions of the
operator $\Delta_\phi^\G :=\div_\phi\ \grg$. The diffusions can be
characterized in terms of a martingale problem and they can be considered
as a Brownian motion on $\G$ perturbed by a singular drift. Another
main result of this paper is the following: if $\mu$ is a canonical
Gibbs measure, then it is extreme (or a ``pure phase'') if and only
if the corresponding weak Sobolev space $\W^/1,2=(\Gamma, \mu)$ on
$\Gamma$ is irreducible. As a consequence we prove that for extreme
canonical Gibbs measures the above mentioned diffusions are time--ergodic.
In particular, this holds for tempered grand canonical Gibbs measures
(``Ruelle measures'') provided the activity constant is small enough.
We also include a complete discussion of the free case (i.e., $\phi \equiv 0 $)
where the underlying space $\R^d$ is even replaced by a Riemannian
manifold $X$.
roeckner@Mathematik.Uni-Bielefeld.DE
885. COLLISION LOCAL TIMES, HISTORICAL STOCHASTIC CALCULUS,
AND COMPETING SUPERPROCESSES
S.N. Evans and E.A. Perkins
Branching measure--valued diffusion models are investigated that
can be regarded as pairs of historical Brownian motions
modified by a competitive interaction
mechanism under which individuals from each population
have their longevity or fertility adversely affected by
collisions with individuals from the other population.
For 3 or fewer spatial dimensions,
such processes are constructed using a new fixed--point
technique as the unique solution of a strong equation
driven by another pair of more explicitly constructible
measure--valued diffusions. This existence
and uniqueness is used to establish
well--posedness of the related martingale problem and
hence the strong Markov property for solutions.
Previous work of the authors has shown that in 4 or more dimensions
models with the
analogous definition do not exist.
The definition of the model and its study require a thorough
understanding of random measures called collision local times.
These gauge the extent to which two measure--valued processes
or an R^d--valued process and a measure--valued process
``collide'' as time evolves. This study and the substantial
amount of related historical stochastic calculus that is developed
are germane to several other problems beyond the
scope of the paper. Moreover, new results are
obtained on the branching
particle systems imbedded in historical processes
and on the existence
and regularity of superprocesses with immigration,
and these are also of independent interest.
evans@stat.berkeley.edu
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information provided by the author
click here.
886. A NON-NONSTANDARD PROOF OF REIMERS' EXISTENCE RESULT FOR HEAT SPDES
Hassan Allouba
In 1989 Reimers gave a nonstandard proof of the existence of a solution to
heat SPDEs, driven by space-time white noise, when the diffusion coefficient
is continuous and satisfies a linear growth condition. Using the martingale
problem approach, we give a non-nonstandard proof of this fact, and with the
aid of Girsanov's theorem for continuous orthogonal martingale measures
(proved in a separate paper by the author), the result is extended to the
case of measurable drift.
allouba@math.duke.edu
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information provided by the author
click here.
887. ACTIVITY PERIODS OF AN INFINITE SERVER QUEUE AND PERFORMANCE
OF CERTAIN HEAVY TAILED FLUID QUEUE
Sidney Resnick and Gennady Samorodnitsky
A fluid queue with {\it ON} periods arriving according to a Poisson
process and having a long--tailed distribution has long range
dependence. As a result, its performance deteriorates. The extent of
this performance deterioration depends on a quantity determined by
the average values of the system parameters. In the case when the
the performance deterioration is the most extreme, we quantify it by
studying the time until the amount of work in the system causes an
overflow of a large
buffer. This turns out to be strongly related to the tail behavior
of the increase in the buffer content during a busy period of the
$M/G/\infty$ queue feeding the buffer. A large deviation approach
provides a powerful method of studying such tail behavior.
sid@orie.cornell.edu gennady@orie.cornell.edu
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information provided by the author
click here.
888. THE STANDARD ADDITIVE COALESCENT
David Aldous and Jim Pitman
Regard an element of the set
$$\Delta := \{(x_1,x_2,\ldots):
x_1 \geq x_2 \geq \ldots \geq 0, \sum_i x_i = 1\}$$
as a fragmentation of unit mass into clusters of masses $x_i$. The
additive coalescent of Evans and Pitman (1997) is the $\Delta$-valued
Markov process in which pairs of clusters of masses $\{x_i,x_j\}$ merge
into a cluster of mass $x_i + x_j$ at rate $x_i+x_j$.
They showed that a version $(X^\infty (t), - \infty < t < \infty)$
of this process arises as a $n \to \infty$ weak limit of the process
started at time $- \frac{1}{2} \log n$ with $n$ clusters of mass $1/n$.
We show this {\em standard additive coalescent} may be constructed
from the continuum random tree of Aldous (1991,1993) by Poisson
splitting along the skeleton of the tree. We describe the distribution
of $X^\infty (t)$ on $\Delta$ at a fixed time $t$. We show that the
size of the cluster containing a given atom, as a process in $t$,
has a simple representation in terms of the stable subordinator of
index $1/2$. As $t \to - \infty$, we establish a Gaussian limit for
(centered and normalized) cluster sizes, and study the size of the
largest cluster.
aldous@stat.berkeley.edu
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information provided by the author
click here.
889. STRICT CONCAVITY OF THE INTERSECTION EXPONENT FOR BROWNIAN
MOTION IN TWO AND THREE DIMENSIONS
Gregory F. Lawler
The intersection exponent for Brownian motion is a measure of
how likely Brownian motion paths in two and three dimensions do not
intersect. We consider the intersection exponent $\xi(\lambda) =
\xi_d(k,\lambda)$ as a function of $\lambda$ and show that $\xi$ has
a continuous, negative second derivative. As a consequence, we improve
some estimates for the intersection exponent; in particular, we give
the first proof that the intersection exponent $\xi_3(1,1)$
is strictly greater than the mean field prediction. The results here
are used in a later paper to analyze the multifractal spectrum
of the harmonic measure of Brownian motion paths.
jose@math.duke.edu
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information provided by the author
click here.
890. THE FRONTIER OF A BROWNIAN PATH IS MULTIFRACTAL
Gregory F. Lawler
We consider the multifractal spectrum of harmonic measure of
a Brownian motion path in two or three dimensions. We show that
the multifractal spectrum is nontrivial and relate the spectrum
to the intersection exponent. As a corollary we show that harmonic
measure on a three dimension Brownian motion path is carried on
a set of Hausdorff dimension strictly less than two.
jose@math.duke.edu
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information provided by the author
click here.
891. AN INVARIANCE PRINCIPLE FOR SEMIMARTINGALE REFLECTING
BROWNIAN MOTIONS IN AN ORTHANT
R. J. Williams
Semimartingale reflecting Brownian motions in an orthant (SRBMs)
are of interest in applied probability
because of their role as heavy traffic approximations for open queueing
networks. It is shown in this paper that a process which satisfies
the definition of an SRBM, except that small
random perturbations in the defining conditions are allowed, is
close in distribution to an SRBM.
This perturbation result is called an invariance principle
by analogy with the
invariance principle of Stroock and Varadhan for diffusions with boundary
conditions. A crucial ingredient in the proof of this result is
an oscillation inequality for solutions of a related
Skorokhod problem.
In a subsequent paper, the invariance principle is used to
give general conditions under which a heavy traffic limit theorem
holds for open multiclass queueing networks.
williams@math.ucsd.edu
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information provided by the author
click here.
892. DIFFUSION APPROXIMATIONS FOR OPEN MULTICLASS QUEUEING NETWORKS:
SUFFICIENT CONDITIONS INVOLVING STATE SPACE COLLAPSE
R. J. Williams
Certain diffusion processes known as
semimartingale reflecting Brownian motions (SRBMs) have been shown to
approximate many single class and some multiclass open queueing
networks under conditions of heavy traffic.
While it is known that not all multiclass networks with feedback
can be approximated
in heavy traffic by SRBMs, one of the outstanding challenges in contemporary
research on queueing networks is to identify broad categories of
networks that can be so approximated and to prove a heavy traffic
limit theorem justifying the approximation.
In this paper, general sufficient conditions are given
under which a heavy traffic limit theorem holds
for open multiclass queueing networks with head-of-the-line (HL)
service disciplines, which in particular require that service
within each class is on a first-in-first-out (FIFO) basis.
The two main conditions that need to be verified are that
(a) the reflection matrix for the SRBM is well defined and
completely-S, and (b) a form of state space collapse holds.
In the current paper, condition (a) is verified
for FIFO networks of Kelly type and for networks with the HLPPS
(head-of-the-line proportional processor sharing) service discipline.
In a companion work, M. Bramson shows that
a multiplicative form of
state space collapse holds for these two families of networks.
These results, when combined with the main theorem of this paper,
yield new heavy traffic limit theorems for
these two families of networks.
williams@math.ucsd.edu
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information provided by the author
click here.
893. STATE SPACE COLLAPSE WITH APPLICATION TO HEAVY TRAFFIC
LIMITS FOR MULTICLASS QUEUEING NETWORKS
Maury Bramson
Heavy traffic limits for queueing networks are a topic of continuing
interest. Presently, the class of networks for which these
limits have been
rigorously derived is restricted. An important ingredient in such work is
the demonstration of state space collapse. Here, we demonstrate state
space collapse for
two families of networks, FIFO queueing networks of Kelly type and HLPPS
queueing networks. We then discuss the ramifications of our techniques for
more general networks. For FIFO networks of Kelly type, the discipline is
first-in, first-out,
and the service rate depends only on the station. For HLPPS networks, the
discipline is head-of-the-line proportional processor sharing, where the
fraction of time spent serving a class present at a station is proportional
to the quantity of the class there, with all of the service going into the
first customer of each class. When the service times are exponentially
distributed, this discipline has queue length process which is equivalent
to that of standard processor sharing, where all customers of a class receive
equal service. To demonstrate state space collapse for both families, we
employ law of large number estimates to show a form of compactness for
appropriately scaled solutions. The limits of these solutions are next
shown to satisfy fluid model equations corresponding to the above queueing
networks. Results from Bramson (1996, 1997) on the asymptotic behavior of
these limits then imply state space collapse. The desired heavy traffic
limits for FIFO and HLPPS networks follow from this and the general
criteria set forth in the companion
paper Williams (1997). State space collapse and the ensuing heavy
traffic limits also hold for more general queueing networks, provided the
solutions of their fluid model equations converge. Partial results are
given for such networks, which include the static priority disciplines.
bramson@math.wisc.edu
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information provided by the author
click here.
894. GENERALIZED MEHLER SEMIGROUPS: THE NON--GAUSSIAN CASE
Marco Fuhrman, Michael R\"ockner
We study generalized Mehler
semigroups, introduced in \cite{BRS} with special emphasis on the non--Gaussian
case. We review and simplify the method of construction. In the general
(non--Gaussian)
case we construct an associated cadlag Markov process in an
appropriate state space obtained as a solution of a stochastic equation which
can be solved ''$\omega$ by $\omega$''. We also show
tightness of the associated $(r,p)$-capacities.
Invariant measures, time regularity and a definition
of the generator are also studied.
roeckner@mathematik.uni-bielefeld.de
895. BROWNIAN SHEET IMAGES AND BESSEL-RIESZ CAPACITY
Davar Khoshnevisan
We show that the image of a 2-dimensional
set under d-dimensional, 2-parameter Brownian sheet
can have positive Lebesgue measure if and only if the
set in question has positive (d/2)-dimensional
Bessel-Riesz capacity. Our methods solve a problem of
J.-P. Kahane.
davar@math.utah.edu (postscript file available)
896. GAUSSIAN MEASURE OF A SMALL BALL AND CAPACITY IN WIENER SPACE
Davar Khoshnevisan and Zhan Shi
We give asymptotic bounds for the Gaussian measure of
a small ball in terms of the hitting probabilities of a suitably
chosen infinite dimensional Brownian motion. Our estimates refine
earlier works of Bruce Erickon.
davar@math.utah.edu (postscript file available)
897. SAMPLE CORRELATIONS OF INFINITE VARIANCE TIME SERIES MODELS:
AN EMPIRICAL AND THEORETICAL STUDY
Jason Cohen, Sidney Resnick and Gennady Samorodnitsky
When the elements of a stationary ergodic time series have finite
variance the
sample correlation function converges (with probability 1) to the
theoretical correlation function.
What happens in the case where the variance is infinite?
In certain cases, the sample correlation function converges in
probability to a constant, but not always.
If within a class of heavy tailed time series the sample correlation
functions do not converge to a constant,
then more care must be taken in making inferences and
in model selection on the basis of sample autocorrelations.
We experimented with simulating various heavy tailed
stationary sequences in an attempt to understand what causes the
sample correlation function to converge or not to converge to a
constant. In two new cases, namely the sum of two independent
moving averages and a random permutation scheme, we are
able to provide theoretical explanations for a random limit
of the sample autocorrelation function as the sample grows.
sid@orie.cornell
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information provided by the author
click here.
898. ON THE INTEGRABILITY CONDITION IN THE MULTIPLICATIVE ERGODIC THEOREM FOR
STOCHASTIC DIFFERENTIAL EQUATIONS
Ludwig Arnold and Peter Imkeller
The multiplicative ergodic theorem is valid under an integrability
condition on the linearized flow with respect to an invariant measure. We
investigate the case where the flow is generated by a stochastic
differential equation and give a criterion in terms of the vector fields
and the (generally non-adapted) invariant measure assuring the validity of
the integrability condition.
imkeller@mathematik.hu-berlin.de
899. ON THE SPATIAL ASYMPTOTIC BEHAVIOUR OF STOCHASTIC FLOWS IN EUCLIDEAN SPACE
Peter Imkeller and Michael Scheutzow
We study asymptotic growth rates of stochastic flows on $\r^d$ and their
derivatives
with respect to the spatial parameter under Lipschitz conditions on the local
characteristics of the generating semimartingales. In a first step these
conditions are
seen to imply moment inequalities for the flow $\p$ of the form\\ $E
\sup_{0\leq t\leq T}
|\p_{0t}(x) - \p_{0t}(y)|^p \leq |x-y|^p\,\,\exp(c p^2)$ for all $p\ge 1$.
In a second
step we deduce the growth rates from an integrated version of these moment
inequalities,
using the continuity lemma of Garsia, Rodemich and Rumsey. We provide two
examples to
show that our results are sharp.
imkeller@mathematik.hu-berlin.de
900. THE ENTRANCE BOUNDARY OF THE MULTIPLICATIVE COALESCENT
David J. Aldous and Vlada Limic
The multiplicative coalescent $X(t)$ is a $l^2$-valued
Markov process representing coalescence of clusters of mass,
where each pair of clusters merges at rate proportional to
product of masses. From random graph asymptotics it is known
(Aldous (1997)) that there exists a {\it standard} version of
this process starting with infinitesimally small clusters at
time $- \infty$.
In this paper, stochastic calculus techniques are used to
describe all versions $(X(t);- \infty < t < \infty)$ of the
multiplicative coalescent. Roughly, an extreme version is
specified by translation and scale parameters, and a vector
$c \in l^3$ of relative sizes of large clusters at time
$- \infty$. Such a version may be characterized in three ways:
via its $t \to - \infty$ behavior, via a representation of
the marginal distribution $X(t)$ in terms of excursion-lengths
of a L{\'e}vy-type process, or via a weak limit of processes
derived from the standard version via a ``coloring" construction.
aldous@stat.berkeley.edu
- To see a preprint or other
information provided by the author
click here.
901. FINITE VOLUME GLAUBER DYNAMICS IN A SMALL MAGNETIC FIELD
Nobuo Yoshida
We consider Glauber dynamics on a finite cube
in the $d$-dimensional lattice ($d \geq 2$),
which is associated with basic Ising model at temperature
$T=1/\b \ll 1$ under a magnetic field $h>0$.
We prove that if the ``effective magnetic field" is positive, then
the relaxation of the Glauber dynamics in the uniform norm
is exponentially fast, uniformly over the size of
underlying cube.
The result covers the case of the free-boundary
condition with arbitrarily small positive magnetic field.
This paper is a continuation of an attempt initiated in
\cite{SY97}( Schonmann, R. H. and Yoshida, N. :
Exponential relaxation of Glauber dynamics with
some special boundary conditions,
to appear in Commun. Math. Phys.)
to shed more light on the relaxation of the finite volume
Glauber dynamics, when the thermodynamic parameter
$(\b,h)$ is so near the phase transition line:
$\{ (\b,h); \b_c <\b \: \& \: h = 0 \}$ that the Dobrushin-Shlosman
mixing condition is no longer available.
nobuo@kusm.kyoto-u.ac.jp (hard copy, latex file, ps.gz file available)
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information provided by the author
click here.
902. THE LOG-SOBOLEV INEQUALITY FOR WEAKLY COUPLED $\phi^{4}$-LATTICE
FIELDS
Nobuo Yoshida
We consider an unbounded spin system on $d$-dimensional lattice
($d \geq 1$) which has ferromagnetically coupled
two-body interactions and
one-body interactions given by a polynomial of degree 4.
Under minimal assumptions on the one-body interaction
(so that arbitrarily deep double wells are allowed),
we prove the log-Sobolev inequality for the Gibbs measures
whenever the coupling constants are small enough.
nobuo@kusm.kyoto-u.ac.jp (hard copy, latex file, ps.gz file available)
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information provided by the author
click here.
903. ON K-CONNECTIVITY FOR A GEOMETRIC RANDOM GRAPH
Mathew D. Penrose
For $n$ points uniformly randomly distributed on the unit cube in $d$
dimensions, with $d > 1$, let $\rho_n$ (respectively $\sigma_n$) denote
the minimum $r$ at which the graph, obtained by adding an edge between
each pair of points distant at most $r$ apart, is $k$-connected
(respectively, has minimum degree $k$). Then $P[\rho_n = \sigma_n] \to 1$
as $n \to \infty$.
Mathew.Penrose@durham.ac.uk
904. A STRONG LAW FOR THE LARGEST NEAREST-NEIGHBOUR LINK
BETWEEN RANDOM POINTS
Mathew D. Penrose
Suppose $X_1,X_2,X_3,\ldots$ are independent random points in $R^d$ with
common density $f$, having compact support $A$ with smooth boundary
$\partial A$, with $f|_A$ continuous. Let $R^n_{i,k}$ denote the
distance from $X_i$ to its $k$-th nearest neighbour amongst the first
$n$ points, and let $M_{n,k} = \max_{i \leq n} R_{i,k}^n$. We
derive an almost sure limit for $ n \theta M_{n,k}^d/\log n $.
We give an analogous result for the case where the points lie in a compact
smooth $d$-dimensional Riemannian manifold.
Mathew.Penrose@durham.ac.uk
905. CONVERGENCE TIME TO EQUILIBRIUM FOR MULTI-PARTICLE MARKOV CHAINS
Anatoli D. Manita
Multi-particle Markov chains ${\cal L}(N)$ are stochastic systems
consisting of $M(N)$ noninteracting particles moving according
to the law of some "one-particle" finite Markov chain ${\cal K}(N)$.
Our goal is to find the convergence time to equilibrium $T(N)$
for the sequence of Markov chains ${\cal L}(N)$ in the situation
when the number of particles $M(N)$ tends to infinity
and the "size" of the one-particle chain ${\cal K}(N)$ grows.
The notion of the convergence time to equilibrium was introduced
in our previous paper (Prob Abstracts 37, abstract 811).
In the present paper we consider wide classes of multi-particle chains
and find $T(N)$ for them as a function of $M(N)$ and $T_{\cal K}(N)$,
where $T_{\cal K}(N)$ is the convergence time to equilibrium
for the sequence of one-particle chains ${\cal K}(N)$.
We apply these results to a discrete analogue of some queueing system
and to a random walk on circle.
manita@mech.math.msu.su
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information provided by the author
click here.
906. A LOWER BOUND FOR THE COVARIANCE OF POSITIVE EVENTS
Massimo Campanino
We prove a lower bound on the covariance of two positive events
in terms of the probabilities of their pivotal events for probability
spaces that verify the conditions of FKG inequality.
campanin@dm.unibo.it
907. POWER-LAW CORRECTIONS TO EXPONENTIAL DECAY OF CONNECTIVITIES
AND CORRELATIONS IN LATTICE MODELS
Kenneth S. Alexander
Consider a translation-invariant bond percolation model on the integer
lattice which has exponential decay of connectivities, that is, the
probability of a connection 0 to x by a path of open bonds decreases like
exp(-m(theta)|x|) for some positive constant m(theta) which may depend on the
direction theta = x/|x|. In two and three dimensions, it is shown that if
the model has the "ratio weak mixing" property and satisfies a special case
of the FKG property, then there is at most a power-law correction to the
exponential decay--there exist A and C such that exp(-m(theta)|x|) \geq P(0
connected to x) \geq A|x|^{-C}exp(-m(theta)|x|) for all nonzero x. In four
or more dimensions, a similar bound holds with |x|^{-C} replaced by
exp(-C(log|x|)^2). In particular the power-law lower bound holds for the
Fortuin-Kasteleyn random cluster model in two dimensions whenever the
connectivity decays exponentially, since ratio weak mixing is known to hold
in that case. Consequently a similar bound holds for correlations in the
Potts model at supercritical temperatures.
alexandr@math.usc.edu
