Probability Abstracts 82

This document contains abstracts 2738-2824. They have been mailed on September 3, 2004.

2738. EXCLUSION PROCESSES WITH MULTIPLE INTERACTIONS 

Yevgeniy Kovchegov

We introduce the mathematical theory of the particle systems that interact
via permutations, where the transition rates are assigned not to the jumps from
a site to a site, but to the permutations themselves. This permutation
processes can be viewed as a generalization of the symmetric exclusion
processes, where particles interact via transpositions. The duality and
coupling techniques for the processes are described, the needed conditions for
them to apply are established. The stationary distributions of the permutation
processes are explored for translation invariant cases.

2739. MULTI-PARTICLE PROCESSES WITH REINFORCEMENTS 

Yevgeniy Kovchegov

The multi-particle generalization of the edge-reinforced random walk is
stated. In one dimension, basic recurrence results are obtained.

2740. LIMIT LAW OF THE STANDARD RIGHT FACTOR OF A RANDOM LYNDON WORD 

Regine Marchand, Elahe Zohoorian Azad

Consider the set of finite words on a totally ordered alphabet with $q$
letters. We prove that the distribution of the length of the standard right
factor of a random Lyndon word with length $n$, divided by $n$, converges to:
$$\mu(dx)=\frac1q \delta_{1}(dx) + \frac{q-1}q \mathbf{1}_{[0,1)}(x)dx,$$ when
$n$ goes to infinity. The convergence of all moments follows. This paper
completes thus the results of \cite{Bassino}, giving the asymptotics of the
mean length of the standard right factor of a random Lyndon word with length
$n$ in the case of a two letters alphabet.

2741. FLUID MODEL FOR A NETWORK OPERATING UNDER A FAIR BANDWIDTH-SHARING POLICY 

F. P. Kelly and R. J. Williams

We consider a model of Internet congestion control that represents the
randomly varying number of flows present in a network where bandwidth is shared
fairly between document transfers. We study critical fluid models obtained as
formal limits under law of large numbers scalings when the average load on at
least one resource is equal to its capacity. We establish convergence to
equilibria for fluid models and identify the invariant manifold.
 The form of the invariant manifold gives insight into the phenomenon of
entrainment whereby congestion at some resources may prevent other resources
from working at their full capacity.

2742. SCHEDULING A MULTI CLASS QUEUE WITH MANY EXPONENTIAL SERVERS: ASYMPTOTIC OPTIMALITY IN HEAVY TRAFFIC 

Rami Atar, Avi Mandelbaum and Martin I. Reiman

We consider the problem of scheduling a queueing system in which many
statistically identical servers cater to several classes of impatient
customers. Service times and impatience clocks are exponential while arrival
processes are renewal. Our cost is an expected cumulative discounted function,
linear or nonlinear, of appropriately normalized performance measures. As a
special case, the cost per unit time can be a function of the number of
customers waiting to be served in each class, the number actually being served,
the abandonment rate, the delay experienced by customers, the number of idling
servers, as well as certain combinations thereof. We study the system in an
asymptotic heavy-traffic regime where the number of servers n and the offered
load r are simultaneously scaled up and carefully balanced: n\approx r+\beta
\sqrtr for some scalar \beta. This yields an operation that enjoys the benefits
of both heavy traffic (high server utilization) and light traffic (high service
levels.)

2743. LOCAL LIMIT THEORY AND LARGE DEVIATIONS FOR SUPERCRITICAL BRANCHING PROCESSES 

Peter E. Ney and Anand N. Vidyashankar

In this paper we study several aspects of the growth of a supercritical
Galton-Watson process {Z_n:n\ge1}, and bring out some criticality phenomena
determined by the Schroder constant. We develop the local limit theory of Z_n,
that is, the behavior of P(Z_n=v_n) as v_n\nearrow \infty, and use this to
study conditional large deviations of {Y_{Z_n}:n\ge1}, where
Y_n satisfies an LDP, particularly of {Z_n^{-1}Z_{n+1}:n\ge1} conditioned on
Z_n\ge v_n.

2744. MODELING CREDIT RISK WITH PARTIAL INFORMATION 

Umut Cetin, Robert Jarrow, Philip Protter and Yildiray Yildirim

This paper provides an alternative approach to Duffie and Lando [Econometrica
69 (2001) 633-664] for obtaining a reduced form credit risk model from a
structural model. Duffie and Lando obtain a reduced form model by constructing
an economy where the market sees the manager's information set plus noise.
 The noise makes default a surprise to the market. In contrast, we obtain a
reduced form model by constructing an economy where the market sees a reduction
of the manager's information set. The reduced information makes default a
surprise to the market. We provide an explicit formula for the default
intensity based on an Azema martingale, and we use excursion theory of Brownian
motions to price risky debt.

2745. STATISTICS OF A VORTEX FILAMENT MODEL 

Franco Flandoli and Massimiliano Gubinelli

A random field composed by Poisson distributed Brownian vortex filaments is
constructed. The filament have a random thickness, length and intensity,
governed by a measure $\gamma$. Under appropriate assumptions on $\gamma$ we
compute the scaling law of the structure function and get the multifractal
scaling as a particular case.

2746. DISTANCES IN RANDOM GRAPHS WITH INFINITE MEAN DEGREES 

Remco van der Hofstad, Gerard Hooghiemstra, Dmitri Znamenski

We study random graphs with an i.i.d. degree sequence of which the tail of
the distribution function $F$ is regularly varying with exponent $\tau\in
(1,2)$. Thus, the degrees have infinite mean. Such random graphs can serve as
models for complex networks where degree power laws are observed.
  The minimal number of edges between two arbitrary nodes, also called the
graph distance or the hopcount, in a graph with $N$ nodes is investigated when
$N\to \infty$. The paper is part of a sequel of three papers. The other two
papers study the case where $\tau \in (2,3)$, and $\tau \in (3,\infty),$
respectively.
  The main result of this paper is that the graph distance converges for
$\tau\in (1,2)$ to a limit random variable with probability mass exclusively on
the points 2 and 3. We also consider the case where we condition the degrees to
be at most $N^{\alpha}$ for some $\alpha>0.$ For
$\tau^{-1}<\alpha<(\tau-1)^{-1}$, the hopcount converges to 3 in probability,
while for $\alpha>(\tau-1)^{-1}$, the hopcount converges to the same limit as
for the unconditioned degrees. Our results give convincing asymptotics for the
hopcount when the mean degree is infinite, using extreme value theory.

2747. DISTANCES IN RANDOM GRAPHS WITH FINITE VARIANCE DEGREES 

Remco van der Hofstad, Gerard Hooghiemstra, Piet Van Mieghem

In this paper we study a random graph with $N$ nodes, where node $j$ has
degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with $\prob(D_j\leq x)=F(x)$. We
assume that $1-F(x)\leq c x^{-\tau+1}$ for some $\tau>3$ and some constant
$c>0$. This graph model is a variant of the so-called configuration model, and
includes heavy tail degrees with finite variance.
  The minimal number of edges between two arbitrary connected nodes, also known
as the graph distance or the hopcount, is investigated when $N\to \infty$. We
prove that the graph distance grows like $\log_{\nu}N$, when the base of the
logarithm equals $\nu=\expec[D_j(D_j -1)]/\expec[D_j]>1$. This confirms the
heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the
random fluctuations around this asymptotic mean $\log_{\nu}{N}$ are
characterized and shown to be uniformly bounded. In particular, we show
convergence in distribution of the centered graph distance along exponentially
growing subsequences.

2748. ANALYTIC URNS 

Philippe Flajolet, Joaquim Gabarr\'o and Helmut Pekari

This article describes a purely analytic approach to urn models of the
generalized or extended P\'olya-Eggenberger type, in the case of two types of
balls and constant ``balance'', i.e., constant row sum. The treatment starts
from a quasilinear first-order partial differential equation associated with a
combinatorial renormalization of the model and bases itself on elementary
conformal mapping arguments coupled with singularity analysis techniques.
Probabilistic consequences in the case of ``subtractive'' urns are new
representations for the probability distribution of the urn's composition at
any time $n$, structural information on the shape of moments of all orders,
estimates of the speed of convergence to the Gaussian limit, and an explicit
determination of the associated large deviation function. In the general case,
analytic solutions involve Abelian integrals over the Fermat curve $x^h+y^h=1$.
Several urn models, including a classical one associated with balanced trees
(2--3 trees and fringe-balanced search trees) and related to a previous study
of Panholzer and Prodinger, as well as all urns of balance 1 or 2 and a
sporadic urn of balance 3, are shown to admit of explicit representations in
terms of Weierstra{\ss} elliptic functions: these elliptic models appear
precisely to correspond to regular tessellations of the Euclidean plane.

2749. CONDITIONAL EQUI-CONCENTRATION OF TYPES 

Marian Grendar

Conditional Equi-concentration of Types on I-projections (ICET) is presented.
It provides an extension of Conditioned Weak Law of Large Numbers to the case
of several I-projections. A novel form of MaxProb/MaxEnt Theorem is recalled,
which permits to state directly a mu-projection variant of the Conditional
Equi-concentration of Types (muCET). Also, Conditional Equi-concentration of
Types on J- and gamma-projections is discussed.

2750. SOLUTION OF MONGE-AMP\`ERE EQUATION ON WIENER SPACE FOR LOG-CONCAVE MEASURES: GENERAL CASE 

Denis Feyel and A.S. Ustunel

We show the existence of the strong solutions of the Monge-Ampere equation
for the log-concave measures without any regularity assumption. In particular
the measures with density of the form $\exp -f$, where $f$ is an $H$-convex
function which may take the value $\infty$ on a set of positive measure are
included.

2751. RANDOM ORIENTED TREES: A MODEL OF DRAINAGE NETWORKS 

Sreela Gangopadhyay, Rahul Roy and Anish Sarkar

Consider the d-dimensional lattice Z^d where each vertex is ``open'' or
``closed'' with probability p or 1-p, respectively. An open vertex v is
connected by an edge to the closest open vertex w such that the dth
co-ordinates of v and w satisfy w(d)=v(d)-1. In case of nonuniqueness of such a
vertex w, we choose any one of the closest vertices with equal probability and
independently of the other random mechanisms. It is shown that this random
graph is a tree almost surely for d=2 and 3 and it is an infinite collection of
distinct trees for d\geq4. In addition, for any dimension, we show that there
is no bi-infinite path in the tree and we also obtain central limit theorems of
(a) the number of vertices of a fixed degree \nu and (b) the number of edges of
a fixed length l.

2752. A CHARACTERIZATION OF HEDGING PORTFOLIOS FOR INTEREST RATE CONTINGENT CLAIMS 

Rene Carmona and Michael Tehranchi

We consider the problem of hedging a European interest rate contingent claim
with a portfolio of zero-coupon bonds and show that an HJM type Markovian model
driven by an infinite number of sources of randomness does not have some of the
shortcomings found in the classical finite-factor models. Indeed, under natural
conditions on the model, we find that there exists a unique hedging strategy,
and that this strategy has the desirable property that at all times it consists
of bonds with maturities that are less than or equal to the longest maturity of
the bonds underlying the claim.

2753. A MIXTURE REPRESENTATION OF \PI WITH APPLICATIONS IN MARKOV CHAIN MONTE CARLO AND PERFECT SAMPLING 

James P. Hobert and Christian P. Robert

Let X={X_n:n=0,1,2,...} be an irreducible, positive recurrent Markov chain
with invariant probability measure \pi. We show that if X satisfies a one-step
minorization condition, then \pi can be represented as an infinite mixture. The
distributions in the mixture are associated with the hitting times on an
accessible atom introduced via the splitting construction of Athreya and Ney
[Trans. Amer. Math. Soc. 245 (1978) 493-501] and Nummelin [Z. Wahrsch. Verw.
Gebiete 43 (1978) 309-318]. When the small set in the minorization condition is
the entire state space, our mixture representation of \pi reduces to a simple
formula, first derived by Breyer and Roberts [Methodol. Comput. Appl. Probab. 3
(2001) 161-177] from which samples can be easily drawn. Despite the fact that
the derivation of this formula involves no coupling or backward simulation
arguments, the formula can be used to reconstruct perfect sampling algorithms
based on coupling from the past

2754. PRACTICAL DRIFT CONDITIONS FOR SUBGEOMETRIC RATES OF CONVERGENCE 

Randal Douc, Gersende Fort, Eric Moulines and Philippe Soulier

We present a new drift condition which implies rates of convergence to the
stationary distribution of the iterates of a \psi-irreducible aperiodic and
positive recurrent transition kernel. This condition, extending a condition
introduced by Jarner and Roberts [Ann. Appl. Probab. 12 (2002) 224-247] for
polynomial convergence rates, turns out to be very convenient to prove
subgeometric rates of convergence. Several applications are presented including
nonlinear autoregressive models, stochastic unit root models and
multidimensional random walk Hastings-Metropolis algorithms.

2755. RUIN PROBABILITIES AND DECOMPOSITIONS FOR GENERAL PERTURBED RISK PROCESSES 

Miljenko Huzak, Mihael Perman, Hrvoje Sikic and Zoran Vondracek

We study a general perturbed risk process with cumulative claims modelled by
a subordinator with finite expectation, with the perturbation being a
spectrally negative Levy process with zero expectation. We derive a
 Pollaczek-Hinchin type formula for the survival probability of that risk
process, and give an interpretation of the formula based on the decomposition
of the dual risk process at modified ladder epochs.

2756. ON THE NEYMAN-PEARSON PROBLEM FOR LAW-INVARIANT RISK MEASURES AND ROBUST UTILITY FUNCTIONALS 

Alexander Schied

Motivated by optimal investment problems in mathematical finance, we consider
a variational problem of Neyman-Pearson type for law-invariant robust utility
functionals and convex risk measures. Explicit solutions are found for
quantile-based coherent risk measures and related utility functionals.
 Typically, these solutions exhibit a critical phenomenon: If the capital
constraint is below some critical value, then the solution will coincide with a
classical solution; above this critical value, the solution is a superposition
of a classical solution and a less risky or even risk-free investment. For
general risk measures and utility functionals, it is shown that there exists a
solution that can be written as a deterministic increasing function of the
price density.

2757. WHEN CAN THE TWO-ARMED BANDIT ALGORITHM BE TRUSTED? 

Damien Lamberton, Gilles Pages and Pierre Tarres

We investigate the asymptotic behavior of one version of the so-called
two-armed bandit algorithm. It is an example of stochastic approximation
procedure whose associated ODE has both a repulsive and an attractive
equilibrium, at which the procedure is noiseless. We show that if the gain
parameter is constant or goes to 0 not too fast, the algorithm does fall in the
noiseless repulsive equilibrium with positive probability, whereas it always
converges to its natural attractive target when the gain parameter goes to zero
at some appropriate rates depending on the parameters of the model. We also
elucidate the behavior of the constant step algorithm when the step goes to 0.
Finally, we highlight the connection between the algorithm and the
 Polya urn. An application to asset allocation is briefly described.

2758. GENERALIZED URN MODELS OF EVOLUTIONARY PROCESSES 

Michel Benaim, Sebastian J. Schreiber and Pierre Tarres

Generalized Polya urn models can describe the dynamics of finite populations
of interacting genotypes. Three basic questions these models can address are:
Under what conditions does a population exhibit growth? On the event of growth,
at what rate does the population increase? What is the long-term behavior of
the distribution of genotypes? To address these questions, we associate a mean
limit ordinary differential equation (ODE) with the urn model. Previously, it
has been shown that on the event of population growth, the limiting
distribution of genotypes is a connected internally chain recurrent set for the
mean limit ODE. To determine when growth and convergence occurs with positive
probability, we prove two results. First, if the mean limit ODE has an
``attainable'' attractor at which growth is expected, then growth and
convergence toward this attractor occurs with positive probability. Second, the
population distribution almost surely does not converge to sets where growth is
not expected

2759. STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY LEVY SPACE-TIME WHITE NOISE 

Arne Lokka, Bernt Oksendal and Frank Proske

In this paper we develop a white noise framework for the study of stochastic
partial differential equations driven by a d-parameter (pure jump) Levy white
noise. As an example we use this theory to solve the stochastic Poisson
equation with respect to Levy white noise for any dimension d. The solution is
a stochastic distribution process given explicitly. We also show that if d\leq
3, then this solution can be represented as a classical random field in L2(\mu
), where \mu is the probability law of the Levy process.
 The starting point of our theory is a chaos expansion in terms of generalized
 Charlier polynomials. Based on this expansion we define Kondratiev spaces and
the Levy Hermite transform.

2760. OPTIMAL SCALING OF MALA FOR NONLINEAR REGRESSION 

Laird Arnault Breyer, Mauro Piccioni and Sergio Scarlatti

We address the problem of simulating efficiently from the posterior
distribution over the parameters of a particular class of nonlinear regression
models using a Langevin-Metropolis sampler. It is shown that as the number N of
parameters increases, the proposal variance must scale as N{-1/3} in order to
converge to a diffusion. This generalizes previous results of Roberts and
Rosenthal [J. R. Stat. Soc. Ser. B Stat. Methodol. 60 (1998) 255-268] for the
i.i.d. case, showing the robustness of their analysis.

2761. EARLIEST-DEADLINE-FIRST SERVICE IN HEAVY-TRAFFIC ACYCLIC NETWORKS 

Lukasz Kruk, John Lehoczky, Steven Shreve and Shu-Ngai Yeung

This paper presents a heavy traffic analysis of the behavior of multi-class
acyclic queueing networks in which the customers have deadlines. We assume the
queueing system consists of J stations, and there are K different customer
classes. Customers from each class arrive to the network according to
independent renewal processes. The customers from each class are assigned a
random deadline drawn from a deadline distribution associated with that class
and they move from station to station according to a fixed acyclic route.
 The customers at a given node are processed according to the
earliest-deadline-first
 (EDF) queue discipline. At any time, the customers of each type at each node
have a lead time, the time until their deadline lapses. We model these lead
times as a random counting measure on the real line. Under heavy traffic
conditions and suitable scaling, it is proved that the measure-valued lead-time
process converges to a deterministic function of the workload process.

2762. ITERATED BROWNIAN MOTION IN AN OPEN SET 

R. Dante DeBlassie

Suppose a solid has a crack filled with a gas. If the crack reaches the
surrounding medium, how long does it take the gas to diffuse out of the crack?
Iterated Brownian motion serves as a model for diffusion in a crack.
  If \tau is the first exit time of iterated Brownian motion from the solid,
then P(\tau>t) can be viewed as a measurement of the amount of contaminant left
in the crack at time t. We determine the large time asymptotics of P(\tau>t)
for both bounded and unbounded sets. We also discuss a strange connection
between iterated Brownian motion and the parabolic operator
{1/8}\Delta^2-\frac{\partial}{\partial t}.

2763. CONCENTRATION OF PERMANENT ESTIMATORS FOR CERTAIN LARGE MATRICES 

Shmuel Friedland, Brian Rider and Ofer Zeitouni

Let A_n=(a_{ij})_{i,j=1}^n be an n\times n positive matrix with entries in
[a,b], 0<a\le b. Let X_n=(\sqrta_{ij}x_{ij})_{i,j=1}^n be a random matrix,
where {x_{ij}} are i.i.d. N(0,1) random variables. We show that for large n,
\det (X_n^TX_n) concentrates sharply at the permanent of A_n, in the sense that
n^{-1}\log (\det(X_n^TX_n)/perA_n)\to_{n\to\infty}0 in probability.

2764. UNIFORM MARKOV RENEWAL THEORY AND RUIN PROBABILITIES IN MARKOV RANDOM WALKS 

Cheng-Der Fuh

Let {X_n,n\geq0} be a Markov chain on a general state space X with transition
probability P and stationary probability \pi. Suppose an additive component
 S_n takes values in the real line R and is adjoined to the chain such that
 {(X_n,S_n),n\geq0} is a Markov random walk. In this paper, we prove a uniform
 Markov renewal theorem with an estimate on the rate of convergence. This
result is applied to boundary crossing problems for {(X_n,S_n),n\geq0}.
 To be more precise, for given b\geq0, define the stopping time
\tau=\tau(b)=inf{n:S_n>b}.
 When a drift \mu of the random walk S_n is 0, we derive a one-term Edgeworth
type asymptotic expansion for the first passage probabilities P_{\pi}{\tau<m}
and P_{\pi}{\tau<m,S_m<c}, where m\leq\infty, c\leq b and P_{\pi} denotes the
probability under the initial distribution \pi. When \mu\neq0, Brownian
approximations for the first passage probabilities with correction terms are
derived.

2765. THE EVOLUTION OF A RANDOM VORTEX FILAMENT 

Hakima Bessaih and Massimiliano Gubinelli and Francesco Russo

We study an evolution problem in the space of continuous loops in
three-dimensional Euclidean space modelled upon the dynamics of vortex lines in
3d incompressible and inviscid fluids. We establish existence of a local
solution starting from H\"older regular loops with index greater than 1/3. When
the H\"older regularity of the initial condition X is smaller or equal 1/2 we
require X to be a rough path in the sense of Lyons. The solution will then live
in an appropriate space of rough paths. In particular we can construct (local)
solution starting from almost every Brownian loop.

2766. AN ALMOST SURE INVARIANCE PRINCIPLE FOR RENORMALIZED INTERSECTION LOCAL TIMES 

Richard F. Bass, Jay Rosen 

Let \beta_k(n) be the number of self-intersections of order k, appropriately
renormalized, for a mean zero random walk X_n in Z^2 with 2+\delta moments. On
a suitable probability space we can construct X_n and a planar Brownian motion
W_t such that for each k\geq 2, |\beta_k(n)-\gamma_k(n)|=O(n^{-a}), a.s. for
some a>0 where \gamma_k(n) is the renormalized self-intersection local time of
order k at time 1 for the Brownian motion W_{nt}/\sqrt n.

2767. BIVARIATE UNIQUENESS AND ENDOGENY FOR RECURSIVE DISTRIBUTIONAL EQUATIONS : TWO EXAMPLES 

Antar Bandyopadhyay

In this work we prove the \emph{bivariate uniqueness} property for two
"max-type" \emph{recursive distributional equations} which then lead to the
proof of \emph{endogeny} for the associated \emph{recursive tree processes}.
Thus providing two concrete instances of the general theory developed by Aldous
and Bandyopadhyay. The first example discussed here deals with the construction
of a frozen percolation process on a infinite regular binary tree. For this we
prove that the construction do not involve any external randomness. It is also
shown that same is true for any $r$-regular tree and more interestingly for any
infinite regular Galton-Watson branching process trees with mild moment
condition on the progeny distribution. The second example is proving the
endogeny for the \emph{Logistic} recursive distributional equation which
appears for studying the asymptotic limit of the random assignment problem
using local-weak convergence method. The two examples are quite unrelated and
hence illustrate a broad range of applicability of the general methods of
Aldous and Bandyopadhyay.

2768. ROUTING COMPLEXITY OF FAULTY NETWORKS 

Omer Angel, Itai Benjamini, Eran Ofek, Udi Wieder

One of the fundamental problems in distributed computing is how to
efficiently perform routing in a faulty network in which each link fails with
some probability. This paper investigates how big the failure probability can
be, before the capability to efficiently find a path in the network is lost.
Our main results show tight upper and lower bounds for the failure probability
which permits routing, both for the hypercube and for the $d-$dimensional mesh.
We use tools from percolation theory to show that in the $d-$dimensional mesh,
once a giant component appears -- efficient routing is possible. A different
behavior is observed when the hypercube is considered. In the hypercube there
is a range of failure probabilities in which short paths exist with high
probability, yet finding them must involve querying essentially the entire
network. Thus the routing complexity of the hypercube shows an asymptotic phase
transition. The critical probability with respect to routing complexity lies in
a different location then that of the critical probability with respect to
connectivity. Finally we show that an oracle access to links (as opposed to
local routing) may reduce significantly the complexity of the routing problem.
We demonstrate this fact by providing tight upper and lower bounds for the
complexity of routing in the random graph $G_{n,p}$.

2769. MALLIAVIN CALCULUS FOR THE STOCHASTIC 2D NAVIER STOKES EQUATION 

Jonathan C. Mattingly, Etienne Pardoux

We consider the incompressible, two dimensional Navier Stokes equation with
periodic boundary conditions under the effect of an additive, white in time,
stochastic forcing. Under mild restrictions on the geometry of the scales
forced, we show that any finite dimensional projection of the solution
possesses a smooth density with respect to Lebesgue measure. We also show that
under natural assumptions the density of such a projection is everywhere
strictly positive. In particular, our conditions are viscosity independent. We
are mainly interested in forcing which excites a very small number of modes.
All of the results rely on the nondegeneracy of the infinite dimensional
Malliavin matrix.

2770. TWO-DIMENSIONAL GIBBSIAN POINT PROCESSES WITH CONTINUOUS SPIN-SYMMETRIES 

Thomas Richthammer

We consider two-dimensional marked point processes which are Gibbsian with a
two-body-potential U. U is supposed to have an internal continuous symmetry. We
show that under suitable continuity conditions the considered processes are
invariant under the given symmetry. We will achieve this by using Ruelle`s
superstability estimates and percolation arguments.

2771. INTERPOLATED INEQUALITIES BETWEEN EXPONENTIAL AND GAUSSIAN, ORLICZ HYPERCONTRACTIVITY AND ISOPERIMETRY 

F. Barthe, P. Cattiaux and C. Roberto

We introduce and study a notion of Orlicz hypercontractive semigroups. We
analyze their relations with general $F$-Sobolev inequalities, thus extending
Gross hypercontractivity theory. We provide criteria for these Sobolev type
inequalities and for related properties. In particular, we implement in the
context of probability measures the ideas of Maz'ja's capacity theory, and
present equivalent forms relating the capacity of sets to their measure. Orlicz
hypercontractivity efficiently describes the integrability improving properties
of the Heat semigroup associated to the Boltzmann measures $\mu_\alpha (dx) =
(Z_\alpha)^{-1} e^{-2|x|^\alpha} dx$, when $\alpha\in (1,2)$. As an application
we derive accurate isoperimetric inequalities for their products. This
completes earlier works by Bobkov-Houdr\'e and Talagrand, and provides a scale
of dimension free isoperimetric inequalities as well as comparison theorems.

2772. ON THE SHAPE OF THE GROUND STATE EIGENFUNCTION FOR STABLE PROCESSES 

Rodrigo Banuelos, Tadeusz Kulczycki and Pedro J. Mendez-Hernandez

We prove that the ground state eigenfunction for symmetric stable processes
of order $\alpha\in (0, 2)$ killed upon leaving the interval $(-1, 1)$ is
concave on $(-{1/2}, {1/2})$. We call this property "mid--concavity." A similar
statement holds for rectangles in $\R^d$, $d>1$. These result follow from
similar results for finite dimensional distributions of Brownian motion and
subordination.

2773. THE EXIT DISTRIBUTION FOR ITERATED BROWNIAN MOTION IN CONES 

Rodrigo Banuelos and Dante DeBlassie

We study the distribution of the exit place of iterated Brownian motion in a
cone, obtaining information about the chance of the exit place having large
magnitude. Along the way, we determine the joint distribution of the exit time
and exit place of Brownian motion in a cone. This yields information on large
values of the exit place (harmonic measure) for Brownian motion. The harmonic
measure for cones has been studied by many authors for many years. Our results
are sharper than any previously obtained

2774. SYMMETRIC STABLE PROCESSES IN PARABOLA--SHAPED REGIONS 

Rodrigo Banuelos and Krzysztof Bogdan

We identify the critical exponent of integrability of the first exit time of
rotation invariant stable L\'evy process from parabola--shaped region.

2775. IMPROVING ASYMPTOTIC VARIANCE OF MCMC ESTIMATORS: NON-REVERSIBLE CHAINS ARE BETTER 

Radford M. Neal

I show how any reversible Markov chain on a finite state space that is
irreducible, and hence suitable for estimating expectations with respect to its
invariant distribution, can be used to construct a non-reversible Markov chain
on a related state space that can also be used to estimate these expectations,
with asymptotic variance at least as small as that using the reversible chain
(typically smaller). The non-reversible chain achieves this improvement by
avoiding (to the extent possible) transitions that backtrack to the state from
which the chain just came. The proof that this modification cannot increase the
asymptotic variance of an MCMC estimator uses a new technique that can also be
used to prove Peskun's (1973) theorem that modifying a reversible chain to
reduce the probability of staying in the same state cannot increase asymptotic
variance. A non-reversible chain that avoids backtracking will often take
little or no more computation time per transition than the original reversible
chain, and can sometime produce a large reduction in asymptotic variance,
though for other chains the improvement is slight. In addition to being of some
practical interest, this construction demonstrates that non-reversible chains
have a fundamental advantage over reversible chains for MCMC estimation.
Research into better MCMC methods may therefore best be focused on
non-reversible chains.

2776. ASYMPTOTIC BEHAVIOR OF RANDOM HEAPS 

J. Ben Hough

We study the asymptotic behavior of a random walk on the locally free group,
and disprove a conjecture concerning the expected number of removeable
generators.

2777. YOUNG INTEGRALS AND SPDES 

Antoine Lejay, Massimiliano Gubinelli, Samy Tindel

In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt +
B(Y_t) dX_t$, where $X$ is a $\gamma$-H\"older continuous function of the time
parameter, with values in a distribution space, and $-A$ the generator of an
analytical semigroup. Then, we will give some sharp conditions on $X$ in order
to solve the above equation in a function space, first in the linear case (for
any value of $\gamma$ in $(0,1)$), and then when $B$ satisfies some Lipschitz
type conditions (for $\gamma>1/2$). The solution of the evolution problem will
be understood in the mild sense, and the integrals involved in that definition
will be of Young type.

2778. BROWNIAN MOTION WITH KILLING AND REFLECTION AND THE "HOT--SPOTS" PROBLEM 

Rodrigo Banuelos, Michael Pang and Mihai Pascu

We investigate the "hot--spots" property for the survival time probability of
Brownian motion with killing and reflection in planar convex domains whose
boundary consists of two curves, one of which is an arc of a circle,
intersecting at acute angles. This leads to the "hot--spots" property for the
mixed Dirichlet--Neumann eigenvalue problem in the domain with Neumann
conditions on one of the curves and Dirichlet conditions on the other

2779. \ALPHA-CONTINUITY PROPERTIES OF STABLE PROCESSES 

R.D. DeBlassie, Pedro J. Mendez-Hernandez

Let $D$ be a domain of finite Lebesgue measure in $\bR^d$ and let $X^D_t$ be
the symmetric $\alpha$-stable process killed upon exiting $D$. Each element of
the set $\{\lambda_i^\alpha\}_{i=1}^\infty$ of eigenvalues associated to
$X^D_t$, regarded as a function of $\alpha\in(0,2)$, is right continuous. In
addition, if $D$ is Lipschitz and bounded, then each $ \lambda_i^\alpha$ is
continuous in $\alpha$ and the set of associated eigenfunctions is precompact.
We also prove that if $D$ is a domain of finite Lebesgue measure, then for all
$0<\alpha<\beta\leq 2$ and $i\geq 1$, \[\lambda_i^\alpha \leq [ 
\lambda^\beta_i]^{\alpha/\beta}.\] Previously, this bound had been known only
for $\beta=2$ and $\alpha$ rational.

2780. ASYMPTOTIC ENUMERATION AND LOGICAL LIMIT LAWS FOR EXPANSIVE MULTISETS AND SELECTIONS 

Boris L. Granovsky and Dudley Stark

A multiset is an unordered sample from a set of object types in which the
number of items is variable, but the total weight of the objects equals a
parameter $n$. The number of types of objects of weight $j$ is $a_j$. Let $c_n$
be the number of multiset representatives of total weight $n$. Then, for
$T_n=\sum_{j=1}^njZ_j$ where $Z_j$ are independent but not identically
distributed negative binomial $\bigotimes$ random variables with appropriate
parameters, $c_n=e^{n\sigma} \prod_{j=1}^n ({1-e^{-\sigma j}})^{-a_j}
P(T_n=n)$, where $\sigma$ is an arbitrary parameter partially determining the
distribution of the $Z_j$.
  When $a_j\asymp j^{r-1} y^j$ for some $r>0$, $y\geq 1$, then we say that the
multiset is {\em expansive}. For expansive multisets we prove a local limit
lemma for $T_n$ under the condition that $\sigma$ is chosen so that $E(T_n)=n$.
Moreover, we prove that $c_n/c_{n+1}\to 1$ and that $c_n/c_{n+1}<1$ for large
enough $n$. This allows us to prove Monadic Second Order Limit Laws for
expansive multisets. $ \bigotimes$ The above results are extended to a class of
expansive multisets with oscillation.
  If the condition $a_j=Kj^{r-1}y^j + O(y^{\nu j})$ is imposed, where $K>0$,
$r>0$, $y>\bigotimes 1$, $\nu\in (0,1)$, we are then able to find an explicit
$\bigotimes$ asymptotic formula for $c_n$.
  $\bigotimes$ In a similar way we study the asymptotic behavior of selections
which are defined as multisets composed of components of distinct sizes.

2781. LAW OF LARGE NUMBERS FOR INCREASING SUBSEQUENCES OF RANDOM PERMUTATIONS AND AN APPROXIMATION RESULT FOR THE UNIFORM MEASURE 

Ross Pinsky

Let the random variable $Z_{n,k}$ denote the number of increasing
subsequences of length $k$ in a random permutation from $S_n$, the symmetric
group of permutations of $\{1,...,n\}$. We show that
$Var(Z_{n,k_n})=o((EZ_{n,k_n})^2)$ as $ n\to\infty$ if and only if
$k_n=o(n^\frac25)$. In particular then, the weak law of large numbers holds for
$Z_{n,k_n}$ if $k_n=o(n^\frac25)$. We also show the following approximation
result for the uniform measure $U_n$ on $S_n$. Define the probability measure
$\mu_{n;k_n}$ on $S_n$ as follows: Consider $n$ cards, numbered from 1 to $n$,
and laid out on a table from left to right in increasing order. Place a mark on
$k_n$ of the cards, chosen at random. Then pick up all the unmarked cards and
randomly insert them between the $k_n$ marked cards that remained on the table.
Denote the resulting distribution on $S_n$ by $\mu_{n;k_n}$. The weak law of
large numbers holds for $Z_{n,k_n}$ if and only if the total variation distance
between $\mu_{n;k_n}$ and $U_n$ converges to 0 as $n\to\infty$.
  In order to evaluate the asymptotic behavior of the second moment, we need to
analyze certain occupation times of certain conditioned two-dimensional random
walks.

2782. THE SEMIGROUP OF THE GLAUBER DYNAMICS OF A CONTINUOUS SYSTEM OF FREE PARTICLES 

Yuri Kondratiev, Eugene Lytvynov, Michael R\"ockner

We study properties of the semigroup $(e^{-tH})_{t\ge 0}$ on the space $L^
2(\Gamma_X,\pi)$, where $\Gamma_X$ is the configuration space over a locally
compact second countable Hausdorff topological space $X$, $\pi$ is a Poisson
measure on $\Gamma_X$, and $H$ is the generator of the Glauber dynamics. We
explicitly construct the corresponding Markov semigroup of kernels $(P_t)_{t\ge
0}$ and, using it, we prove the main results of the paper: the Feller property
of the semigroup $(P_t)_{t\ge 0}$ with respect to the vague topology on the
configuration space $\Gamma_X$, and the ergodic property of $(P_t)_{t\ge 0}$.
Following an idea of D. Surgailis, we also give a direct construction of the
Glauber dynamics of a continuous infinite system of free particles. The main
point here is that this process can start in every $\gamma\in\Gamma_X$, will
never leave $\Gamma_X$ and has cadlag sample paths in $\Gamma_X$.

2783. L\'EVY PROCESSES AND JACOBI FIELDS 

Eugene Lytvynov

We review the recent results on the Jacobi field of a (real-valued) L\'evy
process defined on a Riemannian manifold. In the case where the L\'evy process
is neither Gaussian, nor Poisson, the corresponding Jacobi field acts in an
extended Fock space. We also give a unitary equivalent representation of the
Jacobi field in a usual Fock space. This representation is inspired by a result
by Accardi, Franz, and Skeide (2002).

2784. LIMIT THEOREMS FOR RAREFACTION OF SET OF DIFFUSION PROCESSES BY BOUNDARIES 

Anielllo Fedullo and Vitalii A. Gasanenko

This paper is devoted to the study of the following problem. We have set of
diffusion processes with absorption on boundaries in some region at initial
time $t=0$. It is required to estimate of number of the unabsorbed processes
for the fixed time ~$\tau>0$. The number of initial processes is considered as
function of $\tau$ and $\tau\to\infty$.

2785. LINEAR FILTERING OF SYSTEMS WITH MEMORY 

A. Inoue, Y. Nakano, V. Anh

We study the linear filtering problem for systems driven by continuous
Gaussian processes with memory described by two parameters. The driving
processes have the virtue that they possess stationary increments and simple
semimartingale representations simultaneously. It allows for straightforward
parameter estimations. After giving the semimartingale representations of the
processes by innovation theory, we derive Kalman-Bucy-type filtering equations
for the systems. We apply the result to the optimal portfolio problem for an
investor with partial observations. We illustrate the tractability of the
filtering algorithm by numerical implementations.

2786. THE THEORY OF QUANTUM LEVY PROCESSES 

Uwe Franz

Various recent results on quantum L\'evy processes are presented. The first
part provides an introduction to the theory of L\'evy processes on involutive
bialgebras. The notion of independence used for these processes is tensor
independence, which generalizes the notion of independence used in classical
probability and corresponds to independent observables in quantum physics.
  In quantum probability there exist other notions of independence and L\'evy
processes can also be defined for the five so-called universal independences.
This is the topic of the second part. In particular, it is shown that boolean,
monotone, and anti-monotone independence can be reduced to tensor independence.
  Finally, in the third part, several classes of quantum L\'evy processes of
special interest are considered, e.g., L\'evy processes on real Lie algebras or
Brownian motions on braided spaces. Several applications of these processes are
also presented.

2787. EXISTENCE OF WEAK SOLUTIONS TO STOCHASTIC EVOLUTION INCLUSIONS 

Adam Jakubowski, Mikhail Kamenskii, Paul Raynaud De Fitte
 
We consider the Cauchy problem for a semilinear stochastic differential
inclusion in a Hilbert space. The linear operator generates a strongly
continuous semigroup and the nonlinear term is multivalued and satisfies a
condition which is more heneral than the Lipschitz condition. We prove the
existence of a mild solution to this problem. This solution is not "strong" in
the probabilistic sense, that is, it is not defined on the underlying
probability space, but on a larger one, which provides a "very good extension"
in the sense of Jacod and Memin. Actually, we construct this solution as a
Young measure, limit of approximated solutions provided by the Euler scheme.
The compactness in the space of Young measures of this sequence of approximated
solutions is obtained by proving that some measure of noncompactness equals
zero.

2788. THE AVERAGE AMOUNT OF INFORMATION LOST IN MULTIPLICATION 

Nicholas Pippenger

We show that if X and Y are integers independently and uniformly distributed
in the set {1, ..., N}, then the information lost in forming their product
(which is given by the equivocation H(X,Y | XY)), is of order log log N. We
also prove two extremal results regarding cases in which X and Y are not
necessarily independently or uniformly distributed. First, we note that the
information lost in multiplication can of course be 0. We show that the
condition H(X,Y | XY) = 0 implies that 2log_2 N - H(X, Y) is of order at least
log log N. Furthermore, if X and Y are independent and uniformly distributed on
disjoint sets of primes, it is possible to have H(X,Y | XY) = 0 with log_2 N -
H(X) and log_2 N - H(Y) each of order at most log log N. Second, we show that
however X and Y are distributed, H(X,Y | XY) is of order at most log N/log log
N. Furthermore, there are distributions (in which X and Y are independent and
uniformly distributed over sets of numbers having only small and distinct prime
factors) for which H(X,Y | XY) is of order log log N.

2789. A LOG-SCALE LIMIT THEOREM FOR ONE-DIMENSIONAL RANDOM WALKS IN RANDOM ENVIRONMENTS 

Alexander Roitershtein

We consider a random walk X_n in non-i.i.d. environment and show that the
ratio of log X_n to log n converges in probability to a positive constant.

2790. INFINITE VOLUME LIMITS OF HIGH-DIMENSIONAL SANDPILE MODELS 

Antal A. Jarai and Frank Redig

We study the Abelian sandpile model on Z^d. In d > 4 we prove existence of
the infinite volume addition operator, almost surely w.r.t the infinite volume
limit mu of the uniform measures on recurrent configurations. We prove the
existence of a Markov process with stationary measure mu, and study ergodic
properties of this process. The main techniques we use are a connection between
the statistics of waves and uniform two-component spanning trees and results on
the uniform spanning tree measure on Z^d.

2791. ON THE SHAPE OF THE GROUND STATE EIGENVALUE DENSITY OF A RANDOM HILL'S EQUATION 

Santiago Cambronero, Jose Ramirez, Brian Rider

Consider the Hill's operator $Q = - d^2/dx^2 + q(x)$ in which $q(x)$, $0 \le
x \le 1$, is a White Noise. Denote by $f(\mu)$ the probability density function
of $-\lambda_0(q)$, the negative of the ground state eigenvalue, at $\mu$. We
describe the detailed asymptotics of this density as $\mu \to +\infty$. This
result is based on a precise Laplace analysis of a functional integral
representation for $f(\m)$ established by S. Cambronero and H.P. McKean.

2792. REGENERATIVE PARTITION STRUCTURES 

Alexander Gnedin and Jim Pitman

We consider Kingman's partition structures which are regenerative with
respect to a general operation of random deletion of some part. Prototypes of
this class are the Ewens partition structures which Kingman characterised by
regeneration after deletion of a part chosen by size-biased sampling. We
associate each regenerative partition structure with a corresponding
regenerative composition structure, which (as we showed in a previous paper)
can be associated in turn with a regenerative random subset of the positive
halfline, that is the closed range of a subordinator. A general regenerative
partition structure is thus represented in terms of the Laplace exponent of an
associated subordinator. We also analyse deletion properties characteristic of
the two-parameter family of partition structures.

2793. THE SPECTRUM OF A RANDOM GEOMETRIC GRAPH IS CONCENTRATED 

Sanatan Rai

Consider $n$ points distributed uniformly in $[0,1]^d$. Form a graph by
connecting two points if their mutual distance is no greater than $r(n)$. This
gives a random geometric graph, $\gnrn$, which is connected for appropriate
$r(n)$. We show that the spectral measure of the transition matrix of the
simple random walk (\abbr{srw}) on $\gnrn$ is concentrated, and in fact
converges to that of the graph on the deterministic grid.

2794. BINARY MARKET MODELS WITH MEMORY 

Akihiko Inoue, Yumiharu Nakano and Vo Anh

We construct a binary market model with memory that approximates a
continuous-time market model driven by a Gaussian process equivalent to
Brownian motion. We give a sufficient conditions for the binary market to be
arbitrage-free. In a case when arbitrage opportunities exist, we present the
rate at which the arbitrage probability tends to zero as the number of periods
goes to infinity.

2795. DUAL RANDOM FRAGMENTATION AND COAGULATION AND AN APPLICATION TO THE GENEALOGY OF YULE PROCESSES 

Jean Bertoin and Christina Goldschmidt

The purpose of this work is to describe a duality between a fragmentation
associated to certain Dirichlet distributions and a natural random coagulation.
The dual fragmentation and coalescent chains arising in this setting appear in
the description of the genealogy of Yule processes.

2796. SPDES WITH COLOURED NOISE: ANALYTIC AND STOCHASTIC APPROACHES 

Marco Ferrante and Marta Sanz-Sol\'e

We study strictly parabolic stochastic partial differential equations on
$\R^d$, $d\ge 1$, driven by a Gaussian noise white in time and coloured in
space. Assuming that the coefficients of the differential operator are random,
we give sufficient conditions on the correlation of the noise ensuring H\"older
continuity for the trajectories of the solution of the equation. For
self-adjoint operators with deterministic coefficients, the mild and weak
formulation of the equation are related, deriving path properties of the
solution to a parabolic Cauchy problem in evolution form.

2797. STOCHASTIC LOEWNER EVOLUTION IN MULTIPLY CONNECTED DOMAINS 

Robert O. Bauer, Roland M. Friedrich

We construct radial stochastic Loewner evolution in multiply connected
domains, choosing the unit disk with concentric circular slits as a family of
standard domains. The natural driving function or input is a diffusion on the
associated Teichm\"uller space. The diffusion stops when it reaches the
boundary of the Teichm\"uller space. We show that for this driving function the
family of random growing compacts has a phase transition for $\kappa=4$ and
$\kappa=8$, and that it satisfies locality for $\kappa=6$ and the restriction
property for $\kappa=8/3$.

2798. SOME CONDITIONAL CORRELATION INEQUALITIES FOR PERCOLATION AND RELATED PROCESSES 

Jacob van den Berg, Olle Haggstrom and Jeff Kahn

Consider ordinary bond percolation on a finite or countably infinite graph.
Let s, t, a and b be vertices. An earlier paper proved the (nonintuitive)
result that, conditioned on the event that there is no open path from s to t,
the two events "there is an open path from s to a" and "there is an open path
from s to b" are positively correlated. In the present paper we further
investigate and generalize the theorem of which this result was a consequence.
This leads to results saying, informally, that, with the above conditioning,
the open cluster of s is conditionally positively (self-)associated and that it
is conditionally negatively correlated with the open cluster of t.
  We also present analogues of some of our results for (a) random-cluster
measures, and (b) directed percolation and contact processes, and observe that
the latter lead to improvements of some of the results in a paper of Belitsky,
Ferrari, Konno and Liggett (1997).

2799. ON OCCUPATION TIMES OF STATIONARY EXCURSIONS 

Marina Kozlova and Paavo Salminen

In this paper excursions of a stationary diffusion in stationary state are
studied.

2800. PROBABILISTIC NORMED SPACES WITH NON NECESSARILY CONTINUOUS TRIANGLE FUNCTIONS 

Bernardo Lafuerza-Guillen Jose L. Rodriguez

In this paper we consider probabilistic normed spaces as defined by Alsina,
Sklar and Schweizer, but equipped with non necessarily continuous triangle
functions. Such spaces endow a generalized topology of type ${\mathcal V}_D$
(in the sense of Frechet), which is translation-invariant and countably
generated by radial and circled 0-neighborhoods. We show that in fact all such
generalized topologies are induced by appropriate probabilistic norms. The
proof is based on an analogous result for probabilistic metrics due to Hoehle.
  Our interest on such spaces is motivated by the problem, suggested by Hoehle,
of comparing the notions of $\D$-boundedness (defined by using the
probabilistic radius) and boundedness (defined in terms of the associated
generalized topology).

2801. EIGENVALUE GAPS FOR THE CAUCHY PROCESS AND A POINCAR\'E INEQUALITY 

Rodrigo Banuelos and Tadeusz Kulczycki

A connection between the semigroup of the Cauchy process killed upon exiting
a domain $D$ and a mixed boundary value problem for the Laplacian in one
dimension higher known as the "mixed Steklov problem," was established in a
previous paper of the authors. From this, a variational characterization for
the eigenvalues $\lambda_n$, $n\geq 1$, of the Cauchy process in $D$ was
obtained. In this paper we obtain a variational characterization of the
difference between $\lambda_n$ and $\lambda_1$. We study bounded convex domains
which are symmetric with respect to one of the coordinate axis and obtain lower
bound estimates for $\lambda_* - \lambda_1$ where $\lambda_*$ is the eigenvalue
corresponding to the "first" antisymmetric eigenfunction for $D$. The proof is
based on a variational characterization of $\lambda_* - \lambda_1$ and on a
weighted Poincar\'e--type inequality. The Poincar\'e inequality is valid for
all $\alpha$ symmetric stable processes, $0<\alpha\leq 2$, and any other
process obtained from Brownian motion by subordination. We also prove upper
bound estimates for the spectral gap $\lambda_2-\lambda_1$ in bounded convex
domains.

2802. MONTE CARLO ALGORITHMS FOR OPTIMAL STOPPING AND STATISTICAL LEARNING 

Daniel Egloff

We extend the Longstaff-Schwartz algorithm for approximately solving optimal
stopping problems on high-dimensional state spaces. We reformulate the optimal
stopping problem for Markov processes in discrete time as a generalized
statistical learning problem. Within this setup we apply deviation inequalities
for suprema of empirical processes to derive consistency criteria, and to
estimate the convergence rate and sample complexity. Our results strengthen and
extend earlier results.

2803. ANNEALED DEVIATIONS OF RANDOM WALK IN RANDOM SCENERY 

Nina Gantert, Wolfgang K\"onig, Zhan Shi

Let $(Z_n)_{n\in\N_0}$ be a $d$-dimensional {\it random walk in random
scenery}, i.e.,
  $Z_n=\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in\N_0}$ a random walk in $\Z^d$
and $(Y(z))_{z\in\Z^d}$ an i.i.d. scenery, independent of the walk. The
walker's steps have mean zero and finite variance. We identify the speed and
the rate of the logarithmic decay of $\P(\frac 1n Z_n>b_n)$ for various choices
of sequences $(b_n)_n$ in $[1,\infty)$. Depending on $(b_n)_n$ and the upper
tails of the scenery, we identify different regimes for the speed of decay and
different variational formulas for the rate functions. In contrast to recent
work \cite{AC02} by A. Asselah and F. Castell, we consider sceneries {\it
unbounded} to infinity. It turns out that there are interesting connections to
large deviation properties of self-intersections of the walk, which have been
studied recently by X. Chen \cite{C03}.

2804. SPECIAL MOMENTS 

Greg Kuperberg 

In this note, we show that a linear combination $X$ of $n$ independent,
unbiased Bernoulli random variables $\{X_k\}$ can match the first $2n$ moments
of a random variable $Y$ which is uniform on an interval. More generally, for
each $p \ge 2$, each $X_k$ can be uniform on an arithmetic progression of
length $p$. All values of $X$ lie in the range of $Y$, and their ordering as
real numbers coincides with dictionary order on the vector $(X_1,...,X_n)$.
  The construction involves the roots of truncated $q$-exponential series. It
applies to a construction in numerical cubature using error-correcting codes
[arXiv:math.NA/0402047]. For example, when $n=2$ and $p=2$, the values of $X$
are the 4-point Chebyshev quadrature formula.

2805. ON SOME EXPONENTIAL INTEGRAL FUNCTIONALS OF BM($\MU$) AND BES(3) 

A.N. Borodin and Paavo Salminen

In this paper we derive the Laplace transforms of the integral functionals $$
\int_0^\infty (p(\exp(B^{(\mu)}_t)+1)^{-1}+ q(\exp(B^{(\mu)}_t)+1)^{-2}) dt, $$
$$ \int_0^\infty (p(\exp(R^{(3)}_t)-1)^{-1}+ q(\exp(R^{(3)}_t)-1)^{-2}) dt, $$
where $p$ and $q$ are real numbers, $\{B^{(\mu)}_t: t\geq 0\}$ is a Brownian
motion with drift $\mu>0,$ BM($\mu$), and $\{R^{(3)}_t: t\geq 0\}$ is a
3-dimensional Bessel process, BES(3). The transforms are given in terms of
Gauss' hypergeometric functions and it is seen that the results are closely
related to some functionals of Jacobi diffusions. This work generalizes and
completes some results of Donati--Martin and Yor and Salminen and Yor.

2806. ASYMPTOTIC ANALYSIS OF A PARTICLE SYSTEM WITH A MEAN-FIELD INTERACTION 

A. Manita, V. Shcherbakov

We study a system of $N$ interacting particles on $\bf{Z}$. The stochastic
dynamics consists of two components: a free motion of each particle
(independent random walks) and a pair-wise interaction between particles. The
interaction belongs to the class of mean-field interactions and models a
rollback synchronization in asynchronous networks of processors for a
distributed simulation. First of all we study an empirical measure generated by
the particle configuration on $\bf{R}$. We prove that if space, time and a
parameter of the interaction are appropriately scaled (hydrodynamical scale),
then the empirical measure converges weakly to a deterministic limit as $N$
goes to infinity. The limit process is defined as a weak solution of some
partial differential equation. We also study the long time evolution of the
particle system with fixed number of particles. The Markov chain formed by
individual positions of the particles is not ergodic. Nevertheless it is
possible to introduce relative coordinates and to prove that the new Markov
chain is ergodic while the system as a whole moves with an asymptotically
constant mean speed which differs from the mean drift of the free particle
motion.

2807. THE KINETIC LIMIT OF A SYSTEM OF COAGULATING BROWNIAN PARTICLES 

Alan Hammond and Fraydoun Rezakhanlou

We consider a random model of diffusion and coagulation. A large number of
small particles are randomly scattered at an initial time. Each particle has
some integer mass and moves in a Brownian motion whose diffusion rate is
determined by that mass. When any two particles are close, they are liable to
combine into a single particle that bears the mass of each of them. Choosing
the initial density of particles so that, if their size is very small, a
typical one is liable to interact with a unit order of other particles in a
unit of time, we determine the macroscopic evolution of the system, in any
dimension d \geq 3. The density of particles evolves according to the
Smoluchowski system of PDEs, indexed by the mass parameter, in which the
interaction term is a sum of products of densities. Central to the proof is
establishing the so-called Stosszahlensatz, which asserts that, at any given
time, the presence of particles of two distinct masses at any given point in
macroscopic space is asymptotically independent, as the size of the particles
is taken towards zero.

2808. NON-INTERSECTING, SIMPLE, SYMMETRIC RANDOM WALKS AND THE EXTENDED HAHN KERNEL 

Kurt Johansson

Consider $a$ particles performing simple, symmetric, non-intersecting random
walks, starting at points $2(j-1)$, $1\le j\le a$ at time 0 and ending at
$2(j-1)+c-b$ at time $b+c$. This can also be interpreted as a random rhombus
tiling of an $abc$-hexagon, or as a random boxed planar partition confined to a
rectangular box with side lengths $a$, $b$ and $c$. The positions of the
particles at all times gives a determinantal point process with a correlation
kernel given in terms of the associated Hahn polynomials. In a suitable scaling
limit we obtain non-intersecting Brownian motions which can be related to
Dysons's Hermitian Brownian motion via a suitable transformation.

2809. A FATOU THEOREM FOR $\alpha$-HARMONIC FUNCTIONS IN LIPSCHITZ DOMAINS 

Richard F. Bass and Dahae You

We study $\alpha$-harmonic functions in Lipschitz domains. 
We prove a Fatou theorem when the boundary function is 
bounded and $L^p$-Holder continuous of 
order $\beta$ with $\beta p>1$.

bass@math.uconn.edu    you@math.uconn.edu

  • To see a preprint or other information provided by the author click here.

2810. STABILITY OF PARABOLIC HARNACK INEQUALITIES ON METRIC MEASURE SPACES 

Martin T. Barlow, Richard F. Bass and Takashi Kumagai

Let $(X,d,\mu)$ be a metric measure space with a local regular Dirichlet form. 
We give necessary and sufficient conditions for a parabolic Harnack inequality 
with global space-time scaling exponent $\beta\ge 2$ to hold.  
We show that this parabolic Harnack inequality is stable under rough isometries. 
As a consequence, once such a Harnack inequality is established on a metric 
measure space, then it holds for any uniformly elliptic operator in divergence form 
on a manifold naturally defined from the graph approximation of the space.

barlow@math.ubc.ca  bass@math.uconn.edu  kumagai@kurims.kyoto-u.ac.jp

  • To see a preprint or other information provided by the author click here.
  • Or here.
  • Or here.

2811. INVARIANCE IMPLIES GIBBSIAN: SOME NEW RESULTS 

Vladimir I. Bogachev, Michael Roeckner, Feng-Yu Wang

We investigate stationary distributions of stochastic
gradient systems in Riemannian manifolds and prove that,
under certain assumptions, such distributions are
symmetric. These results are extended to countable products
of finite dimensional manifolds and applied to Gibbs 
distributions in the case where the single spin spaces are
Riemannian manifolds. In particular, we obtain a new 
result concerning the question whether all invariant 
measures are Gibbsian. Actually, we consider a more general 
object: weak elliptic equations for measures, which, on the 
one hand, yields the results obtained stronger than the 
above mentioned statements, and, on the other hand, enables
us to give simpler proofs of more general than previously
known facts. Applications to concrete models of lattice 
systems over $\mathbb{Z}^d$ with not necessarily compact
spin space are presented (also in the case $d\geq 3$).

roeckner@mathematik.uni-bielefeld.de

  • To see a preprint or other information provided by the author click here.
  • Or here.

2812. SAMPLE PATH LARGE DEVIATIONS FOR DIFFUSION PROCESSES ON CONFIGURATION SPACES OVER A RIEMANNIAN MANIFOLD 

Michael Roeckner and Tu-Sheng Zhang

In this paper, we establish a sample path large deviation
principle for a class of diffusion processes on
configuration spaces over a Riemannian manifold. The rate
functional turns out to be the energy of the paths
associated to the $L^2$-Wasserstein distance.

roeckner@mathematik.uni-bielefeld.de

  • To see a preprint or other information provided by the author click here.
  • Or here.

2813. UNIQUENESS OF DIFFUSION GENERATORS FOR TWO TYPES OF PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS 

Yu.G. Kondratiev, A.Yu. Konstantinov and M. Roeckner

For two types of stochastic particle systems in
$\mathbb{R}^d$ we show non-explosion in finite time by
proving that their respective generators are 
$L^1(\mu)$-unique, where $\mu$ is their respective
invariant (in these cases even symmetrizing) measure. We 
also prove the much harder $L^2(\mu)$-uniqueness in both
models.

kondrat@mathematik.uni-bielefeld.de, konst@faust.kiev.ua, roeckner@mathematik.uni-bielefeld.de

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2814. LARGE DEVIATIONS FOR INVARIANT MEASURES OF STOCHASTIC REACTION-DIFFUSION SYSTEMS WITH MULTIPLICATIVE NOISE AND NON-LIPSCHITZ REACTION TERM 

Sandra Cerrai and Michael Roeckner

In this paper we prove a large deviations principle for the
invariant measures of a class of reaction-diffusion systems
in bounded domains of $\mathbb{R}^d$, $d\geq 1$, perturbed
by a noise of multiplicative type. We consider reaction
terms which are not Lipschitz-continuous and diffusion
coefficients in front of the noise which are not bounded
and may be degenerate. This covers for example the case of
Ginzburg-Landau systems with unbounded multiplicative
noise.

sandra.cerrai@dmd.unifi.it, roeckner@mathematik.uni-bielefeld.de

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2815. STRONG SOLUTIONS OF STOCHASTIC EQUATIONS WITH SINGULAR TIME DEPENDENT DRIFT 

Nicolai V. Krylov and Michael Roeckner

We prove existence and uniqueness of strong solutions to
stochastic equations in domains $G\subset \mathbb{R}^d$
with unit diffusion and singular time dependent drift $b$
up to an explosion time. We only assume local
$L_{q-}L_p$-integrability of $b$ in $\mathbb{R}\times G$
with $d/p + 2/q <1$. We also prove strong Feller 
properties in this case. If $b$ is the gradient in $x$ of
a nonnegative function $\phi$ blowing up as 
$G\ni x \to \partial G$, we prove that the conditions
  $2D_t \phi \leq K \phi$,
  $2 D_t \phi + \Delta \phi \leq K e^{\epsilon\phi}$,
  $\epsilon \in [0,1)$,
imply that the explosion time is infinite and the 
distributions of the solution have sub Gaussian tails.

roeckner@mathematik.uni-bielefeld.de

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2816. PDE APPROACH TO INVARIANT AND GIBBS MEASURES WITH APPLICATIONS 

Michael Roeckner

In this paper we give a pedagogical account of the PDE
approach to invariant and Gibbs measures in finite and
infinite dimensions. As an application we describe some
recent new results on the classical problem whether
``invariance implies Gibbsian'' and illustrate how they
apply to a well-studied lattice model from statistical 
mechanics with non-compact single spin spaces.

roeckner@mathematik.uni-bielefeld.de

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2817. INFINITE INTERACTING DIFFUSION PARTICLES I: EQUILIBRIUM PROCESS AND ITS SCALING LIMIT 

Yuri Kondratiev, Eugene Lytvynov and Michael Roeckner

A stochastic dynamics $(X(t))_{t\geq 0}$ of a classical
continuous system is a stochastic process which takes 
values in the space $\Gamma$ of all locally finite subsets
(configurations) in $\mathbb{R}^d$ and which has a Gibbs
measure $\mu$ as an invariant measure. We assume that $\mu$
corresponds to a symmetric pair potential $\phi(x-y)$. An
important class of stochastic dynamics of a classical 
continuous system is formed by diffusions. Till now, only
one type of such dynamics -- the so-called gradient
stochastic dynamics, or interacting Brownian particles --
has been investigated. By using the theory of Dirichlet 
forms from [Ma, Roeckner. Springer, 1992], we construct
and investigate a new type of stochastic dynamics, which 
we call infinite interacting diffusion particles. We 
introduce a Dirichlet form $\Escript_\mu^\Gamma$ on
$L^2(\Gamma; \mu)$, and under general conditions on the
potential $\phi$, prove its closability. For a potential
$\phi$ having a ``weak'' singularity at zero, we also write
down an explicit form of the generator of
$\Escript_\mu^\Gamma$ on the set of smooth cylinder
functions. We then show that, for any Dirichlet form
$\Escript_\mu^\Gamma$, there exists a diffusion process
that is properly associated with it. Finally, in a way
parallel to [Grothaus, Kondratiev, Lytvynov, Roeckner. Ann.
Prob. 31 (2003)], we study a scaling limit of interacting
diffusions in terms of convergence of the corresponding
Dirichlet forms, and we also show that these scaled 
processes are tight in $C( [0,\infty) , \Dscript')$,
where $\Dscript'$ is the dual space of 
$\Dscript := C_0^\infty (\mathbb{R}^d)$.

roeckner@mathematik.uni-bielefeld.de

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2818. L^1 -THEORY FOR THE KOLMOGOROV OPERATORS OF STOCHASTIC GENERALIZED BURGERS EQUATIONS 

Michael Roeckner and Zeev Sobol

This paper contains supplementary results to the recent
paper [Roeckner, Sobol. BiBoS:03-10-129 (2003)] by the two
authors. It focuses on the $L^1$-theory of a class of
Kolmogorov operators $L$ in infinitely many variables which
e.g. are associated to stochastic generalized Burgers
equations. Their $L^1$-theory is developed with respect to
a whole class of reference measures identified in this
paper, which contains in particular infinitesimally
invariant measures for $L$. Essential maximal dissipativity
for $L$ with initial domain given by $C^2$-smooth bounded
cylinder functions is proved to hold on $L^1(\nu)$ for all
measures $\nu$ in this class. The obtained respective
$C_0$-semigroup on $L^1(\nu)$ is proved to come from the
semigroup of kernels constructed in [Roeckner, Sobol.
BiBoS:03-10-129 (2003)]. Finally, a measure is constructed
in this class which is of full topological support, i.e.
charges every non-empty open set of the underlying infinite
dimensional space, which here is $L^2(0,1)$.

roeckner@mathematik.uni-bielefeld.de, sobol@mathematik.uni-bielefeld.de

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2819. KOLMOGOROV EQUATIONS IN INFINITE DIMENSIONS: WELL-POSEDNESS AND REGULARITY OF SOLUTIONS, WITH APPLICATIONS TO STOCHASTIC GENERALIZED BURGERS EQUATIONS 

Michael Roeckner and Zeev Sobol

We develop a new method to uniquely solve a large class of
heat equations, so called Kolmogorov equations in
infinitely many variables. The equations are analyzed in
spaces of sequentially weakly continuous functions weighted
by proper (Lyapunov type) functions. This way for the first
time  the solutions are constructed everywhere without
exceptional sets for equations with possibly non-locally
Lipschitz drifts. Apart from general analytic interest, the
main motivation is to apply this to uniquely solve
martingale problems in the sense of Stroock-Varadhan given
by stochastic partial differential equations from
hydrodynamics, such as the stochastic Navier-Stokes
equations. In this paper this is done in the case of the
stochastic generalized Burgers equation. Uniqueness is
shown in the sense of Markov flows.

roeckner@mathematik.uni-bielefeld.de, sobol@mathematik.uni-bielefeld.de

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2820. GLOBAL GRADIENT BOUNDS FOR DISSIPATIVE DIFFUSION OPERATORS 

Vladimir I. Bogachev, Giuseppe Da Prato, Michael Roeckner and Zeev sobol

Let $L$ be a second order elliptic operator on
$\mathbb{R}^d$ with a constant diffusion matrix and a
dissipative (in a weak sense) drift $b \in L_{loc}^p$ with
some $p>d$. We assume that $L$ possesses a Lyapunov
function, but no local boundedness of $b$ is assumed. It is
known that then there exists a unique probability measure
$\mu$ satisfying the equation $L^* \mu = 0$ and that the
closure of $L$ in $L^1(\mu)$ generates a Markov semigroup
$\{T_t\}_{t\geq 0}$ with the resolvent
$\{G_\lambda\}_{\lambda>0}$. We prove that, for any
Libschitzian function $f\in L^1(\mu)$ and all 
$t,\lambda >0$, the functions $T_t f$ and $G_\lambda f$ are
Lipschitzian and 
  $\sup_{x,t} |\Nabla T_t f(x)| \leq \sup_x |\Nabla f(x)|$
and
  $\sup_x |\Nabla G_\lambda f(x)| 
   \leq \frac{1}{\lambda} \sup_x |\Nabla f(x)|$.
In addition, we show that for every bounded Lipschitzian
function $g$ , the function $G_\lambda g$ is the unique
bounded solution of the equation 
  $\lambda f - L f = g$
in the Sobolev class 
  $H_{loc}^{2,2} (\mathbb{R}^d)$.

roeckner@mathematik.uni-bielefeld.de

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2821. ASYMPTOTIC ANALYSIS OF A PARTICLE SYSTEM WITH A MEAN-FIELD INTERACTION 

Anatoli Manita  and  Vadim Shcherbakov

We study a long-time behavior of a system of $N$ interacting 
particles on $Z$. The stochastic dynamics consists of two 
components: a free motion of each particle (independent 
random walks) and a pair-wise interaction  between particles. 
The interaction belongs to the class of mean-field
interactions and it models a rollback synchronization 
in asynchronous networks of processors for a distributed 
simulation. We consider two different limits: 
     A) $N\rightarrow\infty$, the time and space are changed 
according to an appropiate hydrodynamical scale: 
$(t,x)=(s N^\gamma,y N)$; 
     B) $N$ is fixed,    $t\rightarrow\infty$. 
In the first situation we consider limits of empirical 
distribution of particles and show that they can be described 
by hydrodynamical equations of the first and the second order. 
In the case of the second order PDE, we come 
to the famous Kolmogorov-Petrovski-Piskunov equation. 
For the situation B we prove that  all stochastic 
interacting particles form a single group which moves with 
an asymptotically constant speed. It appears that due to the 
interaction between the particles this speed differs from 
the mean drift of the free particle motion.

manita@mech.math.msu.su   V.Shcherbakov@cwi.nl

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2822. THE SEMIGROUP OF THE GLAUBER DYNAMICS OF A CONTINUOUS SYSTEM OF FREE PARTICLES 

Yuri Kondratiev, Eugene Lytvynov and Michael Roeckner

We study properties of the semigroup $(e^{-tH})_{t\geq 0}$
on the space $L^2(\Gamma_X, \pi)$ where $\Gamma_X$ is the
configuration space over a locally compact second
countable Hausdorff topological space $X$, $\pi$ is a Poisson 
measure on $\Gamma_X$, and $H$ is the generator of the 
Glauber dynamics. We explicitly construct the 
corresponding Markov semigroup of kernels 
$(P_t)_{t\geq 0}$ and, using it, we prove the main results
of the paper: the Feller property of the semigroup
$(P_t)_{t\geq 0}$ with respect to the vague topology on 
the configuration space $\Gamma_X$, and the ergodic 
property of $(P_t)_{t\geq 0}$. Following an idea of 
D. Surgailis, we also give a direct construction of the 
Glauber dynamics of a continuous infinite system of free
particles. The main point here is that this process can
start in every $\gamma\in\Gamma_X$, will never leave
$\Gamma_X$ and has cadlag sample paths in $\Gamma_X$.

kondrat@mathematik.uni-bielefeld.de, e.lytvynov@swansea.ac.uk, roeckner@mathematik.uni-bielefeld.de

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2823. FUNCTIONAL INEQUALITIES FOR PARTICLE SYSTEMS ON POLISH SPACES 

Michael Roeckner and Feng-Yu Wang

Various Poincare-Sobolev type inequalities are studied for
a reaction-diffusion model of particle systems on Polish
spaces. The systems we consider consist of finite particles
which are killed or produced at certain rates, while
particles in the system move on the Polish space
interacting with one another (i.e. diffusion). Thus, the
corresponding Dirichlet form, which we call
reaction-diffusion Dirichlet form, consists of two parts:
the diffusion part induced by certain Markov processes on
the product spaces $E^n$ ($n \geq 1$) which determine the
motion of particles, and the reaction part induced by a
$Q$-process on $\mathbb Z_+$ and a sequence of reference
probability measures, where the $Q$-process determines the
variation of the number of particles and the reference
measures describe the locations of newly produced
particles. We prove that the validity of Poincare and weak
Poincare inequalities are essentially due to the pure
reaction part, i.e. either of these inequalities holds if
and only if it holds for the pure reaction Dirichlet form,
or equivalently, for the corresponding $Q$-process. But
under a mild condition, stronger inequalities rely on both
parts: the reaction-diffusion Dirichlet form satisfies a
super Poincare inequality (e.g. the log-Sobolev inequality)
if and only if so do both the corresponding $Q$-process and
the diffusion part. Explicit estimates of constants in the
inequalities are derived. Finally, some specific examples
are presented to illustrate the main results.

roeckner@mathematik.uni-bielefeld.de

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2824. THE GENERALIZED CANONICAL ENSEMBLE AND ITS UNIVERSAL EQUIVALENCE WITH THE MICROCANONICAL ENSEMBLE 

Marius Costeniuc, Richard S. Ellis, Hugo Touchette and Bruce Turkington

Microcanonical equilibrium macrostates are characterized as the solutions 
of a constrained minimization problem, while canonical equilibrium 
macrostates are characterized as the solutions of a related, 
unconstrained minimization problem.  In Ellis, Haven, and Turkington 
(J. Stat. Phys. 101:999–1064, 2000) the problem of ensemble equivalence 
was completely solved at two separate, but related levels: the level 
of equilibrium macrostates, which focuses on relationships between 
the corresponding sets of equilibrium macrostates, and the thermodynamic 
level, which focuses on when the microcanonical entropy $s$ can be 
expressed as the Legendre-Fenchel transform of the canonical free 
energy.  The present paper extends the results of Ellis et al. 
significantly by addressing the following motivational question.  
If the microcanonical ensemble is nonequivalent with the canonical 
ensemble, then is it possible to replace the canonical ensemble 
with a generalized canonical ensemble that is
equivalent with the microcanonical ensemble?  The generalized 
canonical ensemble that we consider is obtained from the standard 
canonical ensemble by adding an exponential factor involving 
a continuous function $g$ of the Hamiltonian.  As in the paper 
by Ellis et al., we analyze the equivalence of the two ensembles 
at both the level of equilibrium macrostates and the thermodynamic 
level.  A neat but not quite precise statement of the main result 
in the present paper is that the microcanonical and generalized canonical 
ensembles are equivalent at the level of equilibrium macrostates 
if and only if they are equivalent at the thermodynamic level, 
which is the case if and only if the generalized microcanonical 
entropy $s-g$ is concave.  A corollary of this result is that 
if the microcanonical entropy is $C^2$, then there exists a $g$ 
taken from a class of quadratic functions such that the 
microcanonical and generalized canonical ensembles satisfy a strong 
form of equivalence 
which we call universal equivalence.

costeniuc@math.umass.edu   rsellis@math.umass.edu   htouchet@alum.mit.edu   turk@math.umass.edu

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