Probability Abstracts 77

This document contains abstracts 2342-2398. They have been mailed on October 30, 2003.

2342. GLAUBER DYNAMICS ON TREES AND HYPERBOLIC GRAPHS 

Noam Berger, Claire Kenoyn, Elchanan Mossel and Yuval Peres

We study continuous time Glauber dynamics for random configurations with
local constraints (e.g. proper coloring, Ising and Potts models) on finite
graphs with $n$ vertices and of bounded degree. We show that the relaxation
time
 (defined as the reciprocal of the spectral gap $|\lambda_1-\lambda_2|$) for
the dynamics on trees and on planar hyperbolic graphs, is polynomial in $n$.
For these hyperbolic graphs, this yields a general polynomial sampling
algorithm for random configurations. We then show that if the relaxation time
$\tau_2$ satisfies $\tau_2=O(1)$, then the correlation coefficient, and the
mutual information, between any local function (which depends only on the
configuration in a fixed window) and the boundary conditions, decays
exponentially in the distance between the window and the boundary. For the
Ising model on a regular tree, this condition is sharp.

mossel@stat.berkeley.edu

2343. LOOP-ERASED RANDOM WALK ON A TORUS IN DIMENSIONS 4 AND ABOVE 

Itai Benjamini and Gady Kozma

Sharp estimates for the length of loop erased random walk between two
vertices on the [n]^d -torus, d > 4, are established. The mean length is order
n^{d/2} . In dimension 4 we have only an upper bound.

itai@wisdom.weizmann.ac.il

2344. RANDOM WALKS ON THE TORUS WITH SEVERAL GENERATORS 

Timothy Prescott and Francis Edward Su

Our paper gives bounds for the rate of convergence for a class of random
walks on the d-dimensional torus generated by a set of n vectors in R^d/Z^d. We
give bounds on the discrepancy distance from Haar measure; our lower bound
holds for all such walks, and if the generators arise from the rows of a "badly
approximable" matrix, then there is a corresponding upper bound. The bounds are
sharp for walks on the circle.

su@math.hmc.edu

2345. DIFFERENTIAL EQUATIONS FOR DYSON PROCESSES 

Craig A. Tracy and Harold Widom

We call "Dyson process" any process on ensembles of matrices in which the
entries undergo diffusion. We are interested in the distribution of the
eigenvalues (or singular values) of such matrices. In the original Dyson
process it was the ensemble of n by n Hermitian matrices, and the eigenvalues
describe n curves. Given sets X_1,...,X_m the probability that for each k no
curve passes through X_k at time \tau_k is given by the Fredholm determinant of
a certain matrix kernel, the extended Hermite kernel. For this reason we call
this Dyson process the Hermite process. Similarly, when the entries of a
complex matrix undergo diffusion we call the evolution of its singular values
the Laguerre process, for which there is a corresponding extended Laguerre
kernel. Scaling the Hermite process at the edge leads to the Airy process and
in the bulk to the sine process; scaling the Laguerre process at the edge leads
to the Bessel process. Generalizing and strengthening earlier work, we assume
that each X_k is a finite union of intervals and find for the Airy process a
system of partial differential equations, with the end-points of the intervals
of the X_k as independent variables, whose solution determines the probability
that for each k no curve passes through X_k at time \tau_k. Then we find the
analogous systems for the Hermite process (which is more complicated) and also
for the sine process. Finally we find an analogous system of PDEs for the
Bessel process, which is the most difficult.

widom@ucsc.edu

2346. CONVERGENCE OF SYMMETRIC DIFFUSIONS ON WIENER SPACES 

Andrea Posilicano and Tusheng Zhang

We prove convergence of symmetric diffusions on Wiener spaces by using
stopping times arguments and capacity techniques. The drifts of the diffusions
can be singular, we require the densities of the processes to be neither
bounded from above nor away from zero.

andrea.posilicano@uninsubria.it

2347. AN INVARIANT OF FINITARY CODES WITH FINITE EXPECTED SQUARE ROOT CODING LENGTH 

Nate Harvey and Yuval Peres 

Let $p$ and $q$ be probability vectors with the same entropy $h$. Denote by
$B(p)$ the Bernoulli shift indexed by $\Z$ with marginal distribution $p$.
Suppose that $\phi$ is a measure preserving homomorphism from $B(p)$ to $B(q)$.
We prove that if the coding length of $\phi$ has a finite 1/2 moment, then
$\sigma_p^2=\sigma_q^2$, where $\sigma_p^2=\sum_i p_i(-\log p_i-h)^2$ is the
{\dof informational variance} of $p$. In this result, which sharpens a theorem
of Parry (1979), the 1/2 moment cannot be replaced by a lower moment. On the
other hand, for any $\theta<1$, we exhibit probability vectors $p$ and $q$ that
are not permutations of each other, such that there exists a finitary
isomorphism $\Phi$ from $B(p)$ to $B(q)$ where the coding lengths of $\Phi$ and
of its inverse have a finite $\theta$ moment. We also present an extension to
ergodic Markov chains.

neh@uclink.berkeley.edu

2348. ON THE DIRICHLET PROBLEM FOR ASYMMETRIC ZERO-RANGE PROCESS ON INCREASING DOMAINS 

Amine Asselah

We characterize the principal eigenvalue of the generator of the asymmetric
zero-range process in dimensions d>2, with Dirichlet boundary on special
domains. We obtain a Donsker-Varadhan variational representation for the
principal eigenvalue, and show that the corresponding eigenfunction is unique
in a natural class of functions. This allows us to obtain asymptotic hitting
time estimates.

asselah@cmi.univ-mrs.fr

2349. HITTING TIMES FOR SPECIAL PATTERNS IN THE SYMMETRIC EXCLUSION PROCESS 

Amine Asselah and Paolo Dai Pra

We consider the symmetric exclusion process. For special patterns, we
consider the problem of establishing sharp estimates for their occurrence time.
  We present a novel argument based on monotonicity which helps in some cases
to obtain sharp tail asymptotics for occurrence time in a simple way. Also, we
characterize the trajectories conditioned on not entering the pattern up to
large time.

asselah@cmi.univ-mrs.fr

2350. VIRASORO ACTION ON SCHUR FUNCTION EXPANSIONS, SKEW YOUNG TABLEAUX AND RANDOM WALKS 

M. Adler and P. van Moerbeke

It is known that some matrix integrals over U(n) satisfy an sl(2,R)-algebra
of Virasoro constraints. Acting with these Virasoro generators on 2-dimensional
Schur function expansions leads to difference relations on the coefficients of
this expansions. These difference relations, set equal to zero, are precisely
the backward and forward equations for non-intersecting random walks. The
transition probabilities for these random walks appear as the coefficients of
an expansion of U(n)-matrix integrals (of the type above), by inserting in the
integral the product of two Schur polynomials associated with two partitions;
the latter are specified by the initial and final positions of the
non-intersecting random walk. An essential ingredient in this work is the
generalization of the Murnaghan-Nakayama rule to the action of Virasoro on
Schur polynomials.

vanmoerbeke@geom.ucl.ac.be

2351. FAST SIMULATION OF NEW COINS FROM OLD 

Serban Nacu and Yuval Peres

Let $S \subset (0,1)$. Given a function $f:S \to (0,1)$, we consider the
problem of using independent tosses of a coin with probability of heads $p$
(where $p \in S$ is unknown) to simulate a coin with probability of heads
$f(p)$. We prove that if $S$ is a closed interval and $f$ is real analytic on
$S$, then $f$ has a fast simulation on $S$ (the number of $p$-coin tosses
needed has exponential tails). Conversely, if a function $f$ has a fast
simulation on an open set, then it is real analytic on that set.

serban@stat.berkeley.edu

2352. EXTREMAL REVERSIBLE MEASURES FOR THE EXCLUSION PROCESS 

Paul Jung

We give a characterization of the invariant measures for the exclusion
process on the integers with certain reversible transition kernels. Some
examples include all nearest-neighbor kernels with asymptotic mean zero. One
tool used is a necessary and sufficient condition for reversible measures to be
extremal in the set of all invariant measures which is an interesting result in
its own right.

pjung@math.cornell.edu

2353. PERTURBATIONS OF THE SYMMETRIC EXCLUSION PROCESS 

Paul Jung

For the exclusion process with symmetric kernel p(x,y)=p(y,x), the set of
invariant measures has been completely studied. This paper gives results
concerning the invariant measures for exclusion processes where p(x,y)=p(y,x)
except for finitely many (x,y) and p(x,y) corresponds to a transient transition
kernel.

pjung@math.cornell.edu

2354. STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS 

Richard F. Bass

This paper is a survey of uniqueness results for stochastic differential
equations with jumps and regularity results for the corresponding harmonic
functions.

bass@math.uconn.edu

2355. THE CENTER OF MASS OF THE ISE AND THE WIENER INDEX OF TREES 

Svante Janson and Philippe Chassaing

We derive the distribution of the center of mass $S$ of the integrated
superBrownian excursion (ISE) {from} the asymptotic distribution of the Wiener
index for simple trees. Equivalently, this is the distribution of the integral
of a Brownian snake. A recursion formula for the moments and asymptotics for
moments and tail probabilities are derived.

chassain@iecn.u-nancy.fr

2356. SLAB PERCOLATION FOR THE ISING MODEL 

Thierry Bodineau

For the FK representation of the Ising model, we prove that the slab
percolation threshold coincides with the critical temperature in any dimension
larger or equal to three.

bodineau@math.jussieu.fr

2357. STOCHASTIC DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS DRIVEN BY A POISSON NOISE 

Aureli Alabert and Miguel A. Marmolejo

We consider one-dimensional stochastic differential equations with a boundary
condition, driven by a Poisson process. We study existence and uniqueness of
solutions and the absolute continuity of the law of the solution. In the case
when the coefficients are linear, we give an explicit form of the solution and
study the reciprocal process property.

alabert@mat.uab.es

2358. ON THE LARGEST EIGENVALUE OF WISHART MATRICES WITH IDENTITY COVARIANCE WHEN N, P AND P/N TEND TO INFINITY 

Noureddine El Karoui

Let X be a n*p matrix and l_1 the largest eigenvalue of the covariance matrix
X^{*}*X. The "null case" where X_{i,j} are independent Normal(0,1) is of
particular interest for principal component analysis. For this model, when n, p
tend to infinity and n/p tends to gamma in (0,\infty), it was shown in
Johnstone (2001) that l_1, properly centered and scaled, converges to the
Tracy-Widom law. We show that with the same centering and scaling, the result
is true even when p/n or n/p tends to infinity. The derivation uses ideas and
techniques quite similar to the ones presented in Johnstone (2001). Following
Soshnikov (2002), we also show that the same is true for the joint distribution
of the k largest eigenvalues, where k is a fixed integer. Numerical experiments
illustrate the fact that the Tracy-Widom approximation is reasonable even when
one of the dimension is "small".

nkaroui@stanford.edu

2359. SELF-INTERACTING DIFFUSIONS : SYMMETRIC INTERACTIONS 

Michel Benaim and Olivier Raimond

Let $M$ be a compact Riemannian manifold. A {\em self-interacting diffusion}
on $M$ is a stochastic process solution to $$dX_t = dW_t(X_t) -
\frac{1}{t}(\int_0^t \nabla V_{X_s}(X_t)ds)dt$$ where $\{W_t\}$ is a Brownian
vector field on $M$ and $V_x(y) = V(x,y)$ a smooth function. Let $\mu_t =
\frac{1}{t} \int_0^t \delta_{X_s} ds$ denote the normalized occupation measure
of $X_t$. We prove that, when $V$ is symmetric, $\mu_t$ converges almost surely
to the critical set of a certain nonlinear free energy functional $J$.
Furthermore, $J$ has generically finitely many critical points and $\mu_t$
converges almost surely toward a local minimum of $J.$ Each local minimum
having a positive probability to be selected.

olivier.raimond@math.u-psud.fr

2360. ASYMPTOTIC ANALYSIS VIA MELLIN TRANSFORMS FOR SMALL DEVIATIONS IN $L^2$-NORM OF INTEGRATED BROWNIAN SHEETS 

James Allen Fill and Fred Torcaso 

We use Mellin transforms to compute a full asymptotic expansion for the tail
of the Laplace transform of the squared $L^2$-norm of any multiply-integrated
Brownian sheet. Through reversion we obtain corresponding strong
small-deviation estimates.

jimfill@jhu.edu

2361. MAXIMUM WEIGHT INDEPENDENT SETS AND MATCHINGS IN SPARSE RANDOM GRAPHS. EXACT RESULTS USING THE LOCAL WEAK CONVERGENCE METHOD 

David Gamarnik, Tomasz Nowicki and Grzegorz Swirszcz

Let $G(n,c/n)$ and $G_r(n)$ be an $n$-node sparse random graph and a sparse
random $r$-regular graph, respectively, and let ${\cal I}(n,r)$ and ${\cal
I}(n,c)$ be the sizes of the largest independent set in $G(n,c/n)$ and
$G_r(n)$. The asymptotic value of ${\cal I}(n,c)/n$ as $n\to\infty$, can be
computed using the Karp-Sipser algorithm when $c\leq e$. For random cubic
graphs, $r=3$, it is only known that $.432\leq\liminf_n {\cal I}(n,3)/n \leq
\limsup_n {\cal I}(n,3)\leq .4591$ with high probability (w.h.p.) as
$n\to\infty$, as shown by Frieze and Suen and by Bollobas, respectively.
  In this paper we assume in addition that the nodes of the graph are equipped
with non-negative weights, independently generated according to some common
distribution, and we consider instead the maximum weight of an independent set.
Surprisingly, we discover that for certain weight distributions, the limit
$\lim_n {\cal I}(n,c)/n$ can be computed exactly even when $c>e$, and $\lim_n
{\cal I}(n,r)/n$ can be computed exactly for some $r\geq 2$. For example, when
the weights are exponentially distributed with parameter 1, $\lim_n {\cal
I}(n,2e)/n\approx .5517$, and $\lim_n {\cal I}(n,3)/n\approx .6077$. Our
results are established using the recently developed local weak convergence
method further reduced to a certain local optimality property exhibited by the
models we consider.

gamarnik@watson.ibm.com

2362. EXACT SOLUTION OF DISCRETE HEDGING EQUATION FOR EUROPEAN OPTION 

D.E. Yakovlev, D.N. Zhabin

The approach that allows find European option price on the assumption of
hedging at discrete times is proposed. The routine allows find the option price
not for lognormal distribution functions of underlying asset only but for wide
enough classes of distribution functions too. It is shown that there exists a
nonzero possibility that market parameters can take values such that to realize
the hedging policy becomes impossible. This fact is not in contradiction with
Black-Scholes option price model as long as this possibility tends to zero at
the limit of continuous hedging.

zhabin@phys.tsu.ru

2363. THE MARGINALIZATION PARADOX DOES NOT IMPLY INCONSISTENCY FOR IMPROPER PRIORS 

Timothy C. Wallstrom

The marginalization paradox involves a disagreement between two Bayesians who
use two different procedures for calculating a posterior in the presence of an
improper prior. We show that the argument used to justify the procedure of one
of the Bayesians is inapplicable. There is therefore no reason to expect
agreement, no paradox, and no evidence that improper priors are inherently
inconsistent. We show further that the procedure in question can be interpreted
as the cancellation of infinities in the formal posterior. We suggest that the
implicit use of this formal procedure is the source of the observed
disagreement.

tcw@lanl.gov

2364. FUNCTIONALS OF DIRICHLET PROCESSES, THE MARKOV KREIN IDENTITY AND BETA-GAMMA PROCESSES 

Lancelot F. James

This paper describes how one can use the well-known Bayesian prior to
posterior analysis of the Dirichlet process, and less known results for the
gamma process, to address the formidable problem of assessing the distribution
of linear functionals of Dirichlet processes. In particular, in conjunction
with a gamma identity, we show easily that a generalized Cauchy-Stieltjes
transform of a linear functional of a Dirichlet process is equivalent to the
Laplace functional of a class of, what we define as, beta-gamma processes. This
represents a generalization of the Markov-Krein identity for mean functionals
of Dirichlet processes. A prior to posterior analysis of beta-gamma processes
is given that not only leads to an easy derivation of the Markov-Krein
identity, but additionally yields new distributional identities for gamma and
beta-gamma processes. These results give new explanations and intepretations of
exisiting results in the literature. This is punctuated by establishing a
simple distributional relationship between beta-gamma and Dirichlet processes.

lancelot@ust.hk

2365. SECOND-ORDER FLUCTUATIONS AND CURRENT ACROSS CHARACTERISTIC FOR A ONE-DIMENSIONAL GROWTH MODEL OF INDEPENDENT RANDOM WALKS 

Timo Seppalainen 

Fluctuations from a hydrodynamic limit of a one-dimensional asymmetric system
come at two levels. On the central limit scale n^{1/2} one sees initial
fluctuations transported along characteristics and no dynamical noise. The
second order of fluctuations comes from the particle current across the
characteristic. For a system made up of independent random walks we show that
the second order fluctuations appear at scale n^{1/4} and converge to a certain
self-similar Gaussian process. If the system is in equilibrium this limiting
process specializes to fractional Brownian motion with Hurst parameter 1/4.
This contrasts with asymmetric exclusion and Hammersley's process whose second
order fluctuations appear at scale n^{1/3} as has been discovered through
related combinatorial growth models.

seppalai@math.wisc.edu

2366. EXACT SAMPLING FROM PERFECT MATCHINGS OF DENSE NEARLY REGULAR BIPARTITE GRAPHS 

Mark Huber

We present the first algorithm for generating random variates exactly
uniformly from the set of perfect matchings of a bipartite graph with a
polynomial expected running time over a nontrivial set of graphs. Previous
Markov chain approaches obtain approximately uniform variates for arbitrary
graphs in polynomial time, but their general running time is $\Theta(n^{26}
(\ln n)^2).$ Our algorithm employs acceptance/rejection together with a new
upper limit on the permanent of a form similar to Bregman's Theorem. For a
graph with $2n$ nodes where the degree of every node is nearly $\gamma n$ for a
constant $\gamma$, the expected running time is $O(n^{1.5 + .5/\gamma})$. Under
these conditions, Jerrum and Sinclair showed that a Markov chain of Broder can
generate approximately uniform variates in $\Theta(n^{4.5 + .5/\gamma} \ln n)$
time, making our algorithm significantly faster on this class of graph. With
our approach, approximately counting the number of perfect matchings
(equivalent to finding the permanent of a 0-1 matrix and so $\sharp P$
complete) can be done without use of selfreducibility.

mhuber@math.duke.edu

2367. INCREMENTS OF RANDOM PARTITIONS 

Serban Nacu

For any partition of $\{1, 2, ..., n\}$ we define its {\it increments} $X_i,
1 \le i \le n$ by $X_i = 1$ if $i$ is the smallest element in the partition
block that contains it, $X_i = 0$ otherwise. We prove that for partially
exchangeable random partitions (where the probability of a partition depends
only on its block sizes in order of appearance), the law of the increments
uniquely determines the law of the partition. One consequence is that the
Chinese Restaurant Process CRP($\theta$) (the partition with distribution given
by the Ewens sampling formula with parameter $\theta$) is the only exchangeable
random partition with independent increments.

serban@stat.berkeley.edu

2368. THE HARMONIC EXPLORER AND ITS CONVERGENCE TO SLE(4) 

Oded Schramm and Scott Sheffield

The harmonic explorer is a random grid path. Very roughly, at each step the
harmonic explorer takes a turn to the right with probability equal to the
discrete harmonic measure of the left hand side of the path from a point near
the end of the current path. We prove that the harmonic explorer converges to
SLE(4) as the grid gets finer.

sheff@microsoft.com

2369. UNIVERSALITY OF CRITICAL BEHAVIOUR IN A CLASS OF RECURRENT RANDOM WALKS 

Ostap Hryniv and Yvan Velenik

Let X_0=0, X_1, X_2, ..., be an aperiodic random walk generated by a sequence
xi_1, xi_2, ..., of i.i.d. integer-valued random variables with common
distribution p(.) having zero mean and finite variance. For an N-step
trajectory X=(X_0,X_1,...,X_N) and a monotone convex function V: R^+ -> R^+
with V(0)=0, define V(X)= sum_{j=1}^{N-1} V(|X_j|). Further, let I_{N,+}^{a,b}
be the set of all non-negative paths X compatible with the boundary conditions
X_0=a, X_N=b. We discuss asymptotic properties of X in I_{N,+}^{a,b} w.r.t. the
probability distribution P_{N}^{a,b}(X)= (Z_{N}^{a,b})^{-1} exp{-lambda V(X)}
prod_{i=0}^{N-1} p(X_{i+1}-X_i) as N -> infinity and lambda -> 0, Z_{N}^{a,b}
being the corresponding normalization. If V(.) grows not faster than
polynomially at infinity, define H(lambda) to be the unique solution to the
equation lambda H^2 V(H) =1. Our main result reads that as lambda -> 0, the
typical height of X_{[alpha N]} scales as H(lambda) and the correlations along
X decay exponentially on the scale H(lambda)^2. Using a suitable blocking
argument, we show that the distribution tails of the rescaled height decay
exponentially with critical exponent 3/2. In the particular case of linear
potential V(.), the characteristic length H(lambda) is proportional to
lambda^{-1/3} as lambda -> 0.

Yvan.Velenik@univ-rouen.fr

2370. HIERARCHICAL EQUILIBRIA OF BRANCHING POPULATIONS 

D.A. Dawson, L.G. Gorostiza and A. Wakolbinger

The objective of this paper is the study of the equilibrium behavior of a
population on the hierarchical group $\Omega_N$ consisting of families of
individuals undergoing critical branching random walk and in addition these
families also develop according to a critical branching process. Strong
transience of the random walk guarantees existence of an equilibrium for this
two-level branching system. In the limit $N\to\infty$ (called the hierarchical
mean field limit), the equilibrium aggregated populations in a nested sequence
of balls $B^{(N)}_\ell$ of hierarchical radius $\ell$ converge to a backward
Markov chain on $\mathbb{R_+}$. This limiting Markov chain can be explicitly
represented in terms of a cascade of subordinators which in turn makes possible
a description of the genealogy of the population.

ddawson@math.carleton.ca

2371. SHARP THRESHOLDS FOR MONOTONE PROPERTIES IN RANDOM GEOMETRIC GRAPHS 

Ashish Goel, Sanatan Rai and Bhaskar Krishnamachari

Random geometric graphs result from taking $n$ uniformly distributed points
in the unit cube, $[0,1]^d$, and connecting two points if their Euclidean
distance is at most $r$, for some prescribed $r$. We show that monotone
properties for this class of graphs have sharp thresholds by reducing the
problem to bounding the bottleneck matching on two sets of $n$ points
distributed uniformly in $[0,1]^d$. We present upper bounds on the threshold
width, and show that our bound is sharp for $d=1$ and at most a sublogarithmic
factor away for $d\ge2$. Interestingly, the threshold width is much sharper for
random geometric graphs than for Bernoulli random graphs. Further, a random
geometric graph is shown to be a subgraph, with high probability, of another
independently drawn random geometric graph with a slightly larger radius; this
property is shown to have no analogue for Bernoulli random graphs.

sanat@stanford.edu

2372. ELEMENTARY FIXED POINTS OF THE BRW SMOOTHING TRANSFORMS WITH INFINITE NUMBER OF SUMMANDS 

Aleksander M. Iksanov

The branching random walk (BRW) smoothing transform $T$ is defined as
$T:\text{distr}(U_{1})\mapsto \text{distr} (\sum_{i=1}^{L}X_{i}U_{i})$, where
given realizations $\{X_{i}\}_{i=1}^{L}$ of a point process, $U_{1},U_{2},...$
are conditionally independent identically distributed random variables, and
$0\leq \text{Prob}\{L=\infty \}\leq 1$. Given $\alpha \in (0,1]$,
$\alpha$-\emph{elementary} fixed points are fixed points of $T$ whose
Laplace-Stieltjes transforms $\phi$ satisfy $\underset{s\to
+0}{\lim}\dfrac{1-\phi(s)}{s^{\alpha}}=m$, where $m$ is any given positive
number. If $\alpha=1$, these are the fixed points with finite mean. We show
exactly when elementary fixed points exist. In this case these are the only
fixed points of $T$ and are unique up to a multiplicative constant. These
results do not need any moment conditions. In particular, Biggins' martingale
convergence theorem is proved in full generality. Essentially we apply recent
results due to Lyons (1997) and Goldie and Maller (2000) as the key point of
our approach is a close connection between fixed points with finite mean and
perpetuities. As a by-product, we lift from our general results the solution to
a Pitman-Yor problem. Finally, we study the tail behaviour of some fixed points
with finite mean.

iksan@unicyb.kiev.ua

2373. LINEAR SPEED LARGE DEVIATIONS FOR PERCOLATION CLUSTERS 

Yevgeniy Kovchegov and Scott Sheffield

Let C_n be the origin-containing cluster in subcritical percolation on the
lattice (1/n) Z^d, viewed as a random variable in the space Omega of compact,
connected, origin-containing subsets of R^d, endowed with the Hausdorff metric
delta. When d >= 2, and Gamma is any open subset of Omega, we prove: lim_{n \to
\infty}(1/n) \log P(C_n \in \Gamma) = -\inf_{S \in \Gamma} \lambda(S) where
lambda(S) is the one-dimensional Hausdorff measure of S defined using the {\em
correlation norm}: ||u|| := \lim_{n \to \infty} - \frac{1}{n} \log P (u_n \in
C_n) where u_n is u rounded to the nearest element of (1/n)Z^d. Given points
a^1, >..., a^k in R^d, there are finitely many correlation-norm Steiner trees
spanning these points and the origin. We show that if the C_n are each
conditioned to contain the points a^1_n,..., a^k_n, then the probability that
C_n fails to approximate one of these trees decays exponentially in n.

sheff@microsoft.com

2374. INSTABILITY OF SET RECURRENCE AND GREEN'S FUNCTION ON GROUPS WITH THE LIOUVILLE PROPERTY 

Itai Benjamini and David Revelle

Let $\mu$ and $\nu$ be probability measures on a group \Gamma and let G_\mu
and G_\nu denote Green's function with respect to \mu and \nu . The group
\Gamma is said to admit instability of Green's function if there are symmetric,
finitely supported measures $\mu$ and \nu and a sequence \{x_n\} such that
G_\mu(e, x_n)/G_\nu(e,x_n) \to 0, and \Gamma admits instability of recurrence
if there is a set S that is recurrent with respect to \nu but transient with
respect to \mu . We give a number of examples of groups that have the Liouville
property but have both types of instabilities. Previously known groups with
these instabilities did not have the Liouville property.

itai@wisdom.weizmann.ac.il

2375. PROBABILISTIC REPRESENTATIONS OF SOLUTIONS TO THE HEAT EQUATION 

B. Rajeev and S. Thangavelu

In this paper we provide a new (probabilistic) proof of a classical result in
partial differential equations, viz. if $\phi$ is a tempered distribution, then
the solution of the heat equation for the Laplacian, with initial condition
$\phi$, is given by the convolution of $\phi$ with the heat kernel (Gaussian
density). Our results also extend the probabilistic representation of solutions
of the heat equation to initial conditions that are arbitrary tempered
distributions.

brajeev@isibang.ac.in

2376. ZEROS OF THE I.I.D. GAUSSIAN POWER SERIES: A CONFORMALLY INVARIANT DETERMINANTAL PROCESS 

Yuval Peres and Balint Virag

Consider the zero set of a random power series sum a_n z^n with i.i.d.
complex Gaussian coefficients a_n. We show that these zeros form a
determinantal process: more precisely, their joint intensity can be written as
a minor of the Bergman kernel. We show that the number of zeros in a disk of
radius r about the origin has the same distribuion as the sum of independent
indicators X_k where P(X_k=1)=r^(-2k). The repulsion between zeros can be
studied via a dynamic version where the coefficients perform Brownian motion;
we show that this dynamics is conformally invariant.

balint@math.toronto.edu

2377. THICK POINTS FOR THE CAUCHY PROCESS 

Olivier Daviaud

Let $\mathcal{T}(x,\eps)$ denote the occupation measure of an interval of
length $2\eps$ centered at $x$ by the Cauchy process run until it hits
$(-\infty,-1]\cup [1,\infty)$. We prove that $\sup_{|x|\leq
1}\mathcal{T}(x,\eps)/(\eps(\ln\eps)^2)\to 2/\pi$ a.s. as $\eps\to 0$. We also
obtain the multifractal spectrum for thick points, i.e. the Hausdorff dimension
of the set of $\alpha$-thick points $x$ for which $\lim_{\eps \to 0}
\mathcal{T}(x,\eps)/(\eps(\ln\eps)^2) = \alpha > 0$.

odaviaud@stanford.edu

2378. EXCITED RANDOM WALK IN THREE DIMENSIONS HAS POSITIVE SPEED 

Gady Kozma

Excited random walk is a random walk that has a positive drift to the right
when it reaches a vertex it hasn't been to before. We show that in three
dimensions the walk drifts to the right in non-zero speed.

gady@post.tau.ac.il

2379. DIFFUSION IN RANDOM ENVIRONMENT AND THE RENEWAL THEOREM 

Dimitrios Cheliotis

According to a theorem of S. Schumacher and T. Brox, for a diffusion $X$ in a
Brownian environment it holds that $(X_t-b_{\log t})/\log^2t\to 0 $ in
probability, as $t\to\infty$, where $b_{\cdot}$ is a stochastic process having
an explicit description and depending only on the environment. In the first
part of this paper we compute the distribution of the sign changes for $b$ on
an interval $[1,x]$ and study some of the consequences of the computation; in
particular we get the probability of $b$ keeping the same sign on that
interval. These results have been announced in 1999 in a non-rigorous paper by
P. Le Doussal, C. Monthus, and D. Fisher and were treated with a
Renormalization Group analysis. We prove that this analysis can be made
rigorous using a path decomposition for the Brownian environment and renewal
theory. In the second part we consider the case that the environment is a
spectrally one sided stable process and derive results describing the features
of the environment that matter for the study of the process $b$. In particular
we derive the distribution of $b_1$.

demetris@stanford.edu

2380. AN ESTIMATE ABOUT MULTIPLE STOCHASTIC INTEGRALS WITH RESPECT TO A NORMALIZED EMPIRICAL MEASURE 

Peter Major

Let a sequence of iid. random variables $\xi_1,...,\xi_n$ be given on a
measurable space $(X,\cal X)$ with distribution $\mu$ together with a function
$f(x_1,...,x_k)$ on the product space $(X^k,{\cal X}^k)$. Let $\mu_n$ denote
the empirical measure defined by these random variables and consider the random
integral $$ J_{n,k}(f)={{n^{k/2}}\over{k!}}\int' f(u_1,...,u_k)
(\mu_n(du_1)-\mu(du_1))...(\mu_n(du_k)-\mu(du_k)), $$ where prime means that
the diagonals are omitted from the domain of integration. In this work a good
bound is given on the probability $P(|J_{n,k}(f)|>x)$ for all $x>0$. This
result shows that the tail behaviour of the distribution funtcion of the random
integral $J_{n,k}(f)$ and that of the integral of the function $f$ with respect
to a Gaussian random field show a similar behaviour. The proof is based on an
adaptation of some methods of the theory of Wiener--Ito integrals. In
particular, a sort of diagram formula is proved for the random integrals
$J_{n,k}(f)$ together with some of its important properties, a result which may
be interesting in itself. The relation of this estimate to some results about
$U$-statistics is also discussed.

major@renyi.hu

2381. AN ESTIMATE ON THE MAXIMUM OF A NICE CLASS OF STOCHASTIC INTEGRALS 

Peter Major

Let a sequence of iid. random variables $\xi_1,...,\xi_n$ be given on a space
$(X,\cal X)$ with distribution $\mu$ together with a nice class $\cal F$ of
functions $f(x_1,...,x_k)$ of $k$ variables on the product space $(X^k,{\cal
X}^k)$. For all $f\in\cal F$ we consider the random integral $J_{n,k}(f)$ of
the function $f$ with respect to the $k$-fold product of the normalized signed
measure $\sqrt n(\mu_n-\mu)$, where $\mu_n$ denotes the empirical measure
defined by the random variables $\xi_1,...,\xi_n$ and investigate the
probabilities $P(\sup_{f\in {\cal F}}|J_{n,k}(f)|>x)$ for all $x>0$. We show
that for nice classes of functions, for instance if $\cal F$ is a
Vapnik-Cervonenkis class, an almost as good bound can be given for these
probabilities as in the case when only the random integral of one function is
considered.

major@renyi.hu

2382. ON ASYMPTOTICS OF LARGE HAAR DISTRIBUTED UNITARY MATRICES 

Denes Petz and Julia Reffy

Let $U_n$ be an $n \times n$ Haar unitary matrix. In this paper, the
asymptotic normality and independence of $\Tr U_n, \Tr U_n^2, ..., \Tr U_n^k$
are shown by using elementary methods. More generally, it is shown that the
renormalized truncated Haar unitaries converge to a Gaussian random matrix in
distribution.

reffyj@math.bme.hu

2383. LIMITING DISTRIBUTION OF LAST PASSAGE PERCOLATION MODELS 

Jinho Baik

We survey some results and applications of last percolation models of which
the limiting distribution can be evaluated.

baik@umich.edu

2384. STOCHASTIC LOEWNER EVOLUTION IN DOUBLY CONNECTED DOMAINS 

Dapeng Zhan

This paper introduces the annulus SLE$_\kappa$ processes in doubly connected
domains. Annulus SLE$_6$ has the same law as stopped radial SLE$_6$, up to a
time-change. For $\kappa\not=6$, some weak equivalence relation exits between
annulus SLE$_\kappa$ and radial SLE$_\kappa$. Annulus SLE$_2$ is the scaling
limit of the corresponding loop-erased conditional random walk, which implies
that a certain form of SLE$_2$ satisfies the reversibility property. We also
consider the disc SLE$_\kappa$ process defined as a limiting case of the annuls
SLE's. Disc SLE$_6$ has the same law as stopped full plane SLE$_6$, up to a
time-change. Disc SLE$_2$ is the scaling limit of loop-erased random walk, and
is the reversal of radial SLE$_2$.

dapeng@its.caltech.edu

2385. EXPONENTIAL DISTRIBUTION FOR THE OCCURRENCE OF RARE PATTERNS IN GIBBSIAN RANDOM FIELDS 

M. Abadi, J.-R. Chazottes, F. Redig and E. Verbitskiy

We study the distribution of the occurrence of rare patterns in sufficiently
mixing Gibbs random fields on the lattice $\mathbb{Z}^d$, $d\geq 2$. A typical
example is the high temperature Ising model. This distribution is shown to
converge to an exponential law as the size of the pattern diverges. Our
analysis not only provides this convergence but also establishes a precise
estimate of the distance between the exponential law and the distribution of
the occurrence of finite patterns. A similar result holds for the repetition of
a rare pattern. We apply these results to the fluctuation properties of
occurrence and repetition of patterns: We prove a central limit theorem and a
large deviation principle.

jean-rene.chazottes@cpht.polytechnique.fr

2386. SKOROKHOD EMBEDDINGS, MINIMALITY AND NON-CENTRED TARGET DISTRIBUTIONS 

Alexander Cox and David Hobson

In this paper we consider the Skorokhod embedding problem for target
distributions with non-zero mean. In the zero-mean case, uniform integrability
provides a natural restriction on the class of embeddings, but this is no
longer suitable when the target distribution is not centred. Instead we
restrict our class of stopping times to those which are minimal, and we find a
condition involving stopping times which is equivalent to minimality. We then
apply these results, firstly to the problem of embedding non-centred target
distributions in Brownian motion, and secondly to embedding general target laws
in a diffusion. We find an embedding (which reduces to the Azema-Yor embedding
in the zero-target mean case) which maximises the law of \sup_{s \le T} B_s
among the class of minimal embeddings of a general target distribution \mu in
Brownian motion. We construct a minimal embedding of \mu in a diffusion X which
maximises the law of \sup_{s \le T} h(X_s) for a general function h.

mapamgc@maths.bath.ac.uk

2387. UNILATERAL SMALL DEVIATIONS FOR THE INTEGRAL OF FRACTIONAL BROWNIAN MOTION 

G.Molchan and A.Khokhlov

We consider the paths of a Gaussian random process $x(t)$, $x(0)=0$ not
exceeding a fixed positive level over a large time interval $(0,T)$, $T\gg 1$.
The probability $p(T)$ of such event is frequently a regularly varying function
at $\infty$ with exponent $\theta$. In applications this parameter can provide
information on fractal properties of processes that are subordinate to
$x(\cdot)$. For this reason the estimation of $\theta$ is an important
theoretical problem. Here, we consider the process $x(t)$ whose derivative is
fractional Brownian motion with self-similarity parameter $0<H<1$. For this
case we produce new computational evidence in favor of the relations $\log
p(T)=-\theta \log T(1+o(1))$ and $\theta =H(1-H)$. The estimates of $\theta$
are to within 0.01 in the range $0.1\le H\le 0.9$. An analytical result for the
problem in hand is known for the markovian case alone, i.e., for $H=1/2$. We
point out other statistics of $x(t)$ whose small values have probabilities of
the same order as $p(T)$ in the $\log$ scale.

khokhlov@ipgp.jussieu.fr

2388. ANOMALOUS DIFFUSION WITH PERIODICAL INITIAL CONDITIONS ON INTERVAL WITH REFLECTING EDGES 

G. Sh. Tsitsiashvili and A. E. Yashin

Uchaikin suggested a mathematical model of an anomalous diffusion in a space
was suggested. This model origins in an investigation of processes in complex
systems with variable structure: glasses, liquid crystals, biopolymers,
proteins and etc.
  In this model a coordinate of a particle has stable distribution (not normal
one). As a result a density of its distribution function satisfies an analog of
diffusion equation in which second derivative by coordinate is replaced by
partial derivative.
  In this paper the anomalous diffusion with periodic initial conditions on an
interval with reflecting edges is considered. Such problem is important for
example in technical mechanics for an analysis of fuel mixing in straight flow
engine too. As A.A. Borovkov suggested special probability methods are
developed to analyze such a diffusion.

guram@iam.dvo.ru

2389. WAITING FOR A BAT TO FLY BY (IN POLYNOMIAL TIME) 

Itai Benjamini, Gady Kozma, Laszlo Lovasz, Dan Romik and Gabor Tardos

We observe returns of a simple random walk on a finite graph to a fixed node,
and would like to infer properties of the graph, in particular properties of
the spectrum of the transition matrix. This is not possible in general, but at
least the eigenvalues can be recovered under fairly general conditions, e.g.
when the graph has a node-transitive automorphism group. The main result is
that by observing polynomially many returns, it is possible to estimate the
spectral gap of such a graph up to a constant factor.

itai@wisdom.weizmann.ac.il

2390. EUCLIDEAN GIBBS MEASURES ON LOOP LATTICES: EXISTENCE AND A PRIORI ESTIMATES 

Sergio Albeverio, Yuri G. Kondratiev, Tatiana Pasurek and Michael Roeckner

We prove existence and uniform a priori estimates for
Euclidean Gibbs measures corresponding to quantum anharmonic
crystals. Our method is based first on the alternative
characterization of Gibbs measures in terms of their
logarithmic derivatives through integration by parts formulas, 
and second on the choice of appropriate Lyapunov functionals.

albeverio@uni-bonn.de  kondrat@mathematik.uni-bielefeld.de  
pasurek@physik.uni-bielefeld.de  roeckner@.mathematik.uni-bielefeld.de

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

2391. EXISTENCE OF SOLUTIONS TO WEAK PARABOLIC EQUATIONS FOR MEASURES 

Vladimir I. Bogachev, Guiseppe Da Prato and Michael Roeckner

Let $A=(a^{ij})$ be a Borel mapping on $[0,1]\times
\mathbb{R}^d$ with values in the space of
nonnegative operators on $\mathbb{R}^d$ and let $b=(b^i)$
be a Borel mapping on $[0,1]\times \mathbb{R}^d$ with
values in $\mathbb{R}^d$. Let $Lu(t,x)=\partial_{t}u(t,x)+
a^{ij}(t,x)\partial_{x_i}\partial_{x_j}u(t,x)+
b^i(t,x)\partial_{x_i}u(t,x)$, $u\in
C_0^\infty((0,1)\times\mathbb{R}^d)$.
Under broad assumptions on $A$ and $b$, we construct a
family $\mu=(\mu_t)_{t\in [0,1)}$ of probability measures 
$\mu_t$ on $\mathbb{R}^d$ which solves the Cauchy problem
$L^{*}\mu =0$ with initial condition $\mu_0=\nu$, where
$\nu$  is a probability measure on $\mathbb{R}^d$, in the
following weak sense:
$$
\int_0^1\int_{\mathbb{R}^d} Lu(t,x)\, \mu_t(dx)\, dt=0,\quad
u\in C_0^\infty((0,1)\times\mathbb{R}^d),
$$
and
$$
\lim\limits_{t\to 0}
\int_{\mathbb{R}^d} \zeta(x)\, \mu_t(dx)=
\int_{\mathbb{R}^d} \zeta(x)\, \nu(dx),\quad
\zeta \in C_0^\infty(\mathbb{R}^d).
$$
Such an equation is satisfied by transition probabilities
of a diffusion process associated with $A$ and $b$ provided
such a process exists. However, we do not assume the
existence of a process and allow quite singular
coefficients, in particular, $b$ may be locally unbounded
or $A$ may be degenerate. An infinite dimensional analogue 
is discussed as well. Main methods are $L^p$-analysis with 
respect to suitably chosen measures and reduction to the 
elliptic case (studied previously) by piecewise constant 
approximations in time.

vbogach@mech.math.msu.su  daprato@sns.it  roeckner@mathematik.uni-bielefeld.de

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

2392. 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 funtional turns 
out to be the energy of the paths associated to the 
$L^2$-Wasserstein  distance.

roeckner@mathematik.uni-bielefeld.de  tzhang@ma.man.ac.uk

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

2393. SMALL MASS IMPLIES UNIQUENESS OF GIBBS STATES OF A QUANTUM CRYSTAL 

Sergio Albeverio, Yuri G. Kondratiev, Yuri Kozitsky and Michael Roeckner

A model of interacting quantum particles performing
one-dimensional anharmonic oscillations around their
equilibrium positions which form a lattice $\mathbb{Z}^d$ is
considered. For this model, it is proved that the set of
tempered Euclidean Gibbs measures is a singleton provided
the particle mass is less than a certain bound $m_{\ast}$,
which is independent of the temperature $\beta^{-1}$.
This settles a problem that was open for a long time and
is an essential improvement of a similar result proved
before by the same authors [5], where the bound $m_{\ast}$
depended on $\beta$ in such a way that $m_{\ast} (\beta) \to 0$
as $\beta \to +\infty$.

albeverio@uni-bonn.de  kondrat@mathematik.uni-bielefeld.de  
jkozi@golem.umcs.lublin.pl  roeckner@mathematik.uni-bielefeld.de

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

2394. INVARIANCE IMPLIES GIBBSIAN: SOME NEW RESULTS 

Vladimir I. Bogachev, Michael Roeckner and 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 $d\ge 3$).

vbogach@mech.math.msu.su  roeckner@mathematik.uni-bielefeld.de  wangfy@bnu.edu.cn

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

2395. ASYMPTOTICS FOR THE STAIRCASE AND CHACON'S TRANSFORMATION 

Vincent Chamoître and Michal Kupsa

We prove that either in Chacon's weak mixing system or 
in the classical staircase mixing one, almost surely, 
asymptotics for return time exist and equal the distribution 
function $F$ defined by $F(t)=1_{[1,+\infty [}(t)$. 

This gives the first known example of a mixing system with 
non exponential asymptotics. 

vincent.chaumoitre@u-picardie.fr kupsa@cpt.univ-mrs.fr kupsa@kti.mff.cuni.cz

2396. SYSTEMS OF EQUATIONS DRIVEN BY STABLE PROCESSES 

Richard F. Bass and Zhen-Qing Chen

Let $Z^j_t$, $j=1, \ldots, d$, be independent 
one-dimensional symmetric stable processes of 
index $\alpha\in(0,2)$. We consider the system of 
stochastic differential equations
$$
dX^i_t=\sum_{j=1}^d A_{ij}(X_{t-}) dZ^j_t, 
    \quad i=1, \ldots, d,
$$
where the matrix $A(x)=(A_{ij}(x))_{1\leq i, j \leq d}$  
is continuous in $x$ and nondegenerate for each $x$. 
We prove existence and uniqueness of a weak solution to 
this system. The approach of this paper uses the martingale 
problem method. For this, we establish some  estimates for 
pseudodifferential operators with singular state-dependent 
symbols. Let $\lambda_2>\lambda_1>0$. We show that for any
two vectors $a, b\in \R^d$ with $|a|, |b| \in (\lambda_1,
 \lambda_2)$ and $p$ sufficiently large, the $L^p$-norm of
the operator whose Fourier multiplier is 
$(|u\cdot a|^\alpha - |u \cdot b|^\alpha )/\sum_{j=1}^d |u_i|^\alpha $
is bounded by a constant multiple of $|a-b|^\theta$
for some $\theta >0$, where $u=(u_1, \ldots, u_d) \in R^d$.
We deduce from this the $L^p$-boundedness of pseudodifferential 
operators with symbols of the form 
$\psi(x,u)=|u\cdot a(x)|^\alpha/ \sum_{j=1}^d |u_i|^\alpha$,
where $u=(u_1, \ldots, u_d)$ and $a$ is a continuous function 
on $R^d$ with $|a(x)|\in (\lambda_1, \lambda_2)$ for all $x$.

bass@math.uconn.edu   zchen@math.washington.edu

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

2397. SDDEs LIMITS SOLUTIONS TO SUBLINEAR REACTION-DIFFUSION SPDEs 

Hassan Allouba

We start by introducing a new definition of solutions to
heat-based SPDEs driven by space-time white noise: SDDEs
(stochastic differential-difference equations) limits 
solutions.  In contrast to the standard direct definition 
of SPDEs solutions; this new notion, which builds on and 
refines our SDDEs approach to SPDEs from earlier work, is 
entirely based on the approximating SDDEs.  It is 
applicable to, and gives a multiscale view of, a variety 
of SPDEs.  We extend this approach in related work to 
other heat-based SPDEs (Burgers, Allen-Cahn, and others) 
and to the difficult case of SPDEs with multi-dimensional 
spatial variable.  We focus here on one-spatial-
dimensional reaction-diffusion SPDEs; and we prove the 
existence of a SDDEs limit solution to these equations 
under less-than-Lipschitz conditions on the drift and the 
diffusion coefficients, thus extending our earlier 
SDDEs work to the nonzero drift case.   The regularity of 
this solution is obtained  as a by-product of the 
existence estimates.  The uniqueness in law of our SPDEs 
follows, for a large class of such drifts/diffusions, as a 
simple extension of our recent Allen-Cahn uniqueness 
result.  We also examine briefly, through order parameters 
$\epsilon_1$ and $\epsilon_2$ multiplied by the Laplacian 
and the noise, the effect of letting $\epsilon_1,\epsilon_2
\to0$ at different speeds.  More precisely, it is shown 
that the ratio $\epsilon_2/\epsilon_1^{1/4}$ determines 
the behavior as $\epsilon_1,\epsilon_2\to0$.

allouba@math.kent.edu

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

2398. HOLDER NORM ESTIMATES FOR ELLIPTIC OPERATORS ON FINITE AND INFINITE DIMENSIONAL SPACES 

Siva Athreya, Richard F. Bass, and Ed Perkins

We introduce a new method for proving the estimate
$$\left\Vert{\frac{\del^2 u}{\del x_i \del x_j}}\right\Vert_{C^\al}
\leq c \norm{f}_{C^\al},$$
where $u$ solves the equation $ \Delta u-\lambda u=f$.
The method can be applied to the Laplacian on $\R^\infty$.
It also allows us to obtain similar estimates when we replace
the Laplacian by an infinite dimensional Ornstein-Uhlenbeck
operator or other elliptic operators. These operators arise
naturally in  martingale problems arising from measure-valued
branching diffusions and from stochastic partial differential
equations.

athreya@isid.ac.in  bass@math.uconn.edu  perkins@math.ubc.ca

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