Probability Abstracts 53

1354. PHASE TRANSITIONS ON NONAMENABLE GRAPHS

Russell Lyons

We survey known results about phase transitions in various models of
statistical physics when the underlying space is a nonamenable graph. Most
attention is devoted to transitive graphs and trees.

rdlyons@indiana.edu

• This article is available from the xxx mathematics archive as math.PR/9908177.

1355. PERTURBATION OF THE EQUILIBRIUM FOR A TOTALLY ASYMMETRIC STICK PROCESS IN ONE DIMENSION

Timo Seppalainen

We study the evolution of a small perturbation of the equilibrium of a
totally asymmetric one-dimensional interacting system. The model we take as
example is Hammersley's process as seen from a tagged particle, which can be
viewed as a process of interacting positive-valued stick heights on the sites
of Z. It is known that under Euler scaling (space and time scale n) the
empirical stick profile obeys the Burgers equation. We refine this result in
two ways: If the process starts close enough to equilibrium, then over times
n^\nu for 1\le\nu<3, and up to errors that vanish in hydrodynamic scale, the
dynamics merely translates the initial stick configuration. A time evolution
for the perturbation is visible under a particular family of scalings: over
times n^\nu, 1<\nu<3/2, a perturbation of order n^(1-\nu) from equilibrium
follows the inviscid Burgers equation. The results for the stick model are
derived from asymptotic results for tagged particles in Hammersley's process.

seppalai@iastate.edu

• This article is available from the xxx mathematics archive as math.PR/9909011.

1356. MEASURING THE MAGNITUDE OF SUMS OF INDEPENDENT RANDOM VARIABLES

Pawel Hitczenko, Stephen Montgomery-Smith

This paper considers how to measure the magnitude of the sum of independent
random variables in several ways. We give a formula for the tail distribution
for sequences that satisfy the so called Levy property. We then give a
connection between the tail distribution and the pth moment, and between the
pth moment and the rearrangement invariant norms.

stephen@cauchy.math.missouri.edu

• This article is available from the xxx mathematics archive as math.PR/9909054.

1357. A NOTE ON WETTING TRANSITION FOR GRADIENT FIELDS

Pietro Caputo and Yvan Velenik

We prove existence of a wetting transition for two types of gradient fields:
1) Continuous SOS models in any dimension and 2) Massless Gaussian model in two
dimensions. Combined with a recent result showing the absence of such a
transition for Gaussian models above two dimensions by Bolthausen et al, this
shows in particular that absolute-value and quadratic interactions can give
rise to completely different behaviors.

caputo@math.tu-berlin.de

• This article is available from the xxx mathematics archive as math.PR/9909057.

1358. SINGULARITY OF SOME RANDOM CONTINUED FRACTIONS

Russell Lyons

We prove singularity of some distributions of random continued fractions that
correspond to iterated function systems with overlap and a parabolic point.
These arose while studying the conductance of Galton-Watson trees.

rdlyons@indiana.edu

• This article is available from the xxx mathematics archive as math.PR/9909097.

1359. RANDOM WORDS, QUANTUM STATISTICS, CENTRAL LIMITS, RANDOM MATRICES

Greg Kuperberg 

Recently Tracy and Widom conjectured [math.CO/9904042] and Johansson proved
[math.CO/9906120] that the expected shape \lambda of the semi-standard tableau
produced by a random word in k letters is asymptotically the spectrum of a
random traceless k by k GUE matrix. In this article we give two arguments for
this fact. In the first argument, we realize the random matrix itself as a
quantum random variable on the space of random words, if this space is viewed
as a quantum state space. In the second argument, we show that the distribution
of \lambda is asymptotically given by the usual local limit theorem, but the
resulting Gaussian is disguised by an extra polynomial weight and by reflecting
walls. Both arguments more generally apply to an arbitrary finite-dimensional
representation V of an arbitrary simple Lie algebra g. In the original
question, V is the defining representation of g = su(k).

greg@math.ucdavis.edu

• This article is available from the xxx mathematics archive as math.PR/9909104.

1360. EXACT ESTIMATES FOR MOMENTS OF RANDOM BILINEAR FORMS

R. Ibragimov, Sh. Sharakhmetov and A. Cecen

The present paper concentrates on the analogues of Rosenthal's inequalities
for ordinary and decoupled bilinear forms in symmetric random variables. More
specifically, we prove the exact moment inequalities for these objects in terms
of moments of their individual components. As a corollary of these results we
obtain the explicit expressions for the best constant in the analogues of
Rosenthal's inequality for ordinary and decoupled bilinear forms in identically
distributed symmetric random variables in the case of the fixed number of
random variables.

ibrag1r@mail.cmich.edu

• This article is available from the xxx mathematics archive as math.PR/9909111.

1361. INTEGRATION OF BROWNIAN VECTOR FIELDS

Yves Le Jan and Olivier Raimond

  Using the Wiener chaos decomposition, we show that strong solutions of non
Lipschitzian S.D.E.'s are given by random Markovian kernels. The example of
Sobolev flows is studied in some detail, exhibiting interesting phase
transitions.

raimond@cristal.math.u-psud.fr

• This article is available from the xxx mathematics archive as math.PR/9909147.

1362. NUMERICAL CHARACTERISTICS OF GROUPS AND THEIR INTERRELATIONS

Anatoly M. Vershik

The entropy of the random walk on the discrete contable group could be used
for comparison of the system of the generators. Fundamental inequality between
growth, entropy and escape gives the possibility to define "the best" system of
the generators. We formulate a new circle of the problems related to the
various growth and asymptotics on the groups.

matob@pdmi.ras.ru

• This article is available from the xxx mathematics archive as math.PR/9909167.

1363. PHASE TRANSITION AND PERCOLATION IN GIBBSIAN PARTICLE MODELS

H.-O. Georgii

We discuss the interrelation between phase transitions in interacting lattice
or continuum models, and the existence of infinite clusters in suitable
random-graph models. In particular, we describe a random-geometric approach to
the phase transition in the continuum Ising model of two species of particles
with soft or hard interspecies repulsion. We comment also on the related
area-interaction process and on perfect simulation.

georgii@rz.mathematik.uni-muenchen.de

• This article is available from the xxx mathematics archive as math.PR/9910005.

1364. PERCOLATION AND NUMBER OF PHASES IN THE 2D ISING MODEL

H.-O. Georgii, Y. Higuchi

We reconsider the percolation approach of Russo, Aizenman and Higuchi 
for showing that there exist only two phases in the Ising model on the square 
lattice. We give a fairly short alternative proof which is only based 
on FKG monotonicity and avoids the use of GKS-type inequalities originally 
needed for some background results. Our proof extends to the Ising model 
on other planar lattices such as the triangular and honeycomb lattice. 
We can also treat the Ising antiferromagnet in an external field 
and the hard-core lattice gas model on $Z^2$. 

georgii@rz.mathematik.uni-muenchen.de

• This article is available from the xxx mathematics archive as math.PR/9907186.

1365. ESTIMATING THE J FUNCTION WITHOUT EDGE CORRECTION

Adrian Baddeley, Martin Kerscher, Katja Schladitz, and Bryan T. Scott

The interaction between points in a spatial point process can be measured by
its empty space function F, its nearest-neighbour distance distribution
function G, and by combinations such as the J-function $J = (1-G)/(1-F)$. The
estimation of these functions is hampered by edge effects: the uncorrected,
empirical distributions of distances observed in a bounded sampling window W
give severely biased estimates of F and G. However, in this paper we show that
the corresponding {\em uncorrected} estimator of the function $J=(1-G)/(1-F)$
is approximately unbiased for the Poisson case, and is useful as a summary
statistic. Specifically, consider the estimate $\hat{J}_W$ of J computed from
uncorrected estimates of F and G. The function $J_W(r)$, estimated by
$\hat{J}_W$, possesses similar properties to the J function, for example
$J_W(r)$ is identically 1 for Poisson processes. This enables direct
interpretation of uncorrected estimates of J, something not possible with
uncorrected estimates of either F, G or K. We propose a Monte Carlo test for
complete spatial randomness based on testing whether $J_W(r)(r)\equiv 1$.
Computer simulations suggest this test is at least as powerful as tests based
on edge corrected estimators of J.

kerscher@theorie.physik.uni-muenchen.de

• This article is available from the xxx mathematics archive as math.PR/9910011.

1366. HOW TO COUPLE FROM THE PAST USING A READ-ONCE SOURCE OF RANDOMNESS

David B. Wilson

We give a new method for generating perfectly random samples from the
stationary distribution of a Markov chain. The method is related to coupling
from the past (CFTP), but only runs the Markov chain forwards in time, and
never restarts it at previous times in the past. The method is also related to
an idea known as PASTA (Poisson arrivals see time averages) in the operations
research literature. Because the new algorithm can be run using a read-once
stream of randomness, we call it read-once CFTP. The memory and time
requirements of read-once CFTP are on par with the requirements of the usual
form of CFTP, and for a variety of applications the requirements may be
noticeably less. Some perfect sampling algorithms for point processes are based
on an extension of CFTP known as coupling into and from the past; for
completeness, we give a read-once version of coupling into and from the past,
but it remains unpractical. For these point process applications, we give an
alternative coupling method with which read-once CFTP may be efficiently used.

dbwilson@microsoft.com

• This article is available from the xxx mathematics archive as math.PR/9910050.

1367. EFFICIENT SPHERE-COVERING AND CONVERSE MEASURE CONCENTRATION VIA GENERALIZED CODING THEOREMS

Ioannis Kontoyiannis

Suppose A is a finite set equipped with a probability measure P and let M be
a ``mass'' function on A. We give a probabilistic characterization of the most
efficient way in which A^n can be almost-covered using spheres of a fixed
radius. An almost-covering is a subset C_n of A^n, such that the union of the
spheres centered at the points of C_n has probability close to one with respect
to the product measure P^n. An efficient covering is one with small mass
M^n(C_n); n is typically large. With different choices for M and the geometry
on A our results give various corollaries as special cases, including Shannon's
data compression theorem, a version of Stein's lemma (in hypothesis testing),
and a new converse to some measure concentration inequalities on discrete
spaces. Under mild conditions, we generalize our results to abstract spaces and
non-product measures.

yiannis@stat.purdue.edu

• This article is available from the xxx mathematics archive as math.PR/9910062.

1368. TRANSVERSAL FLUCTUATIONS FOR INCREASING SUBSEQUENCES ON THE PLANE

Kurt Johansson

Consider a realization of a Poisson process in R^2 with intensity 1 and take
a maximal up/right path from the origin to (N,N) consisting of line segments
between the points, where maximal means that it contains as many points as
possible. The number of points in such a path has fluctuations of order N^chi,
where chi=1/3 by a result of Baik-Deift-Johansson. Here we show that typical
deviations of a maximal path from the diagonal x=y is of order N^xi with
xi=2/3. This is consistent with the scaling identity chi=2xi-1, which is
believed to hold in many random growth models.

kurtj@math.kth.se

• This article is available from the xxx mathematics archive as math.PR/9910146.

1369. MONOTONICITY OF CONDITIONAL DISTRIBUTIONS AND GROWTH MODELS ON TREES

Thomas M. Liggett

We consider a sequence of probability measures $\nu_n$
obtained by conditioning a random vector $X=(X_1,..., X_d)$
with nonnegative integer valued components on  
$$X_1+\cdots +X_d=n-1,$$ 
and give several sufficient conditions on the distribution 
of $X$ for $\nu_n$ to be stochastically increasing in $n$. 
The problem is motivated by an interacting particle system
on the homogeneous tree in which each vertex has $d+1$ 
neighbors. This system is a variant of the contact process, 
and was studied recently by A. Puha. She showed that the 
critical value for this process is $\frac 14$ if $d=2$, 
and gave a conjectured expression for the critical value 
for all $d$. Our results confirm her conjecture, 
by showing that certain $\nu_n$'s
defined in terms of $d$-ary Catalan numbers
are stochastically increasing. The proof uses certain
combinatorial identities satisfied by the
$d$-ary Catalan numbers.

tml@math.ucla.edu

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

1370. DYNAMIC SCHEDULING OF A SYSTEM WITH TWO PARALLEL SERVERS IN HEAVY TRAFFIC WITH COMPLETE RESOURCE POOLING

S. L. Bell and R. J. Williams 

This paper concerns a dynamic scheduling problem for a queueing system
that has two streams of arrivals to infinite capacity buffers and two
(non-identical) servers working in parallel. One server can only process
jobs from one buffer, whereas the other server can process jobs from
either buffer. The service time distribution may depend on the buffer
being served and the server providing the service. The system manager
dynamically schedules waiting jobs onto available servers. We consider
a parameter regime in which the system satisfies both a heavy traffic
condition and a resource pooling condition. Our cost function is a mean
cumulative discounted cost of holding jobs in the system, where the
(undiscounted) cost per unit time is a linear function of normalized (with
heavy traffic scaling) queue length. We first review the analytic solution
of the Brownian control problem (formal heavy traffic approximation) for
this system. We `interpret' this solution by proposing a `continuous
review' threshold control policy for use in the original parallel server
system. We show that this policy is asymptotically optimal in the heavy
traffic limit and the limiting cost is the same as the optimal cost in the
Brownian control problem. The techniques developed here are expected
to be useful for analyzing the performance of threshold-type policies in
more complex multiserver systems. 

williams@math.ucsd.edu

1371. SUPERPROCESSES OVER A STOCHASTIC FLOW

Georgios Skoulakis and Robert J. Adler 

We study a specific particle system in which particles 
undergo random  branching and spatial motion. Such systems 
are best described, mathematically, via measure valued 
stochastic processes. As is now quite standard, we study 
the so called superprocess limit of such a system as both 
the number of particles in the system and the branching
rate tend to infinity. What differentiates our system from 
the classical superprocess case, in which the particles move
independently of each other, is that  the motions of our
particles are affected by the presence of a global 
stochastic flow. 

We establish weak convergence to the solution of a 
well-posed martingale problem. Using the particle picture 
formulation of the flow superprocess, we study some of its
properties. We give formulae for its first two moments and
consider two macroscopic quantities describing its average
behavior, properties that have been studied in some detail 
previously in the pure flow situation, where branching was 
absent.

Explicit formulae for these quantities are given
and graphs are presented for a specific example of a linear
flow of Ornstein-Uhlenbeck type.

eorgios@email.unc.edu robert@ieadler.technion.ac.il

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

1372. SMALL DEVIATIONS FOR SOME MULTI-PARAMETER GAUSSIAN PROCESSES

David M. Mason and Zhan Shi

We prove some general lower bounds for the probability that
a multi-parameter Gaussian process has very small values.
These results, when applied to a certain class of fractional
Brownian sheets, yield the exact rate for their so-called
small ball probability. We show by example how to use such
results to compute the Hausdorff dimension of some
exceptional sets determined by maximal increments.

davidm@math.udel.edu  zhan@proba.jussieu.fr

1373. DIVERGENCE FORM OPERATORS ON FRACTAL-LIKE DOMAINS

Martin Barlow and Richard Bass

We consider elliptic operators $\cal L$ in divergence form 
on certain domains in $R^d$ with fractal volume growth. 
The domains we look at are pre-Sierpinski carpets, which are
derived from higher dimensional Sierpinski carpets. We 
prove a Harnack inequality for nonnegative $\cal L$-harmonic
functions on these domains and establish upper and lower 
bounds for the corresponding heat equation. 

barlow@math.ubc.ca  bass@math.uconn.edu

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

1374. STOCHASTIC BILLIARDS ON GENERAL TABLES

Steven N. Evans

We consider stochastic analogues of classical billiard 
systems. A particle moves at unit speed with constant 
direction in the interior of a bounded, $d$--dimensional
region with continuously differentiable boundary.
The boundary need not be connected; that is, the ``table'' 
may have interior ``obstacles''. When the particle strikes 
the boundary, a new direction is chosen uniformly at random
from the directions that point back into the interior of
the region and the motion continues. Such chains are
closely related to those that appear in shake--and--bake 
simulation algorithms.

For the discrete time Markov chain that records the locations
of successive hits on the boundary, we show that, uniformly
in the starting point, there is exponentially fast
total variation convergence to an invariant distribution.
By analysing an associated non--linear, first--order PDE,
we investigate which regions are such that this chain is 
reversible with respect to surface measure on the boundary.
We also establish a result on uniform total variation 
C\'esaro convergence to equilibrium for the continuous 
time Markov process that tracks the position and direction 
of the particle.
 
evans@stat.berkeley.edu

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

1375. RIGHT INVERSES OF LEVY PROCESSES AND STATIONARY STOPPED LOCAL TIMES

Steven N. Evans

If $X$ is a symmetric L\'evy process on the line, then 
there exists a non--decreasing, c\`adl\`ag process $H$
such that $X(H(x)) = x$ for all $x \ge 0$ if and only if 
$X$ is recurrent and has a non--trivial Gaussian component.
The minimal such $H$ is a subordinator $K$.  The law of $K$
is identified and shown to be the same as that of a multiple
of the inverse local time at $0$ of $X$. When $X$ is 
Brownian motion, $K$ is just the usual ladder times process
and this result extends the classical result of L\'evy 
that the maximum process has the same law as the local time
at $0$.

Write $G_t$ for last point in the range of $K$ prior to $t$.
In a parallel with classical fluctuation theory, the
process $Z := (X_t - X_{G_t})_{t \ge 0}$ is Markov
with local time at $0$ given by $(X_{G_t})_{t \ge 0}$.
The transition kernel and excursion measure of $Z$ are 
identified.  A similar programme is outlined for L\'evy 
processes on the circle.  This leads to the construction of
a stopping time such that the stopped local times constitute
a stationary process indexed by the circle.

evans@stat.berkeley.edu

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

1376. WHERE DID THE BROWNIAN PARTICLE GO?

Robin Pemantle, Yuval Peres, Jim Pitman and Marc Yor

Consider the radial projection onto the unit sphere
of the path a  $d$-dimensional Brownian motion $W$,
started at the center of the sphere and run for unit time. 
Given the occupation measure $\mu$ of this projected path, 
what can be said about the terminal point $W(1)$, or about 
the range of the original path?  In any dimension, for each 
Borel set $A$ of $S^{d-1}$, the conditional probability that 
the projection of $W(1)$ is in $A$, given $\mu(A)$, is just 
$\mu(A)$.  Nevertheless, in dimension $d>2$, both the range 
and the terminal point of $W$ can be recovered with 
probability one from $\mu$. In particular, for $d>2$, the 
conditional law of the projection of $W(1)$ given $\mu$ is 
not $\mu$. In dimension 2 we conjecture that the projection 
of $W(1)$ cannot be recovered almost surely from $\mu$, and 
show that the conditional law of the projection of $W(1)$ 
given $\mu$ is not $\mu$.

pemantle@math.wisc.edu peres@math.huji.ac.il pitman@stat.berkeley.edu

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

1377. EXISTENCE OF $L^q$ DIMENSIONS AND ENTROPY DIMENSION FOR SELF-CONFORMAL MEASURES

Yuval Peres and Boris Solomyak

We prove the existence of limits in the definitions of 
$L^q$ dimensions (for all positive $q$ different from 1)
as well as the entropy dimension, for any self-conformal 
measure, without any separation assumptions.
  We also show the existence of order-two densities for a 
class of self-similar measures with overlap. 

 peres@math.huji.ac.il  solomyak@math.washington.edu

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

1378. GENERALIZED ORNSTEIN-UHLENBECK SEMIGROUPS: LITTLEWOOD-PALEY-STEIN INEQUALITIES AND THE P.A. MEYER EQUIVALENCE OF NORMS

Anna Chojnowska-Michalik and Beniamin Goldys

Let $\mu$ be a 
nondegenerate Gaussian measure on a Hilbert space $H$. For 
an arbitrary selfadjoint nonnegative operator 
$A$ we consider the semigroup 
$e^{tL}=\Gamma\left(e^{-tA}\right)$ on $L^p(\mu )$, 
where $\Gamma$ stands for the 
second quantization operator. We provide an explicit 
characterization of the domains of 
$(I-L)^{m/2}$ in $L^p(\mu )$ in terms 
of Gaussian Sobolev spaces thus extending the P. A. 
Meyer result on equivalence of norms.  The main tool are 
the Littlewood-Paley-Stein inequalities which are proved  
under minimal assumptions by a purely analytic method 
following Stein. 

B.Goldys@unsw.edu.au

1379. PARAMETER ESTIMATION FOR CONTROLLED SEMILINEAR STOCHASTIC SYSTEMS: IDENTIFIABILITY AND CONSISTENCY

Beniamin Goldys and Bohdan Maslowski

We consider a controlled stochastic semilinear evolution 
equation with the drift depending on the unknown parameter. 
We show that the Maximum Likelihood Estimator is strongly 
consistent for a class of bounded predictable controls.

B.Goldys@unsw.edu.au

1380. STRONG APPROXIMATION OF SPATIAL RANDOM WALK IN RANDOM SCENERY

Pal Revesz and Zhan Shi

We prove a strong approximation for the spatial
Kesten-Spitzer random walk in random scenery by a Wiener
process. 

revesz@ci.tuwien.ac.at  zhan@proba.jussieu.fr

1381. ON THE MARTINGALE PROBLEM FOR SUPER-BROWNIAN MOTION

Richard Bass and Edwin Perkins

The law of super-Brownian motion can be characterized 
as the solution to a certain martingale problem.  We 
give a new proof of this fact that uses only basic
stochastic calculus and some simple facts about weak 
convergence.

bass@math.uconn.edu  perkins@heron.math.ubc.ca

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

1382. SPECTRAL GAP FOR KAC'S MODEL OF BOLTZMANN EQUATION

Elise Janvresse

We consider a random walk on $S^{n-1}$, the standard 
sphere of dimension $n-1$, generated by random rotations 
on randomly selected coordinate planes $i, j$ with 
$1 \le i <  j \le n$. 
This dynamics was used by  Marc Kac as a model for a 
Boltzmann equation. 
We prove that the spectral gap on $S^{n-1}$ is $n^{-1}$ 
up to a constant independent of $n$.

Elise.Janvresse@univ-rouen.fr

1383. LONG STRANGE SEGMENTS OF A STOCHASTIC PROCESS AND LONG RANGE DEPENDENCE

P. Mansfield, S. Rachev and  G. Samorodnitsky

We study long strange intervals in a linear stationary stochastic
process with regularly varying tails. It turns out that the length of
the longest strange interval grows, as a function of the sample size,
at different rates in different parts of the parameter space. We argue
that this phenomenon  may be viewed in a fruitful
way as a phase transition between short and long range
dependence. We prove a limit theorem that may form a basis for 
statistical detection of long range dependence.

gennady@orie.cornell.edu

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

1384. UNIQUENESS OF INVARIANT MEASURES AND MAXIMAL DISSIPATIVITY OF DIFFUSION OPERATORS ON L^1

Vladimir I. Bogachev, Michael Röckner and Wilhelm Stannat

It is proved that there exists at most one probability
measure $\mu$ on $R^d$, so that $L^\ast\mu=0$, where
$L=a^{ij}\partial_i\partial_j+b^i\partial_i$, provided
$(L,C^\infty_0(R))$ is maximally dissipative on
$L^1(R^d,\nu)$ for at least one $\nu$, so that $L^\ast\nu=0$.
Here it is assumed that $(a^{ij})$ is non-degenerate,
$a^{ij}\in H^{p,1}_{loc}$, and $b^i\in L^p_{loc}$. We also
present a whole class of examples (even for $a^{ij}=
\delta^{ij}$), where $L^\ast\mu=0$ has more than one
solution. Furthermore, recent related results are reviewed.

bogachev@vbogach.mccme.ru roeckner@mathematik.uni-bielefeld.de 
stannat@mathematik.uni-bielefeld.de

1385. INFINITELY MANY CONTACT PROCESS TRANSITIONS ON A TREE

Marcia Salzano

We continue our study of the ergodic behavior of the contact
process on infinite connected graphs of bounded degree. Examples
are provided of trees on which, as the infection parameter increases,
complete convergence alternates between holding and failing infinitely
many times.

msalzano@math.ucla.edu

1386. THIN POINTS FOR BROWNIAN MOTION

Amir Dembo, Yuval Peres, Jay Rosen and Ofer Zeitouni

Let T(x,r) denote the occupation measure of the ball 
of radius r centered at x for Brownian motion {W_t}
in dimension d>1, run for unit time. Denote
$h(r)=r^2/|\log r|$. We prove that for any analytic 
set E in [0,1], the infimum over t in E of  
$liminf_{r \to 0} T(W_t,r)/h(r)$ equals the reciprocal 
of the packing dimension of E. We deduce that for any a>1, 
the Hausdorff dimension of the set of ``thin points'' x 
for which $\liminf_{r \to 0} T(x,r)/h(r)=a$, is almost 
surely 2-2/a; this is the correct scaling to obtain a 
nondegenerate ``multifractal spectrum'' for the ``thin'' 
part of Brownian occupation measure. The methods of 
this paper differ considerably from those of our work 
on Brownian thick points, due to the high degree of 
correlation in the present case. To prove our results, 
we establish general criteria for determining which 
deterministic setsare hit by random fractals of 
`limsup type' in the presence of long-range correlations. 
The hitting criteria then yield lower bounds on Hausdorff 
dimension. This refines previous work of Khoshnevisan, Xiao 
and the second author, that required decay of correlations. 

amir@stat.stanford.edu peres@math.huji.ac.il jrosen3@idt.net zeitouni@ee.technion.ac.il

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

1387. ANISOTROPIC CONTACT PROCESS ON HOMOGENEOUS TREES

Irene Hueter

The existence of a weak survival region is established for 
the symmetric anisotropic contact process on a homogeneous
tree ${\cal T}_{d+1}$ of degree  $d+1 \geq 3:$ For parameter 
values in a certain connected region of positive Lebesgue 
measure, the population survives forever with positive 
probability but ultimately vacates every finite subset of 
the tree with probability one. In this phase, infection 
trails must converge to the geometric boundary $\Omega$ 
of the tree. The random subset $\Lambda$ of the boundary 
consisting of all ends of the tree in which the infection 
survives, called the limit set of the process, is shown to 
have Hausdorff dimension no larger than one half the 
Hausdorff dimension of the entire geometric boundary. In 
addition, there is strict inequality at the transition between
weak and strong survival except when the contact process is 
isotropic. It is further shown that in all cases there is a 
distinguished probability measure $\mu,$ supported by $\Omega,$
such that the Hausdorff dimension of $\Lambda \cap \Omega_{\mu},$ 
where $\Omega_{\mu}$ is the set of $\mu$-generic points of 
$\Omega,$ converges to one half the Hausdorff dimension of 
$\Omega_{\mu}$ at the phase separation points. Exact formulae 
for the Hausdorff dimension of $\Lambda$ and $\Lambda \cap 
\Omega_{\mu}$ are obtained. We also prove that the contact 
process at the transition between extinction and weak survival 
does not survive. The method developed shows that at the phase 
transition to strong survival the contact process survives 
weakly for $d \geq 2$ and survives strongly on $Z.$ 

hueter@math.ufl.edu

1388. ASYMMETRIC CRITICAL CONTACT PROCESS ON HOMOGENEOUS TREES

Irene Hueter

We consider the asymmetric anisotropic contact process on 
a homogeneous tree ${\cal T}_{d+1}$ of degree $d+1 \geq 2$ 
and give algebraic conditions that characterize the three 
phases ${\cal R}_1,$ ${\cal R}_2,$ and ${\cal R}_3,$ that 
is, extinction, weak survival, and strong survival, along 
with their boundaries. Features of ${\cal R}_2$ are studied
by means of the continuity principle. It is proven that on 
$Z_+$ (a) $\overline{\cal R}_1 \cap \overline{\cal R}_2 \subset 
{\cal R}_1,$ (b) $\overline{\cal R}_1 \cap \overline{\cal R}_3 
\subset {\cal R}_3,$ (c) $\overline{\cal R}_2 \cap 
\overline{\cal R}_3 \subset {\cal R}_3,$ and on ${\cal T}_{d+1}$ 
for $d > 1,$ (d) $\overline{\cal R}_1 \cap \overline{\cal R}_2 
\subset {\cal R}_1,$ (e) $\overline{\cal R}_1 \cap 
\overline{\cal R}_3 \subset {\cal R}_1,$ (f) $\overline{\cal R}_2 
\cap \overline{\cal R}_3 \subset {\cal R}_2.$ In the survival 
region that is free of singularities, the random subset of the 
boundary of the tree consisting of all ends of the tree in which 
the infection survives, called the limit set of the process, is 
shown to have a Hausdorff dimension that never exceeds a certain 
real multiple of the Hausdorff dimension of the entire geometric 
boundary. In addition, there is strict inequality at the 
discontinuity set, which traverses the weak and strong survival 
regions, except when the contact process has identical backwards 
infection rates. Perhaps surprisingly, this shows that the latter
rates determine whether the exponential rates of decay of the 
hitting probability functions at infinity are equal regardless 
of the choice of the forward infection rates. Furthermore, the 
distribution of the limiting points of the process is investigated 
as well as the exponential rate of growth in space-time on the 
event of survival and the exponential rate $\eta$ of decay in 
time $t$  of the probability that the initial infected site is 
infected at time $t.$  

hueter@math.ufl.edu

1389. ASYMMETRIC CONTACT PROCESS ON A TREE: ISOTROPIC BACKTRACKING

Irene Hueter

We consider the asymmetric anisotropic contact process on a 
homogeneous tree ${\cal T}_{d+1}$ of degree $d+1 \geq 2$
that has identical backwards infection rates. It was
previously proven by the author that isotropic backtracking 
of the infection leads to equality of the exponential rates 
$\beta_i$ of decay of the hitting probability functions at 
infinity, regardless of the choice of the forward infection 
rates. The main point here is to highlight the interplay 
between the phase transitions and the discontinuity set,
which traverses the weak and strong survival regions, for 
this special yet elucidating example, which perhaps 
surprisingly carries formulae that bridge to the infection 
rates. Specifically, if $t_*$ denotes the square root of the
ratio of the backwards infection rate and the sum of the 
forward infection rates, then the value of the function 
$\beta$ is shown to equal $\min(1/d, t_*)$ at the transition
to extinction and to equal $t_*$ at the discontinuity set 
unless the latter is hidden in the strong survival phase.

hueter@math.ufl.edu

1390. GUES AND QUEUES

Yuliy Baryshnikov

Consider an infinite series of identical queues, all 
buffers but the  first one initally empty, at the first 
queue infinitely many identical customers. Let l(i,j) 
be the time when the i-th customer leaves j-th server. 
The limiting process 

    D(i)=lim (l(i,j)-e j)/sqrt(v j),

(here e is the expectation and v the variance of the
service time) was studied by Glynn and Whitt. It describes
inter alia also the behavior of the totally asymmetric 
exclusion process `near the front edge' for initial 
configuration

    ...11110000... 

The main result of this paper is that the process 
D(i) has the law of the process of the largest 
eigenvalues of the main minors of an infinite random 
Hermitean matrix drawn from Gaussian Unitary Ensemble.

baryshnikov@eurandom.tue.nl

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

1391. PATHWISE UNIQUENESS FOR PURE JUMP STOCHASTIC DIFFERENTIAL EQUATIONS

Richard Bass

We consider pure jump stochastic differential equations
driven by compensated Poisson point processes. Under an 
integrability assumption that guarantees the paths will 
be of bounded variation, we show that pathwise uniqueness 
holds whenever weak uniqueness holds. We also give an analog
of the Yamada-Watanabe condition for pathwise uniqueness. 
This says that under a smoothness condition on the 
coefficients pathwise uniqueness holds.

bass@math.uconn.edu

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

1392. FROM BRUNN-MINKOWSKI TO BRASCAMP-LIEB AND TO LOGARITHMIC SOBOLEV INEQUALITIES

Sergey Bobkov and Michel Ledoux

We develop several applications of the Brunn-Minkowki inequality
in the Pr\'ekopa-Leindler form. In particular, we show that an argument of B. Maurey
may be adapted to deduce from the Pr\'ekopa-Leindler theorem
the Brascamp-Lieb inequality for stricly convex potentials.
We deduce similarly the logarithmic Sobolev inequality for uniformly
convex potentials for which we deal more generally with arbitrary norms and
obtain some new results in this context. Applications to transportation
cost and  to concentration on uniformly convex bodies complete the exposition.

ledoux@cict.fr

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

1393. ORNSTEIN-ZERNIKE THEORY FOR THE BERNOULLI BOND PERCOLATION ON Z^d

Massimo Campanino and Dmitry Ioffe

We derive precise Ornstein-Zernike asymptotic formula for the decay of the
two-point function $\Pp (0\lra x)$ of the Bernoulli bond percolation
on the integer lattice $\Zd$ in any dimension $d\geq 2$, in
any direction $x$   and for any
sub-critical value
of  $p < p_c (d)$.

campanin@dm.unibo.it ieioffe@ie.technion.ac.il

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

1394. UNIQUENESS IN LAW FOR THE ALLEN-CAHN SPDE VIA CHANGE OF MEASURE

Hassan Allouba

We start by first using change of measure to prove the transfer of 
uniqueness in law among pairs of parabolic SPDEs differing only by
a drift function, under an almost sure $L^2$ condition on the 
drift/diffusion ratio.  This is a considerably weaker condition than 
the usual Novikov one, and it allows us to prove uniqueness in law for
the Allen-Cahn SPDE driven by space-time white noise with diffusion 
function $a(t,x,u)=Cu^\gamma$, $1/2<\gamma\le1$ and $C\ne0$.  The same 
transfer result is also valid for ordinary SDEs and hyperbolic SPDEs.

allouba@math.umass.edu

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

1395. BROWNIAN-TIME PROCESSES: THE PDE CONNECTION AND THE HALF-DERIVATIVE GENERATOR

Hassan Allouba and Weian Zheng

We introduce a class of interesting stochastic processes based on Brownian-time
processes.  These are obtained by taking Markov processes and replacing the time 
parameter with the modulus of Brownian motion.  They generalize the iterated 
Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they 
introduce new interesting examples.  After defining Brownian-time processes, we 
relate them to fourth order parabolic PDEs.  We then study their exit problem  
as they exit nice domains in $\Rd$, and connect it to elliptic PDEs.  We show that 
these processes, including Burdzy's IBM, have the peculiar property that they 
solve fourth order parabolic PDEs, but their exit distribution solves the usual 
second order Dirichlet problem.  We recover fourth order PDEs in the elliptic setting 
by encoding the iterative nature of the Brownian time process, through its exit time, 
in a standard Brownian motion.  We also show that it is possible to assign a formal 
generator to these non-Markovian processes by giving such a generator in the 
half-derivative sense. 

allouba@math.umass.edu  wzheng@math.uci.edu

1396. BOUNDARY CROSSINGS AND THE DISTRIBUTION FUNCTION OF THE MAXIMUM OF BROWNIAN SHEET

E. Csaki, D. Khoshnevisan & Z. Shi 

Our main intention is to describe the behavior of the 
(cumulative) distribution function of the random variable
$M_{0,1} := \sup_{0\le s,t\le 1} W(s,t)$ near $0$, where $W$ denotes
one--dimensional, two--parameter Brownian sheet.
A remarkable result of Florit and Nualart
asserts that $M_{0,1}$ has a smooth density
function with respect to Lebesgue's measure.
Our estimates, in turn, seem to imply that the behavior of 
the density function of $M_{0,1}$ near 0 is quite exotic and,
in particular, there is no clear-cut notion of a two--parameter 
reflection principle.

We also consider the supremum of Brownian sheet over rectangles
that are away from the origin. We apply our estimates to get an
infinite dimensional analogue of Hirsch's theorem for Brownian 
motion. 

davar@math.utah.edu  (Available in Pdf and Postscript)

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

1397. CAPACITY ESTIMATES FOR BOUNDARY CROSSINGS OF THE ORNSTEIN-UHLENBECK PROCESS IN WIENER SPACE

E. Csaki, D. Khoshnevisan & Z. Shi

Let $T_1$ denote the first passage time to $1$ of
a standard Brownian motion. It is well known that
as $\lambda\to\infty$, 
\[
   \P\{ T_1 > \lambda\} \sim c \lambda^{-1/2},
\]
where $c=(2/\pi)^{1/2}$. The goal of this note is to
establish a quantitative, infinite dimensional version
of this result. Namely, we will prove the existence of
positive and finite constants $K_1$ and $K_2$ such that for
all $\lambda>e^e$, 
\[
   K_1 \lambda^{-1/2} \le \mathsf{Cap}\{ T_1 > \lambda\} 
        \le K_2 \lambda^{-1/2} \log^3(\lambda) \cdot \log \log (\lambda),
\]
where `$\log$' denotes the natural logarithm, and 
$\mathsf{Cap}$ is the Fukushima--Malliavin capacity on the
space of continuous functions.

davar@math.utah.edu  (Available in Pdf and Postscript)

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

1398. CONTINUITY OF SOME $l_p$-VALUED GAUSSIAN RANDOM FUNCTIONS

X.Fernique

Let $X = \{X_n(t), t\in\bf {R}, n\in\bf{N}\}$ be a $\bf{R^N}$-valued 
stationary Gaussian random function with independent components, 
we studie the existence of a modification of X with continuous paths 
in $l_p, 1\le p < \infty$ ; we present a necessary condition ; 
this condition is also sufficient if the laws of the components are 
Ornstein-Uhlenbeck ; we present also a characterization assuming 
$p \le 2$ and the metrics associated with the laws of the components 
to be nondecreasing ; finally for $p = 2$, we purpose a general 
conjecture of characterization.

fernique@math.u-strasbg.fr

1399. SOME PROPERTIES OF THE ARC SINE LAW RELATED TO ITS INVARIANCE UNDER A FAMILY OF RATIONAL MAPS

Jim Pitman and Marc Yor

This paper shows how the invariance of the arc sine distribution on (0,1) 
under a family of rational maps is related on the one hand to various integral 
identities with probabilistic interpretations involving random variables 
derived from Brownian motion with arc sine, Gaussian, Cauchy and other 
distributions, and on the other hand to results in the analytic theory of 
iterated rational maps. 

pitman@stat.berkeley.edu

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

1400. ON THE DISTRIBUTION OF RANKED HEIGHTS OF EXCURSIONS OF A BROWNIAN BRIDGE

Jim Pitman and Marc Yor

The distribution of the sequence of ranked maximum and minimum values attained 
during excursions of a standard Brownian bridge is described.  The height of 
the jth highest maximum M_j over a positive excursion of the bridge has the 
same distribution as M_1/j, where the distribution of M_1 is given by Levy's 
formula P( M_1 > x ) = e^{-2x^2}.  The probability density of the height of 
the jth highest maximum of excursions of the reflecting Brownian bridge 
is given by a modification of the known theta-function series for the 
density of the maximum absolute value of the bridge.  These results are 
obtained from a more general description of the distribution of ranked values 
of a homogeneous functional of excursions of the standardized bridge of a 
self-similar recurrent Markov process.

pitman@stat.berkeley.edu

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

1401. TWO COALESCENTS DERIVED FROM THE RANGES OF STABLE SUBORDINATORS

Jean Bertoin and Jim Pitman

Let M_a be the closure of the range of a stable subordinator of
index a between 0 and 1.
There are two natural constructions of the M_a's simultaneously for
all a so that M_a is a subset of M_b for 0< a < b < 1: one based on the 
intersection of independent regenerative sets and one based on Bochner's 
subordination.  We compare the corresponding two coalescent processes defined 
by the lengths of complementary intervals of [0,1]\M_{1-r} for 0 < r < 1.
In particular, we identify the coalescent based on the subordination scheme
with the coalescent recently introduced by Bolthausen and Sznitman.

jbe@ccr.jussieu.fr     pitman@stat.berkeley.edu

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

1402. PROBABILITY LAWS RELATED TO THE JACOBI THETA AND RIEMANN ZETA FUNCTIONS, AND BROWNIAN EXCURSIONS

Philippe Biane and Jim Pitman and Marc Yor

This paper reviews known results which connect Riemann's integral 
representations of his zeta function, involving Jacobi's theta function and 
its derivatives, to some particular probability laws governing sums
of independent exponential variables. These laws are related to one-dimensional Brownian motion and  to higher dimensional Bessel processes. We present some 
characterizations of these probability laws, and some approximations of 
Riemann's zeta function which are related to these laws.

Philippe.Biane@ens.fr   pitman@stat.berkeley.edu

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