Probability Abstracts 73
This document contains abstracts 2145-2176.
They have been mailed on February 27, 2003.
2145. ROTATIONS AND TANGENT PROCESSES ON WIENER SPACE
M. Zakai
The paper considers (a) Representations of measure preserving transformations
(``rotations'') on Wiener space, and (b) The stochastic calculus of variations
induced by parameterized rotations $\{T_\theta w, 0 \le \theta \le \eps\}$:
``Directional derivatives'' $(dF(T_\theta w)/d \theta)_{\theta=0}$, ``vector
fields'' or ``tangent processes'' $(dT_\theta w /d\theta)_{\theta=0}$ and flows
of rotations.
zakai@ee.technion.ac.il
2146. RANDOM WALKS ON SUPERCRITICAL PERCOLATION CLUSTERS
Martin T. Barlow
We obtain Gaussian upper and lower bounds on the transition density
$q_t(x,y)$ of the continuous time simple random walk on a supercritical
percolation cluster $C_\infty$ in the Euclidean lattice. The bounds, analogous
to Aronsen's bounds for uniformly elliptic divergence form diffusions, hold
with constants $c_i$ depending only on $p$ (the percolation probability) and
$d$. The irregular nature of the medium means that the bound for $q_t(x,\cdot)$
only holds for $t$ greater than $S_x(\omega)$, where the constant $S_x(\omega)$
depends on the percolation configuration $\omega$.
barlow@math.ubc.ca
2147. A NOISY SYSTEM WITH A FLATTENED HAMILTONIAN AND MULTIPLE TIME SCALES
Natella V. O'Bryant
We consider a two-dimensional weakly dissipative dynamical system with
time-periodic drift and diffusion coefficients. The average of the drift is
governed by a degenerate Hamiltonian whose set of critical points has an
interior. The dynamics of the system is studied in the presence of three time
scales. Using the martingale problem approach and separating the time scales,
we average the system to show convergence to a Markov process on a stratified
space. The averaging combines the deterministic time averaging of periodic
coefficients, and the stochastic averaging of the resulting system. The
corresponding strata of the reduced space are a two-sphere, a point and a line
segment. Special attention is given to the description of the domain of the
limiting generator, including the analysis of the gluing conditions at the
point where the strata meet. These gluing conditions, resulting from the
effects of the hierarchy of time scales, are similar to the conditions on the
domain of skew Brownian motion and are related to the description of spider
martingales.
nobryant@math.uci.edu
2148. A SYSTEM OF DIFFERENTIAL EQUATIONS FOR THE AIRY PROCESS
Craig A. Tracy and Harold Widom
The Airy process is characterized by its finite-dimensional distribution
functions. We show that each finite-dimensional distribution function is
expressible in terms of a solution to a system of differential equations.
tracy@math.ucdavis.edu
2149. GIRSANOV THEOREM FOR FILTERED POISSON PROCESSES
L. Decreusefond and N. Savy
Shot-noise and fractional Poisson processes are instances of filtered Poisson
processes. We here prove Girsanov theorem for this kind of processes and give
an application to an estimate problem.
laurent.decreusefond@enst.fr
2150. STOCHASTIC INTEGRATION WITH RESPECT TO VOLTERRA PROCESSES
L. Decreusefond
We construct the basis of a stochastic calculus for so-called Volterra
processes, i.e., processes which are defined as the stochastic integral of a
time-dependent kernel with respect to a standard Brownian motion. For these
processes which are natural generalization of fractional Brownian motion, we
construct a stochastic integral and show some of its main properties:
regularity with respect to time and kernel, transformation under an absolutely
continuous change of probability, possible approximation schemes and Ito
formula.
laurent.decreusefond@enst.fr
2151. SUPERCRITICAL MULTITYPE BRANCHING PROCESSES IN CONTINUOUS TIME: THE
ANCESTRAL TYPES OF TYPICAL INDIVIDUALS
Hans-Otto Georgii and Ellen Baake
For supercritical multitype branching processes in continuous time, we
investigate the evolution of types along those lineages that survive up to some
time t. We establish almost-sure convergence theorems for both time and
population averages of ancestral types (conditioned on non-extinction), and
identify the mutation process describing the type evolution along typical
lineages. An important tool is a representation of the family tree in terms of
a suitable size-biased tree with trunk. As a by-product, this representation
allows a `conceptual proof' (in the sense of Kurtz, Lyons, Pemantle, Peres
1997) of the continuous-time version of the Kesten-Stigum theorem.
ebaake@uni-greifswald.de
2152. THE CLAIRVOYANT DEMON HAS A HARD TASK
Peter Gacs
For some m \ge 4, let us color each column of the integer lattice L = Z^2
independently and uniformly into one of m colors. We do the same for the rows,
independently from the columns. A point of L will be called blocked if its row
and column have the same color. We say that this random configuration
percolates if there is a path in L starting at the origin, consisting of
rightward and upward unit steps, and avoiding the blocked points. As a problem
arising in distributed computing, it has been conjectured that for m \ge 4, the
configuration percolates with positive probability. This has now been proved
(in a later paper), for large m. Here, we prove that the probability that there
is percolation to distance n but not to infinity is not exponentially small in
n. This narrows the range of methods available for proving the conjecture.
gacs@cs.bu.edu
2153. BALLISTIC RANDOM WALKS IN RANDOM ENVIRONMENT AT LOW DISORDER
Christophe Sabot
We consider random walks in a random environment of the type
$p_0+\gamma\w_z$, where $p_0$ denotes the transition probabilities of a
stationary random walk on $\BZ^d$, to nearest neighbors, and $\w_z$ is an iid
random perturbation. We give an explicit development, for small $\gamma$, of
the asymptotic speed of the random walk under the annealed law, up to order 2.
As an application, we construct, in dimension $d\ge 2$, a walk which goes
faster than the stationary walk under the mean environment.
sabot@ccr.jussieu.fr
2154. GIRSANOV'S TRANSFORMATION FOR SLE(KAPPA,RHO) PROCESSES, INTERSECTION
EXPONENTS AND HIDING EXPONENTS
Wendelin Werner
We relate the formulas giving Brownian (and other) intersection exponents to
the absolute continuity relations between Bessel process of different
dimensions, via the two-parameter family of Schramm-Loewner Evolution processes
SLE(kappa,rho) introduced in arXiv:math.PR/0209343. This allows also to compute
the value of some new exponents (``hiding exponents'') related to SLEs and
planar Brownian motions.
wendelin.werner@math.u-psud.fr
2155. HYDRODYNAMIC LIMIT OF A DISORDERED LATTICE GAS
A.Faggionato and F.Martinelli
We consider a model of lattice gas dynamics in the d-dimensional cubic
lattice in the presence of disorder. If the particle interaction is only mutual
exclusion and if the disorder field is given by i.i.d. bounded random
variables, we prove the almost sure existence of the hydrodynamical limit in
dimension d>2. The limit equation is a non linear diffusion equation with
diffusion matrix characterized by a variational principle.
fmartin@eecs.berkeley.edu
2156. NEGATIVE ASSOCIATION IN UNIFORM FORESTS AND CONNECTED GRAPHS
G. R. Grimmett and S. N. Winkler
We consider three probability measures on subsets of edges of a given finite
graph $G$, namely those which govern, respectively, a uniform forest, a uniform
spanning tree, and a uniform connected subgraph. A conjecture concerning the
negative association of two edges is reviewed for a uniform forest, and a
related conjecture is posed for a uniform connected subgraph. The former
conjecture is verified numerically for all graphs $G$ having eight or fewer
vertices, or having nine vertices and no more than eighteen edges, using a
certain computer algorithm which is summarised in this paper. Negative
association is known already to be valid for a uniform spanning tree. The three
cases of uniform forest, uniform spanning tree, and uniform connected subgraph
are special cases of a more general conjecture arising from the random-cluster
model of statistical mechanics.
g.r.grimmett@statslab.cam.ac.uk
2157. INTEGRALS, PARTITIONS, AND CELLULAR AUTOMATA
Alexander E. Holroyd, Thomas M. Liggett, Dan Romik
We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{\pi^2}{3ab}$$ where $f(x)$
is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When
$a$ is an integer and $b=a+1$ we deduce several combinatorial results. These
include an asymptotic formula for the number of integer partitions not having
$a$ consecutive parts, and a formula for the metastability thresholds of a
class of threshold growth cellular automaton models related to bootstrap
percolation.
holroyd@Math.Berkeley.EDU
2158. REFLECTED PLANAR BROWNIAN MOTIONS, INTERTWINING RELATIONS AND CROSSING
PROBABILITIES
Julien Dubedat
Prompted by an example arising in critical percolation, we study some
reflected Brownian motions in symmetric planar domains and show that they are
intertwined with one-dimensional diffusions. In the case of a wedge, the
reflected Brownian motion is intertwined with the 3-dimensional Bessel process.
This implies some simple hitting distributions and sheds some light on the
formula proposed by Watts for double-crossing probabilities in critical
percolation.
dubedat@clipper.ens.fr
2159. JUGGLING PROBABILITIES
Gregory S. Warrington
The act of a person juggling can be viewed as a Markov process if we assume
that the juggler throws to random heights. I make this association for the
simplest reasonable model of random juggling and compute the steady state
probabilities in terms of the Stirling numbers of the second kind. I also
explore several alternate models of juggling.
warrington@math.umass.edu
2160. EXCITED RANDOM WALK
Itai Benjamini and David B. Wilson
A random walk on Z^d is excited if the first time it visits a vertex there is
a bias in one direction, but on subsequent visits to that vertex the walker
picks a neighbor uniformly at random. We show that excited random walk on Z^d,
is transient iff d>1.
dbwilson@microsoft.com
2161. A PDE FOR THE JOINT DISTRIBUTIONS OF THE AIRY PROCESS
Mark Adler and Pierre van Moerbeke
In this paper, we answer a question posed by Kurt Johansson, to find a PDE
for the joint distribution of the Airy Process. The latter is a continuous
stationary process, describing the motion of the outermost particle of the
Dyson Brownian motion, when the number of particles get large, with space and
time appropriately rescaled. The question reduces to an asymptotic analysis on
the equation, governing the joint probability of the eigenvalues of coupled
Gaussian Hermitian matrices.
vanmoerbeke@geom.ucl.ac.be
2162. MOMENT GENERATING FUNCTION OF THE INVERSE OF INTEGRAL
OF GEOMETRIC BROWNIAN MOTION
Kyounghee Kim
In this paper we obtain the integral form of the moment genetationg function
of the inverse of the random variable defined by
$ A^{(\nu)}_t := \int ^t_0 exp (2B_s + 2 \nu s) ds$, where ($B_s : s>0$)
is a one dimensional Brownian motion starting from 0. In case $\nu = 1$,
the moment generation function has a particularly simple form.
kimkh@indiana.edu
2163. THE SELF-SIMILAR AND MULTIFRACTAL NATURE OF A NETWORK TRAFFIC MODEL
K. Maulik, S.I. Resnick
We look at a family of models for Internet traffic with
increasing input rates and consider approximation models
which exhibit self-similarity at large time scales and
multifractality at small time scales. Depending on whether
the input rate is fast or slow, the total cumulative input
traffic can be approximated by a self-similar stable motion
or a self-similar Gaussian process. The stable limit does
not depend on the behavior of the individual transmission
schedules but the Gaussian limit does. Also, the models and
their approximations show multifractal behavior at small
time scales.
maulik@eurandom.tue.nl sid@orie.cornell.edu
- To see a preprint or other
information provided by the author
click here.
2164. LENSES IN SKEW BROWNIAN FLOW
Krzysztof Burdzy and Haya Kaspi
We consider a stochastic flow in which individual particles follow skew
Brownian motions, with each one of these processes driven by the same
Brownian motion. Due to the lack of the simultaneous strong uniqueness for
the whole system of stochastic differential equations, the flow contains
lenses, i.e., pairs of skew Brownian motions which start at the same
point, bifurcate, and then coalesce in a finite time. The paper contains
qualitative and quantitative (distributional) results on the geometry of
the flow and lenses.
burdzy@math.washington.edu iehaya@tx.technion.ac.il
- To see a preprint or other
information provided by the author
click here.
2165. ON A CLASS OF GENEALOGICAL AND INTERACTING METROPOLIS MODELS
Pierre Del Moral and Arnaud Doucet
A genealogical tree based particle model for drawing
approximate samples from the conditional path-distributions
of a Markov chain with respect to its terminal values is
presented. This path-particle evolution model can be
interpreted as the historical process associated to a
sequence of interacting Metropolis Markov chains.
This novel class of interacting models can also be
used to obtain approximate samples from a given target
distribution which is only known up to a normalizing
constant. We design an original Feynman-Kac modeling
technique for studying the asymptotic analysis of these
path-particle and Metropolis type simulation models.
We provide precise convergence results as the time or the
size of the systems tends to infinity.
In contrast to the traditional Metropolis model we show
that the decays to equilibrium do not depend on the nature
of the desired limiting distribution.
delmoral@stat.purdue.edu ad2@eng.cam.ac.uk
- To see a preprint or other
information provided by the author
click here.
2166. m-ORDER INTEGRALS AND GENERALIZED ITO's FORMULA;
THE CASE OF A FRACTIONAL BROWNIAN MOTION WITH ANY
HURST INDEX
Mihai Gradinaru, Ivan Nourdin, Francesco Russo and Pierre Vallois
Given an integer m, a probability measure $\nu$ on [0,1],
a process X and a real function g, we define the m-order
$\nu$-integral having as integrator X and as integrand g(X).
In the case of the fractional Brownian motion $B^H$ (H being
the Hurst index), for any locally bounded function g,
the corresponding integral vanishes for all odd indices m > 1/ 2H and
any symmetric $\nu$. One consequence is a Itô-Stratonovich type expansion
for the fractional Brownian motion with arbitrary Hurst index H
belonging in ]0,1[.
On the other hand we show that the classical Itô-Stratonovich
formula holds if and only if H > 1/6.
russo@math.univ-paris13.fr gradinar@iecn.u-nancy.fr
nourdin@iecn.u-nancy.fr vallois@iecn.u-nancy.fr
- To see a preprint or other
information provided by the author
click here.
2167. SELF-INTERSECTION LOCAL TIME: CRITICAL EXPONENT,
LARGE DEVIATIONS, AND LAWS OF THE ITERATED LOGARITHM
Richard F. Bass and Xia Chen
If $\beta_t$ is renormalized self-intersection local time for
planar Brownian motion, we characterize when
$\E e^{\gamma \beta_1}$
is finite or infinite in terms of the best constant of
a Gagliardo-Nirenberg inequality. We obtain large deviations
estimates for $\beta_1$ and $-\beta_1$. We establish
limsup and liminf laws of the iterated
logarithm for $\beta_t$ as $t\to \infty$.
bass@math.uconn.edu xchen@math.utk.edu
- To see a preprint or other
information provided by the author
click here.
2168. A LINEARIZED KURAMOTO-SIVASHINSKY PDE
VIA AN IMAGINARY-BROWNIAN-TIME-BROWNIAN-ANGLE PROCESS
Hassan Allouba
We introduce a new imaginary-Brownian-time-Brownian-angle process,
which we also call the linear-Kuramoto-Sivashinsky process (LKSP).
Building on our techniques in two recent articles involving the
connection of Brownian-time processes to fourth order PDEs, we give
an explicit solution to a linearized Kuramoto-Sivashinsky
PDE in $d$-dimensional space:
$u_t=-\frac18\Delta^2u-\frac12\Delta u-\frac12u$.
The solution is given in terms of a functional of our LKSP.
allouba@math.kent.edu
- To see a preprint or other
information provided by the author
click here.
2169. HOMOGENIZATION ON FINITELY RAMIFIED FRACTALS
Takashi Kumagai
Let $X_t$ be a continuous time Markov chain on some finitely ramified fractal
graph given by putting i.i.d. random resistors on each cell. We prove that under
an assumption that a renormalization map of resistors has a non-degenerate
fixed point, $\alpha^{-n}X_{\tau^nt}$ converges in law to a non-degenerate
diffusion process on the fractal as $n\to\infty$, where $\alpha$ is a spatial
scale and $\tau$ is a time scale of the fractal.
Especially, when the fixed point of the renormalization map is unique, the diffusion
is a constant time change of Brownian motion on the fractal. These results
improve and extend our former results in Kumagai-Kusuoka (PTRF, 1996).
kumagai@kurims.kyoto-u.ac.jp
- To see a preprint or other
information provided by the author
click here.
2170. NORMED CONVERGENCE PROPERTY FOR HYPERGROUPS ADMITTING
AN INVARIANT MEASURE
C. R. E. Raja
We prove that a hypergroup admitting a countable basis and an invariant
Haar measure has normed convergence property if and only if it is compact.
creraja@isibang.ac.in
2171. LIMITING SHAPE FOR DIRECTED PERCOLATION MODELS
James B. Martin
We consider directed first-passage and last-passage percolation
on the non-negative lattice $Z_+^d$, $d\geq 2$, with i.i.d. weights
at the vertices. Under certain moment conditions on the common distribution
of the weights, the limits $g(x)=\lim_{n\to\infty} n^{-1} T(\lfloor n x \rfloor)$
exist and are constant a.s. for $x \in R_+^d$, where $T(z)$ is the passage
time to the vertex $z \in Z_+^d$. We show that this "shape function" $g$
is continuous on $R_+^d$, in particular at the boundaries. In two dimensions,
we give more precise asymptotics for the behaviour of $g$ near the boundaries;
these asymptotics depend on the common weight distribution only through
its mean and variance. In addition we discuss growth models which are naturally
associated to the percolation processes, giving a shape theorem and
illustrating various possible types of behaviour with output from simulations.
martin@liafa.jussieu.fr
2172. INFINITE SYSTEMS OF NON-COLLIDING BROWNIAN PARTICLES
Makoto Katori, Taro Nagao and Hideki Tanemura
Non-colliding Brownian particles in one dimension is studied.
$N$ Brownian particles start from the origin at time 0 and then
they do not collide with each other until finite time $T$. We
derive the determinantal expressions for the multitime correlation
functions using the self-dual quaternion matrices. We consider the
scaling limit of the infinite particles $N \to \infty$ and the
infinite time interval $T \to \infty$. Depending on the scaling,
two limit theorems are proved for the multitime correlation functions,
which may define temporally inhomogeneous infinite particle systems.
katori@phys.chuo-u.ac.jp
2173. A POWER LAW FOR THE FREE ENERGY IN TWO DIMENSIONAL PERCOLATION
Yu Zhang
Consider bond percolation on the square lattice and site percolation
on the triangular lattice. Let $\kappa(p)$ be the free energy at the
zero field. If we assume the existence of the critical exponents for
the three arm and four arm paths and these critical exponents are
-2/3 and -5/4, respectively, then we can show the following power law
for the free energy function
$\kappa(p)$: {eqnarray*} &&\kappa'''(p)= +(1/2-p)^{-1/3+\delta(|1/2-p|)}
{for} p < 1/2 &&\kappa'''(p)= -(1/2-p)^{-1/3+\delta(|1/2-p|)}
{for} p > 1/2, {eqnarray*} where $\delta(x)$ goes to zero as $x\to 0$.
Note that the critical exponents for four arm and three arm paths indeed
are proven to exist and equal -5/4 and -2/3 on the triangular lattice
and the above power law for $\kappa(p)$ therefore holds for the triangular
lattice. Note that the above power law for $\kappa(p)$ implies $\kappa(p)$
is not third differentiable at the critical point of the triangular
lattice. This answers a long time conjecture that $\kappa(p)$ has
a singularity at 1/2 since 1964 affirmatively.
yzhang@math.uccs.edu
2174. ISOPERIMETRY AND HEAT KERNEL DECAY ON PERCOLATION CLUSTERS
Pierre Mathieu and Elisabeth Remy
We prove that the heat kernel on the infinite Bernoulli percolation
cluster in Z^d almost surely decays faster than t^{-d/2}. We also
derive estimates on the mixing time for the random walk confined
to a finite box. Our approach is based on local isoperimetric inequalities.
pierre.mathieu@cmi.univ-mrs.fr
2175. SCALING LIMIT, NOISE, STABILITY
Boris Tsirelson
Linear functions of many independent random variables lead to
classical noises (white, Poisson, and their combinations) in the
scaling limit. Some singular stochastic flows and some models
of oriented percolation involve very nonlinear functions and
lead to nonclassical noises. Two examples are examined, Warren's
`noise made by a Poisson snake' and the author's `Brownian web
as a black noise'. Classical noises are stable, nonclassical are not.
A new framework for the scaling limit is proposed. Old and new
results are presented about noises, stability, and spectral measures.
tsirel@math.tau.ac.il
2176. COUPLINGS OF UNIFORM SPANNING FORESTS
Lewis Bowen
We prove the existence of an automorphism-invariant coupling
for the wired and the free uniform spanning forests on Cayley graphs
of finitely generated residually amenable groups.
lbowen@math.ucdavis.edu
