Probability Abstracts 72
This document contains abstracts 2094-2144.
They have been mailed on December 30, 2002.
2094. POISSON-KINGMAN PARTITIONS
Jim Pitman
This paper presents some general formulas for random partitions of a finite
set derived by Kingman's model of random sampling from an interval partition
generated by subintervals whose lengths are the points of a Poisson point
process. These lengths can be also interpreted as the jumps of a subordinator,
that is an increasing process with stationary independent increments. Examples
include the two-parameter family of Poisson-Dirichlet models derived from the
Poisson process of jumps of a stable subordinator. Applications are made to the
random partition generated by the lengths of excursions of a Brownian motion or
Brownian bridge conditioned on its local time at zero.
pitman@stat.berkeley.edu
2095. DIFFERENTIAL OPERATORS AND SPECTRAL DISTRIBUTIONS
OF INVARIANT ENSEMBLES FROM THE CLASSICAL ORTHOGONAL
POLYNOMIALS. PART I: THE CONTINUOUS CASE
Michel Ledoux
Following the investigation by U. Haagerup and
S. Thorbjornsen, we present a simple differential approach
to the limit theorems for spectral distributions of complex
random matrices from the Gaussian, Laguerre and Jacobi Unitary Ensembles.
Using the framework of abstract Markov diffusion
operators, we derive in a direct way differential equations for
Laplace transforms and recurrence equations for moments. These recurrence
relations are then used to describe sharp, non
asymptotic, bounds on the largest eigenvalues at the rate given
by the Tracy-Widom distribution.
ledoux@math.ups-tlse.fr
- To see a preprint or other
information provided by the author
click here.
2096. ONSAGER RELATIONS AND EULERIAN HYDRODYNAMICS FOR SYSTEMS WITH SEVERAL
CONSERVATION LAWS
Balint Toth, Benedek Valko
We present the derivation of the hydrodynamic limit under Eulerian scaling
for a general class of one-dimensional interacting particle systems with two or
more conservation laws. Following Yau's relative entropy method it turns out
that in case of more than one conservation laws, in order that the system
exhibit hydrodynamic behaviour, some particular identities reminiscent of
Onsager's reciprocity relations must hold. We check validity of these
identities for a wide class of models. It also follows that, as a general rule,
the equilibrium thermodynamic entropy (as function of the densities of the
conserved variables) is a globally convex Lax entropy of the hyperbolic systems
of conservation laws arising as hydrodynamic limit. The Onsager relations
arising in this context and its consequences seem to be novel. As concrete
examples we also present a number of models modeling deposition (or domain
growth) phenomena.
balint@renyi.hu
2097. MIXING TIME OF THE RUDVALIS SHUFFLE
David Bruce Wilson
We extend a technique for lower-bounding the mixing time of card-shuffling
Markov chains, and use it to bound the mixing time of the Rudvalis Markov
chain, as well as two variants considered by Diaconis and Saloff-Coste. We show
that in each case Theta(n^3 log n) shuffles are required for the permutation to
randomize, which matches (up to constants) previously known upper bounds. In
contrast, for the two variants, the mixing time of an individual card is only
Theta(n^2) shuffles.
dbwilson@microsoft.com
2098. LINEAR PHASE TRANSITION IN RANDOM LINEAR CONSTRAINT SATISFACTION PROBLEM
David Gamarnik
Our model is a generalized linear programming relaxation of a much studied
random K-SAT problem. Specifically, a set of linear constraints C on K
variables is fixed. From a pool of n variables, K variables are chosen
uniformly at random and a constraint is chosen from C also uniformly at random.
This procedure is repeated m times independently. We ask the following
question: is the resulting linear programming problem feasible? We show that
the feasibility property experiences a linear phase transition, when n diverges
to infinity and m=cn for some constant c. Namely, there exists a critical value
c* such that, when c<c*, the system is feasible or is asymptotically almost
feasible, as n increases, but, when c>c*, the "distance" from feasibility is at
least a positive constant independent of n. Our results are obtained using
powerful local weak convergence methods developed by Aldous and Steele.
gamarnik@watson.ibm.com
2099. ENTROPY AND THE LAW OF SMALL NUMBERS
Ioannis Kontoyiannis, Peter Harremoes, and Oliver Johnson
We introduce two new information-theoretic methods for establishing Poisson
approximation inequalities. Both methods are based on simple subadditivity
properties for appropriately defined versions of the relative entropy and
Fisher information. If $S_n$ is the sum of an (possibly dependent) binary
random variables $X_i$ with $E(X_i)=p_i$ and $E(S_n)=\lambda$, then $$
D(P_{S_n}\|{Po}(\lambda))\leq \sum_{i=1}^n p_i^2 +
\Big[\sum_{i=1}^nH(X_i) - H(X_1,X_2,..., X_n)\Big] $$ where
$D(P_{S_n}\|{Po}(\lambda))$ is the relative entropy between the distribution of
$S_n$ and the Poisson($\lambda$) distribution. The first term in this bound
measures the individual smallness of the $X_i$ and the second term measures
their dependence. This result can be thought of as a ``maximum entropy''
statement: Under suitable conditions, the distribution of $S_n$ converges to
the distribution which has maximal entropy among all those that can be obtained
as sums of independent binary random variables with fixed mean $\lambda$. We
outline a general method for obtaining corresponding bounds when approximating
the distribution of a sum of general discrete random variables by an infinitely
divisible distribution. Second, in the particular case when the $X_i$ are
independent, we obtain a sharper bound in total variation,
$$\left\|P_{S_n}-{Po}(\lambda)\right\|_\stv^2 \leq \frac{2}{\lambda}
\sum_{i=1}^n \frac{p_i^3}{1-p_i}.$$
yiannis@dam.brown.edu
2100. HARD RODS: STATISTICS OF PARKING CONFIGURATIONS
Francois Dunlop, Thierry Huillet
We compute the correlation function in the equilibrium version of R\'enyi's
{\sl parking problem}. The correlation length is found to diverge as
$2^{-1}\pi^{-2}(1-\rho)^{-2}$ when $\rho\nearrow1$ (maximum density) and as
$\pi^{-2}(2\rho-1)^{-2}$ when $\rho\searrow1/2$ (minimum density).
thierry.huillet@ptm.u-cergy.fr
2101. A GROWTH MODEL IN MULTIPLE DIMENSIONS AND THE HEIGHT OF A RANDOM PARTIAL
ORDER
Timo Seppalainen
We introduce a model of a randomly growing interface in multidimensional
Euclidean space. The growth model incorporates a random order model as an
ingredient of its graphical construction, in a way that replicates the
connection between the planar increasing sequences model and the
one-dimensional Hammersley process. We prove a hydrodynamic limit for the
height process, and a limit which says that certain perturbations of the random
surface follow the characteristics of the macroscopic equation. By virtue of
the space-time Poissonian construction, we know the macroscopic velocity
function explicitly up to a constant factor.
seppalai@math.wisc.edu
2102. FIRST ORDER ASYMPTOTICS OF MATRIX INTEGRALS ; A RIGOROUS APPROACH
TOWARDS THE UNDERSTANDING OF MATRIX MODELS
Alice Guionnet
We investigate the limit behaviour of the spectral measures of matrices
following the Gibbs measure for the Ising model on random graphs, Potts model
on random graphs, matrices coupled in a chain model or induced QCD model. For
most of these models, we prove that the spectral measures converge almost
surely and describe their limit via solutions to an Euler equation for
isentropic flow with negative pressure $p(\rho)=-3^{-1}\pi^2 \rho^3$.
alice.guionnet@umpa.ens-lyon.fr
2103. CONCENTRATION OF NORMS AND EIGENVALUES OF RANDOM MATRICES
Mark W. Meckes
We prove concentration results for $\ell_p^n$ operator norms of rectangular
random matrices and eigenvalues of self-adjoint random matrices. The random
matrices we consider have bounded entries which are independent, up to a
possible self-adjointness constraint. Our results are based on an isoperimetric
inequality for product spaces due to Talagrand.
mwm2@po.cwru.edu
2104. VARIATIONAL PRINCIPLE FOR GENERALIZED GIBBSIAN MEASURES
Christof Kuelske, Arnaud Le Ny and Frank Redig
We study the thermodynamic formalism for generalized Gibbs measures, such as
renormalization group transformations of Gibbs measures or joint measures of
disordered spin systems. We first show existence of the relative entropy
density and obtain a familiar expression in terms of entropy and relative
energy for "almost Gibbsian measures" (almost sure continuity of conditional
probabilities). We also describe these measures as equilibrium states and
establish an extension of the usual variational principle. As a corollary, we
obtain a full variational principle for quasilocal measures. For the joint
measures of the random field Ising model, we show that the weak Gibbs property
holds, with an almost surely rapidly decaying translation invariant potential.
For these measures we show that the variational principle fails as soon as the
measures loses the almost Gibbs property. These examples suggest that the class
of weakly Gibbsian measures is too broad from the perspective of a reasonable
thermodynamic formalism.
leny@eurandom.tue.nl
2105. THE SPEED OF BIASED RANDOM WALK ON PERCOLATION CLUSTERS
Noam Berger, Nina Gantert, Yuval Peres
We consider biased random walk on supercritical percolation clusters in
$\Z^2$. We show that the random walk is transient and that there are two speed
regimes: If the bias is large enough, the random walk has speed zero, while if
the bias is small enough, the speed of the random walk is positive.
noam@stat.berkeley.edu
2106. ON THE MULTIRESOLUTION STRUCTURE OF INTERNET TRAFFIC TRACES
Konstantinos Drakakis, Dragan Radulovic
Internet traffic on a network link can be modeled as a stochastic process.
After detecting and quantifying the properties of this process, using
statistical tools, a series of mathematical models is developed, culminating in
one that is able to generate ``traffic'' that exhibits --as a key feature-- the
same difference in behavior for different time scales, as observed in real
traffic, and is moreover indistinguishable from real traffic by other
statistical tests as well. Tools inspired from the models are then used to
determine and calibrate the type of activity taking place in each of the time
scales. Surprisingly, the above procedure does not require any detailed
information originating from either the network dynamics, or the decomposition
of the total traffic into its constituent user connections, but rather only the
compliance of these connections to very weak conditions.
drakakis@princeton.edu
2107. THE DIMENSION OF THE SLE CURVES
Vincent Beffara
Let $\gamma$ be the curve generating a Schramm-Loewner Evolution (SLE)
process, with parameter $\kappa$ positive and different from 4. We prove that,
with probability one, the Hausdorff dimension of $\gamma$ is equal to min(2,
1+\kappa/8).
vincent.beffara@math.u-psud.fr
2108. SCALING LIMIT OF STOCHASTIC DYNAMICS IN CLASSICAL CONTINUOUS SYSTEMS
Martin Grothaus, Yuri G. Kondratiev, Eugene Lytvynov, Michael Roeckner
We investigate a scaling limit of gradient stochastic dynamics associated to
Gibbs states in classical continuous systems on ${\mathbb R}^d, d \ge 1$. The
aim is to derive macroscopic quantities from a given micro- or mesoscopic
system. The scaling we consider has been investigated in \cite{Br80},
\cite{Ro81}, \cite{Sp86}, and \cite{GP86}, under the assumption that the
underlying potential is in $C^3_0$ and positive. We prove that the Dirichlet
forms of the scaled stochastic dynamics converge on a core of functions to the
Dirichlet form of a generalized Ornstein--Uhlenbeck process. The proof is based
on the analysis and geometry on the configuration space which was developed in
\cite{AKR98a}, \cite{AKR98b}, and works for general Gibbs measures of Ruelle
type. Hence, the underlying potential may have a singularity at the origin,
only has to be bounded from below, and may not be compactly supported.
Therefore, singular interactions of physical interest are covered, as e.g. the
one given by the Lennard--Jones potential, which is studied in the theory of
fluids. Furthermore, using the Lyons--Zheng decomposition we give a simple
proof for the tightness of the scaled processes. We also prove that the
corresponding generators, however, do not converge in the $L^2$-sense. This
settles a conjecture formulated in \cite{Br80}, \cite{Ro81}, \cite{Sp86}.
lytvynov@wiener.iam.uni-bonn.de
2109. THE HEAT SEMIGROUP ON CONFIGURATION SPACES
Yuri Kondratiev, Eugene Lytvynov, Michael Roeckner
In this paper, we study properties of the heat semigroup of configuration
space analysis. Using a natural ``Riemannian-like'' structure of the
configuration space $\Gamma_X$ over a complete, connected, oriented, and
stochastically complete Riemannian manifold $X$ of infinite volume, the heat
semigroup $(e^{-tH^\Gamma})_{t\in\R_+}$ was introduced and studied in [{\it J.
Func. Anal.} {\bf 154} (1998), 444--500]. Here, $H^\Gamma$ is the Dirichlet
operator of the Dirichlet form ${\cal E}^\Gamma$ over the space
$L^2(\Gamma_X,\pi_m)$, where $\pi_m$ is the Poisson measure on $\Gamma_X$ with
intensity $m$--the volume measure on $X$. We construct a metric space
$\Gamma_\infty$ that is continuously embedded into $\Gamma_X$. Under some
conditions on the manifold $X$ and we prove that $\Gamma_\infty$ is a set of
full $\pi_m$ measure. The central results of the paper are two types of Feller
properties for the heat semigroup. Next, we give a direct construction of the
independent infinite particle process on the manifold $X$, which is a
realization of the Brownian motion on the configuration space. The main point
here is that we prove that this process can start in every
$\gamma\in\Gamma_\infty$, will never leave $\Gamma_\infty$, and has continuous
sample path in $\Gamma_\infty$, provided $\operatorname{dim}X\ge2$. In this
case, we also prove that this process is a strong Markov process whose
transition probabilities are given by the $\P_{t,\gamma}(\cdot)$ above.
Furthermore, we discuss the necessary changes to be done for constructing the
process in the case $\operatorname{dim}X=1$. Finally, as an easy consequence we
get a ``path-wise'' construction of the independent particle process on
$\Gamma_\infty$ from the underlying Brownian motion.
lytvynov@wiener.iam.uni-bonn.de
2110. MARTIN BOUNDARY THEORY OF SOME QUANTUM RANDOM WALKS
Benoit Collins
In this paper we define a general setting for Martin boundary theory
associated to quantum random walks, and prove a general representation theorem.
We show that in the dual of a simply connected Lie subgroup of U(n), the
extremal Martin boundary is homeomorphic to a sphere. Then, we investigate
restriction of quantum random walks to Abelian subalgebras of group algebras,
and establish a Ney-Spitzer theorem for an elementary random walk on the fusion
algebra of SU(n), generalizing a previous result of Biane. We also consider the
restriction of a quantum random walk on $SU_q(n)$ introduced by Izumi to two
natural Abelian subalgebras, and relate the underlying Markov chains by
classical probabilistic processes. This result generalizes a result of Biane.
benoit.collins@ens.fr
2111. A MONTE CARLO METHOD FOR EXPONENTIAL HEDGING OF CONTINGENT CLAIMS
M. R. Grasselli and T. R. Hurd
Utility based methods provide a very general theoretically consistent
approach to pricing and hedging of securities in incomplete financial markets.
Solving problems in the utility based framework typically involves dynamic
programming, which in practise can be difficult to implement. This article
presents a Monte Carlo approach to optimal portfolio problems for which the
dynamic programming is based on the exponential utility function U(x)=-exp(-x).
The algorithm, inspired by the Longstaff-Schwartz approach to pricing American
options by Monte Carlo simulation, involves learning the optimal portfolio
selection strategy on simulated Monte Carlo data. It shares with the LS
framework intuitivity, simplicity and flexibility.
grasselli@icarus.math.mcmaster.ca
2112. STICKY FLOWS ON THE CIRCLE
Yves Le Jan and Olivier Raimond
The purpose of this note is to give an example of stochastic flows of
kernels, which naturally interpolates between the Arratia coalescing flow
associated with systems of coalescing independent Brownian particles on the
circle and the deterministic diffusion flow (actually, the results are given in
the slightly more general framework of symmetric Levy processes for which
points are not polar). The construction is performed using Dirichlet form
theory and the extension of De Finetti's theorem given in the previous work by
the authors. The sticky flows of kernels are associated with systems of sticky
independent Levy particles on the circle, for some fixed parameter of
stickyness. Some elementary asymptotic properties of the flow are also given.
olivier.raimond@math.u-psud.fr
2113. CRAMER'S ESTIMATE FOR THE EXPONENTIAL FUNCTIONAL OF A LEVY PROCESS
Mejane Olivier
We consider the exponential functional $A_{\infty}=\int_0^{\infty} e^{\xi_s}
ds$ associated to a Levy process $(\xi_t)_{t \geq 0}$. We find the asymptotic
behavior of the tail of this random variable, under some assumptions on the
process $\xi$, the main one being Cramer's condition, that asserts the
existence of a real $\chi >0$ such that ${\Bbb E}(e^{\chi \xi_1})=1$. Then
there exists $C>0$ satisfying, when $t \to +\infty$ : $$ {\Bbb P} (A_{\infty}>
t) \sim C t^{-\chi} \quad . $$ This result can be applied for example to the
process $\xi_t = at - S_{\alpha}(t)$ where $S_{\alpha}$ stands for the stable
subordinator of index $\alpha$ ($0 < \alpha < 1$), and $a$ is a positive real
(we have then $\chi=a^{1/(\alpha -1)}$).
olivier.mejane@math.ups-tlse.fr
2114. HIGHER ORDER EXPANSIONS FOR THE OVERLAP OF THE SK MODEL
X.Bardina, D.Marquez-Carreras, C.Rovira, S.Tindel
In this note, the Sherrington Kirkpatrick model of interacting spins is under
consideration. In the high temperature region, we give an asymptotic expansion
for the expected value of some genereral polynomial of the overlap of the
system when the size $N$ grows to infinity. Some of the coefficients obtained
are shown to be vanishing, while the procedure to get the nontrivial ones has
to be performed by a computer program, due to the great amount of computation
involved.
rovira@mat.ub.es
2115. SUPER-BROWNIAN MOTION WITH EXTRA BIRTH AT ONE POINT
Klaus Fleischmann and Carl Mueller
A super-Brownian motion in two and three dimensions is constructed where
``particles'' give birth at a higher rate, if they approach the origin. Via a
log-Laplace approach, the construction is based on Albeverio et al. (1995) who
calculated the fundamental solutions of the heat equation with one-point
potential in dimensions less than four.
cmlr@math.rochester.edu
2116. TREES AND MATCHINGS FROM POINT PROCESSES
Alexander E. Holroyd and Yuval Peres
A factor graph of a point process is a graph whose vertices are the points of
the process, and which is constructed from the process in a deterministic
isometry-invariant way. We prove that the d-dimensional Poisson process has a
one-ended tree as a factor graph. This implies that the Poisson points can be
given an ordering isomorphic to the usual ordering of the integers in a
deterministic isometry-invariant way. For d \geq 4 our result answers a
question posed by Ferrari, Landim and Thorisson. We prove also that any
isometry-invariant ergodic point process of finite intensity in Euclidean or
hyperbolic space has a perfect matching as a factor graph provided all the
inter-point distances are distinct.
holroyd@math.berkeley.edu
2117. ON A RELATION BETWEEN STOCHASTIC INTEGRATION AND GEOMETRIC MEASURE
THEORY
Franco Flandoli, Mariano Giaquinta, Massimiliano Gubinelli,
Vicenzo M. Tortorelli
Two problems are addressed for the path of certain stochastic processes: a)
do they define currents? b) are these currents of a classical type? A general
answer to question a) is given for processes like semimartingales or with
Lyons-Zheng structure. As to question b), it is shown that H\"{o}lder
continuous paths with exponent $\gamma > 1/2$ define integral flat chains.
mgubi@cibs.sns.it
2118. BOUNDS FOR COVARIANCES AND VARIANCES OF TRUNCATED RANDOM VARIABLES
N. Hemachandra and V. Cheriyan
We show that a lower bound for covariance of $\min(X_1,X_2)$ and
$\max(X_1,X_2)$ is $\cov{X_1}{X_2}$ and an upper bound for variance of \\
$\min(X_2,\max(X,X_1))$ is $\var{X} + \var{X_1} +\var{X_2}$ generalizing
previous results. We also characterize the cases where these bounds are sharp.
nh@me.iitb.ac.in
2119. SLE AND TRIANGLES
Julien Dubedat
By analogy with Carleson's observation on Cardy's formula describing crossing
probabilities for the scaling limit of critical percolation, we exhibit
``privileged geometries'' for Stochastic Loewner Evolutions with various
parameters, for which certain hitting distributions are uniformly distributed.
Consequences for limiting probabilities of events concerning various critical
plane discrete models are then examined.
dubedat@clipper.ens.fr
2120. RANDOMNESS AS AN EQUILIBRIUM. POTENTIAL AND PROBABILITY DENSITY
M. Grendar, Jr., M. Grendar
Randomness is viewed through an analogy between a physical quantity, density
of gas, and a mathematical construct -- probability density. Boltzmann's
deduction of equilibrium distribution of ideal gas placed in an external
potential field than provides a way of viewing probability density from a
perspective of forces/potentials, hidden behind it.
umergren@savba.sk
2121. ASYMPTOTIC FLUX ACROSS HYPERSURFACES FOR DIFFUSION PROCESSES
Andrea Posilicano, Stefania Ugolini
We suggest a rigorous definition of the pathwise flux across the boundary of
a bounded open set for transient finite energy diffusion processes. The
expectation of such a flux has the property of depending only on the current
velocity $v$, the nonsymmetric (as regards time reversibility) part of the
drift. In the case the diffusion has a limiting velocity we then define the
asymptotic (as $R\uparrow\infty$) flux across subsets of the sphere of radius
$R$ and compute its expectation, again in terms of $v$.
andrea.posilicano@uninsubria.it
2122. LIMITING BEHAVIOR OF LIMITING BEHAVIOR OF RELATIVE R\'{E}NYI ENTROPY IN
A NON-REGULAR LOCATION SHIFT FAMILY
Masahito Hayashi
We calculate the limiting behavior of relative Renyi entropy when the first
probability distribution is close to the second one in a non-regular
location-shift family which is generated by a probability distribution whose
support is an interval or a half-line. This limit can be regarded as a
generalization of Fisher information, and plays an important role in large
deviation theory.
masahito@brain.riken.go.jp
2123. AN ASYMPTOTIC EXPANSION FOR THE DISCRETE HARMONIC POTENTIAL
Gady Kozma and Ehud Schreiber
We give two algorithms that allow to get arbitrary precision asymptotics for
the harmonic potential of a random walk.
gadyk@wisdom.weizmann.ac.il
2124. ASYMPTOTICS FOR THE NUMBER OF GROUPS AT THE EQUILIBRIUM OF REVERSIBLE
COAGULATION -FRAGMENTATION PROCESSES: THE CASE OF PARAMETRIC FUNCTION WITH
POLYNOMIAL RATE OF GROWTH
Michael Erlihson, Boris Granovsky
We establish the central limit theorem for the number of groups at the
equilibrium of a coagulation-fragmentation process given by a parameter
function with polynomial rate of growth. The result obtained is compared with
the one for random combinatorial structures obeying the logarithmic condition.
mar18aa@techunix.technion.ac.il
2125. AN OPTIMAL SKOROKHOD EMBEDDING FOR DIFFUSIONS
A. M. G. Cox and D. G. Hobson
Given a Brownian motion $B_t$ and a general target law $\mu$ (not necessarily
centered or even integrable) we show how to construct an embedding of $\mu$ in
$B$. This embedding is an extension of an embedding due to Perkins, and is
optimal in the sense that it simultaneously minimises the distribution of the
maximum and maximises the distribution of the minimum among all embeddings of
$\mu$. The embedding is then applied to regular diffusions, and used to
characterise the target laws for which a $H^p$-embedding may be found.
mapamgc@maths.bath.ac.uk
2126. THE P-SPIN INTERACTION MODEL WITH EXTERNAL FIELD
X. Bardina, D. Marquez, C. Rovira, S. Tindel
This paper is devoted to a detailed study of a p-spins interaction model with
external field, including some sharp bounds on the speed of self averaging of
the overlap as well as a central limit theorem for its fluctuations, the
thermodynamical limit for the free energy and the definition of an
Almeida-Thouless type line. Those results show that the external field
dominates the tendency to disorder induced by the increasing level of
interaction between spins, and our system will share many of its features with
the SK model, which is certainly not the case when the external magnetic field
vanishes.
tindel@math.univ-paris13.fr
2127. THE JACOBI FIELD OF A L\'EVY PROCESS
Yuri M. Berezansky, Eugene Lytvynov, Dmytro A. Mierzejewski
We derive an explicit formula for the Jacobi field that is acting in an
extended Fock space and corresponds to an ($\R$-valued) L\'evy process on a
Riemannian manifold. The support of the measure of jumps in the
L\'evy--Khintchine representation for the L\'evy process is supposed to have an
infinite number of points. We characterize the gamma, Pascal, and Meixner
processes as the only L\'evy processes whose Jacobi field leaves the set of
finite continuous elements of the extended Fock space invariant.
lytvynov@wiener.iam.uni-bonn.de
2128. THE MEAN DISTANCE TO THE N-TH NEIGHBOUR IN A UNIFORM DISTRIBUTION OF
RANDOM POINTS: AN APPLICATION OF PROBABILITY THEORY
Pratip Bhattacharyya, Bikas K. Chakrabarti
We study different ways of determining the mean distance r_n between a
reference point and its n-th neighbour among random points distributed with a
uniform density in a D-dimensional Euclidean space. First we present a
heuristic method; though this method provides only a crude mathematical result,
it shows a simple physical way of estimating r_n. Next we describe two
alternative means of deriving the exact expression of r_n, the one by using
absolute probability and the other by using conditional probability. Finally we
obtain an approximation to r_n from the mean volume between the reference point
and its n-th neighbour and compare it with the heuristic and exact results.
pratip@cmp.saha.ernet.in
2129. ABSOLUTE CONTINUITY OF AUTOPHAGE MEASURES ON FINITE-DIMENSIONAL VECTOR
SPACES
C. R. E. Raja
We consider a class of measures called autophage which was introduced and
studied by Szekely for measures on the real line. We show that the autophage
measures on finite-dimensional vector spaces over real or p-adic field are
infinitely divisible without idempotent factors and are absolutely continuous
with bounded continuous density. We also show that certain semistable measures
on such vector spaces are absolutely continuous.
creraja@isibang.ac.in
2130. THE NOISE OF A BROWNIAN STICKY FLOW IS BLACK
Yves Le Jan and Olivier Raimond
In this note, it is proved that the noise (in the sense of Tsirelson)
generated by a Brownian sticky flow (as defined in math.PR/0211387) is black.
olivier.raimond@math.u-psud.fr
2131. SMALL VALUES OF THE MAXIMUM FOR THE INTEGRAL OF FRACTIONAL BROWNIAN
MOTION
G.M.Molchan and A.V.Khokhlov
We consider the integral of fractional Brownian motion (IFBM) and its
functionals $\xi_T$ on the intervals $(0,T)$ and $(-T,T)$ of the following
types: the maximum $M_T$, the position of the maximum, the occupation time
above zero etc. We show how the asymptotics of $P(\xi_T<1)=p_T, T\to \infty$,
is related to the Hausdorff dimension of Lagrangian regular points for the
inviscid Burgers equation with FBM initial velocity. We produce computational
evidence in favor of a power asymptotics for $p_T$. The data do not reject the
hypothesis that the exponent $\theta$ of the power law is related to the
similarity parameter $H$ of fractional Brownian motion as follows: $\theta
=-(1-H)$ for the interval $(-T,T)$ and $\theta =-H(1-H)$ for $(0,T)$. The point
0 is special in that IFBM and its derivative both vanish there.
khokhlov@ipgp.jussieu.fr
2132. KRENGEL-LIN DECOMPOSITION FOR PROBABILITY MEASURES ON HYPERGROUPS
C. R. E. Raja
A Markov operator $P$ on a $\sigma$-finite measure space $(X, \Sigma, m)$
with invariant measure $m$ is said to have Krengel-Lin decomposition if $L^2
(X) = E_0 \oplus L^2 (X,\Sigma_d)$ where $E_0 = \{f \in L^2 (X) \mid ||P^n (f)
|| \ra 0 \}$ and $\Sigma_d$ is the deterministic $\sigma $-field of $P$. We
consider convolution operators and we show that a measure $\lam$ on a
hypergroup has Krengel-Lin decomposition if and only if the sequence $(\check
\lam ^n *\lam ^n)$ converges to an idempotent or $\lam$ is scattered. We verify
this condition for probabilities on Tortrat groups, on commutative hypergroups
and on central hypergroups. We give a counter-example to show that the
decomposition is not true for measures on discrete hypergroups which is in
contrast to the discrete groups case.
creraja@isibang.ac.in
2133. CRITICAL REGION FOR DROPLET FORMATION IN THE TWO-DIMENSIONAL ISING MODEL
Marek Biskup, Lincoln Chayes, Roman Kotecky
We study the formation/dissolution of equilibrium droplets in finite systems
at parameters corresponding to phase coexistence. Specifically, we consider the
2D Ising model in volumes of size $L^2$, inverse temperature $\beta>\betac$ and
overall magnetization conditioned to take the value $\mstar L^2-2\mstar v_L$,
where $\betac^{-1}$ is the critical temperature, $\mstar=\mstar(\beta)$ is the
spontaneous magnetization and $v_L$ is a sequence of positive numbers. We find
that the critical scaling for droplet formation/dissolution is when $v_L^{3/2}
L^{-2}$ tends to a definite limit. Specifically, we identify a dimensionless
parameter $\Delta$, proportional to this limit, a non-trivial critical value
$\Deltac$ and a function $\lambda_\Delta$ such that the following holds: For
$\Delta<\Deltac$, there are no droplets beyond $\log L$ scale, while for
$\Delta>\Deltac$, there is a single, Wulff-shaped droplet containing a fraction
$\lambda_\Delta\ge\lamc=2/3$ of the magnetization deficit and there are no
other droplets beyond the scale of $\log L$. Moreover, $\lambda_\Delta$ and
$\Delta$ are related via a universal equation that apparently is independent of
the details of the system.
biskup@math.ucla.edu
2134. A RESISTANCE BOUND VIA AN ISOPERIMETRIC INEQUALITY
Itai Benjamini and Gady Kozma
An isoperimetric upper bound on the resistance is given. As a corollary we
resolve two problems, regarding mean commute time on finite graphs and
resistance on percolation clusters. Further conjectures are presented.
itai@wisdom.weizmann.ac.il
2135. OPERATOR SEMI-SELFDECOMPOSABLE MEASURES AND RELATED NESTED SUBCLASSES OF
MEASURES ON VECTOR SPACES
C. R. E. Raja
$T$-semi-selfdecomposability and subclasses $L_m(b, Q)$ and $\tilde L_m(b,
Q)$ of measures on complete separable metric vector spaces are introduced and
basic properties are proved. In particular, we show that $\mu$ is
$T$-semi-selfdecomposable if and only if $\mu = T(\mu) \nu$ where $\nu$ is
infinitely divisible and $\mu$ is operator selfdecomposable if and only if $\mu
\in L_0(b, Q)$ for all $0< b < 1$.
creraja@isibang.ac.in
2136. SCALING LIMIT OF LOOP ERASED RANDOM WALK - A NAIVE APPROACH
Gady Kozma
We give an alternative proof of the existence of the scaling limit of loop
erased random walk which does not use Lowner's differential equation.
gadyk@wisdom.weizmann.ac.il
2137. EMBEDDING A MARKOV CHAIN INTO A RANDOM WALK ON A PERMUTATION GROUP
Steven N. Evans
Using representation theory, we obtain a necessary and sufficient
condition for a discrete--time Markov chain on a finite
state space $E$ to be representable as
$\Psi_n \Psi_{n-1} \cdots \Psi_1 z$,
$n \ge 0$, for any $z \in E$, where the $\Psi_i$
are independent, identically distributed random permutations
taking values in some given transitive group of permutations on $E$.
The condition is particularly simple when the group is $2$-transitive
on $E$.
evans@stat.berkeley.edu
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information provided by the author
click here.
- Or
here.
2138. THE INFINITE VOLUME LIMIT OF DISSIPATIVE ABELIAN SANDPILES
Christian Maes, Frank Redig and Ellen Saada
We construct the thermodynamic limit of the
stationary measures of the Bak-Tang-Wiesenfeld sandpile model
with a dissipative toppling matrix (sand grains may disappear
at each toppling). We prove uniqueness and mixing properties of this
measure and we obtain an infinite volume ergodic Markov process
leaving it invariant. We show how to extend the Dhar formalism
of the `abelian group of toppling operators' to infinite volume
in order to obtain a compact abelian group with a unique Haar measure
representing the uniform distribution over the recurrent configurations
that create finite avalanches.
Christian.Maes@fys.kuleuven.ac.be
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information provided by the author
click here.
2139. SOLUTIONS OF $\DELTA U=4U^2$ WITH NEUMANN'S CONDITIONS USING THE BROWNIAN SNAKE
Romain Abraham and Jean-Francois Delmas
We consider a Brownian snake $(W_s,s\geq 0)$ with underlying
process a reflected Brownian motion in a bounded domain $D$.
We construct a continuous additive functional $(L_s, s\geq 0)$
of the Brownian snake which counts the time spent by the end
points $\hat W_s$ of the Brownian snake paths on $\partial D$.
The random measure $Z=\int\delta_{\hat W_s} dL_s$ is supported
by $\partial D$. Then we represent the solution $v$ of
$\Delta u=4u^2$ in $D$ with weak Neumann boundary condition
$\varphi\geq 0$ by using exponential moment of $(Z,\varphi)$
under the excursion measure of the Brownian snake. We then
derive an integral equation for $v$. For small $\varphi$ it is
then possible to describe negative solution of $\Delta u=4u^2$
in $D$ with weak Neumann boundary condition $\varphi$. We also
consider the properties of $Z$. In particular we show it is
absolutely continuous with respect to the surface measure on
$\partial D$ for dimension $2$ and $3$. Let us note that $Z$
is more regular than the exit measure of the Brownian snake
out of $D$.
romain.abraham@math-info.univ-paris5.fr delmas@cermics.enpc.fr
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information provided by the author
click here.
- Or
here.
2140. SUPERPOSITION OPERATORS ON DIRICHLET SPACES
P. J. Fitzsimmons
In the context of a strongly local Dirichlet form
(E,D), we show that if K : R -> R is a measurable
function with K(0)=0 such that K(u)
(functional composition) is an element of D whenever
u is an element of D, then K is necessarily
locally Lipschitz continuous. If, in addition,
D contains unbounded elements, then K must be
globally Lipschitz continuous. The proofs rely on
a co-area formula for condenser potentials.
pfitzsim@ucsd.edu
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information provided by the author
click here.
2141. THICK POINTS OF SUPER-BROWNIAN MOTION
Jochen Blath and Peter Morters
We determine the dimension spectrum of thick points
of the state of a super-Brownian motion in dimension
$d\ge 3$. Our method also yields improvements
of a law of the iterated logarithm of Dawson and Perkins
and a result of Barlow, Evans and Perkins about the most
visited sites of super-Brownian motion.
All these results involve a constant which can be
characterized in terms of the upper tails of the
associated equilibrium Palm distribution.
blath@mathematik.uni-kl.de maspm@bath.ac.uk
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information provided by the author
click here.
2142. NONLINEAR DIFFUSIONS, HYPERCONTRACTIVITY AND THE OPTIMAL
$L^p$-EUCLIDEAN LOGARITHMIC SOBOLEV INEQUALITY
Manuel Del Pino, Jean Dolbeault, Ivan Gentil
The equation $u_t=\Delta_p\big(u^{1/(p-1)}\big)$ for $p>1$
is a nonlinear generalization of the heat equation which is
also homogeneous, of degree 1. For large time asymptotics,
its links with the optimal $L^p$-Euclidean logarithmic
Sobolev inequality have recently been investigated. Here
we focuse on the existence and the uniqueness of the
solutions to the Cauchy problem and on the regularization
properties (hypercontractivity and ultracontractivity) of
the equation using the $L^p$-Euclidean logarithmic Sobolev
inequality. A large deviation result based on a
Hamilton-Jacobi equation and also related to the
$L^p$-Euclidean logarithmic Sobolev inequality is then stated.
delpino@dim.uchile.cl dolbeaul@ceremade.dauphine.fr ivan.gentil@hec.ca
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information provided by the author
click here.
- Or
here.
2143. STRONG APPROXIMATIONS OF ADDITIVE FUNCTIONALS
OF A PLANAR BROWNIAN MOTION
Endre Csaki, Antonia Foldes, Yueyun Hu
This paper is devoted to the study of the additive functional
$t \to\int_0^t f(W(s)) ds$, where $f$ denotes a measurable function and
$W$ is a planar Brownian motion. Kasahara and Kotani (1979) have
obtained its second-order asymptotic behaviors, by using the
skew-product representation of $W$ and the ergodicity of the angular
part. We prove that the vector $(\int_0^\cdot f_j(W(s)) ds)_{1\le j \le n}$
can be strongly approximated by a multi-dimensional Brownian motion
time changed by an independent inhomogeneous L\'evy process. This
strong approximation yields central limit theorems and almost sure
behaviors for additive functionals. We also give their applications to
winding numbers and to symmetric Cauchy process.
csaki@renyi.hu afoldes@gc.cuny.edu hu@ccr.jussieu.fr
- To see a preprint or other
information provided by the author
click here.
- Or
here.
2144. PM-TECHNIQUES IN SOLVING THE DIRICHLET PROBLEM
Alice Vatamanelu
Using the fact that the set of random variables valued
in a global NPC space is an E-space and that any
Markov operator acting on such random variables is a
contraction in the probabilistic metric sense , we
apply probabilistic metric techniques to solve the
Dirichlet problem for functions valued in global NPC
spaces.
watamanelu@yahoo.com
