Probability Abstracts 69
This document contains abstracts 1960-2007.
They have been mailed on June 30, 2002.
1960. HAUSDORFF DIMENSION IN STOCHASTIC DISPERSION
Dmitry Dolgopyat, Vadim Kaloshin, Leonid Koralov
We consider the evolution of a connected set in Euclidean space carried by a
periodic incompressible stochastic flow. While for almost every realization of
the random flow at time t most of the particles are at a distance of order
sqrt{t} away from the origin [DKK1], there is an uncountable set of measure
zero of points, which escape to infinity at the linear rate [CSS1]. In this
paper we prove that this set of linear escape points has full Hausdorff
dimension.
kaloshin@math.mit.edu
1961. A LIMIT SHAPE THEOREM FOR PERIODIC STOCHASTIC DISPERSION
Dmitry Dolgopyat, Vadim Kaloshin, Leonid Koralov
We consider the evolution of a connected set on the plane carried by a
periodic incompressible stochastic flow. While for almost every realization of
the random flow at time t most of the particles are at a distance of order
sqrt{t} away from the origin, there is a measure zero set of points, which
escape to infinity at the linear rate. We study the set of points visited by
the original set by time t, and show that such a set, when scaled down by the
factor of t, has a limiting non random shape.
kaloshin@math.mit.edu
1962. ON THE INTEGRAL OF GEOMETRIC BROWNIAN MOTION
Michael Schr\"oder
This paper studies the law of any power of the integral of geometric Brownian
motion over any finite time interval. As its main results, two integral
representations for this law are derived. This is by enhancing the Laplace
transform ansatz of Yor (1992) with complex analytic methods, which is the main
methodological contribution of the paper. The one of our integrals has a
similar structure to that obtained by Yor, while the other is in terms of
Hermite functions as those of Dufresne (2001). Performing or not performing a
certain Girsanov transformation is identified as the source of these two forms
of the laws. While our results specialize for exponents equal to 1 to those
obtained by Yor, they yield on specialization representations for the exponent
equal to minus 1 laws which are markedly different from those obtained by
Dufresne.
schroder@euklid.math.uni-mannheim.de
1963. RANDOM METRIC SPACES AND THE UNIVERSAL URYSOHN SPACE.2
A. Vershik
We continue to study the set of all metric spaces in terms of the cone of
distance matrices which was suggested in the previous papers (see
math.GT/0203008), and consider topological and probabilistic problems connected
with this object.
Here we prove of that the generic Polish space in the sense of this model is
the so called universal Urysohn space which was defined by P.S.Urysohn in the
1920-th. Then we introduce the set of metric spaces with measures (metric
triples) and define a complete invariant of its - matrix distribution. We give
an intrinsic characterization of matrix distribution and using the ergodic
theorem give a new proof of Gromov's reconstruction theorem.
A natural construction of a wide class of measures on the cone $\cal R$ is
given and for these we show that with probability one the random Polish space
is again the Urysohn space. All the needed notions are defined in the text.
vershik@pdmi.ras.ru
1964. POISSON PROCESS PARTITION CALCULUS WITH APPLICATIONS TO EXCHANGEABLE
MODELS AND BAYESIAN NONPARAMETRICS
Lancelot F. James
This article discusses the usage of a partiton based Fubini calculus for
Poisson processes. The approach is an amplification of Bayesian techniques
developed in Lo and Weng for gamma/Dirichlet processes. Applications to models
are considered which all fall within an inhomogeneous spatial extension of the
size biased framework used in Perman, Pitman and Yor. Among some of the
results; an explicit partition based calculus is then developed for such
models, which also includes a series of important exponential change of measure
formula. These results are applied to obtain results for Levy-Cox models,
identities related to the two-parameter Poisson-Dirichlet process and other
processes, generalisations of the Markov-Krein correspondence, calculus for
extended Neutral to the Right processes, among other things.
lancelot@ust.hk
1965. BROWNIAN BRIDGE AND SELF-AVOIDING RANDOM WALK
Yevgeniy Kovchegov
We establish the Brownian bridge asymptotics for a scaled self-avoiding walk
conditioned on arriving to a far away point
$n \vec{a}$ for $\vec{a}$ in $Z^d$, as $n$ increases to infinity.
yevgeniy@math.stanford.edu
1966. PRICING RULE BASED ON NON-ARBITRAGE ARGUMENTS FOR RANDOM VOLATILITY AND
VOLATILITY SMILE
Nikolai Dokuchaev
We consider a generic market model with a single stock and with random
volatility. We assume that there is a number of tradable options for that stock
with different strike prices. The paper states the problem of finding a pricing
rule that gives Black-Scholes price for at-money options and such that the
market is arbitrage free for any number of tradable options, even if there are
two Brownian motions only: one drives the stock price, the other drives the
volatility process. This problem is reduced to solving a parabolic equation.
ndokuch@uwimona.edu.jm
1967. SPEEDING UP THE FMMR PERFECT SAMPLING ALGORITHM: A CASE STUDY REVISITED
Robert P. Dobrow, James Allen Fill
In a previous paper by the second author,two Markov chain Monte Carlo perfect
sampling algorithms -- one called coupling from the past (CFTP) and the other
(FMMR) based on rejection sampling -- are compared using as a case study the
move-to-front (MTF) self-organizing list chain. Here we revisit that case study
and, in particular, exploit the dependence of FMMR on the user-chosen initial
state. We give a stochastic monotonicity result for the running time of FMMR
applied to MTF and thus identify the initial state that gives the
stochastically smallest running time; by contrast, the initial state used in
the previous study gives the stochastically largest running time. By changing
from worst choice to best choice of initial state we achieve remarkable speedup
of FMMR for MTF; for example, we reduce the running time (as measured in Markov
chain steps) from exponential in the length n of the list nearly down to n when
the items in the list are requested according to a geometric distribution. For
this same example, the running time for CFTP grows exponentially in n.
jimfill@jhu.edu
1968. AN UPPER BOUND ON THE FLUCTUATIONS OF A SECOND CLASS PARTICLE
Timo Seppalainen
This note proves an upper bound for the fluctuations of a second-class
particle in the totally asymmetric simple exclusion process. The proof needs a
lower tail estimate for the last-passage growth model associated with the
exclusion process. A stronger estimate has been proved for the corresponding
discrete time model, so we take the needed estimate as a hypothesis. The
process is initially in local equilibrium with a slowly varying macroscopic
profile. The macroscopic initial profile is smooth in a neighborhood of the
origin where the second-class particle starts off, and the forward
characteristic from the origin is not a shock. Given these assumptions, the
result is that the typical fluctuation of the second-class particle is not of
larger order than n^{2/3}(log n)^{1/3}, where n is the ratio of the macroscopic
and microscopic space scales. The conjectured correct order should be n^{2/3}.
Landim et al. have proved a lower bound of order n^{5/8} for more general
asymmetric exclusion processes in equilibrium. Fluctuations in the case of
shocks and rarefaction fans are covered by earlier results of Ferrari-Fontes
and Ferrari-Kipnis.
seppalai@math.wisc.edu
1969. A LOCAL LIMIT THEOREM FOR A FAMILY OF NON-REVERSIBLE MARKOV CHAINS
Elizabeth L. Wilmer
By proving a local limit theorem for higher-order transitions, we determine
the time required for necklace chains to be close to stationarity. Because
necklace chains, built by arranging identical smaller chains around a directed
cycle, are not reversible, have little symmetry, do not have uniform stationary
distributions, and can be nearly periodic, prior general bounds on rates of
convergence of Markov chains either do not apply or give poor bounds. Necklace
chains can serve as test cases for future techniques for bounding rates of
convergence.
elizabeth.wilmer@oberlin.edu
1970. TOWARDS A PRACTICAL, THEORETICALLY SOUND ALGORITHM FOR RANDOM GENERATION
IN FINITE GROUPS
Gene Cooperman
This work presents a new, simple O(log^2|G|) algorithm, the Fibonacci cube
algorithm, for producing random group elements in black box groups. After the
initial O(log^2|G|) group operations, epsilon-uniform random elements are
produced using O((log 1/epsilon)log|G|) operations each. This is the first
major advance over the ten year old result of Babai [Babai91], which had
required O(log^5|G|) group operations. Preliminary experimental results show
the Fibonacci cube algorithm to be competitive with the product replacement
algorithm.
The new result leads to an amusing reversal of the state of affairs for
permutation group algorithms. In the past, the fastest random generation for
permutation groups was achieved as an application of permutation group
membership algorithms and used deep knowledge about permutation
representations. The new black box random generation algorithm is also valid
for permutation groups, while using no knowledge that is specific to
permutation representations. As an application, we demonstrate a new algorithm
for permutation group membership that is asymptotically faster than all
previously known algorithms.
gene@ccs.neu.edu
1971. ASYMPTOTIC STABILITY OF THE WONHAM FILTER FOR ERGODIC AND NONERGODIC
SIGNALS
P. Baxendale, P. Chigansky, R. Liptser
We discuss stability of nonlinear filters with respect to initial condition.
A simple representation for the filtering error is proposed, which is used in
the study of the Wonham filter equations in non ergodic case. We also point to
a gap in the proof of the classic result by H.Kunita (1971), which remains a
conjecture.
pavelm@eng.tau.ac.il
1972. THE RANDOM-CLUSTER MODEL
Geoffrey Grimmett
The class of random-cluster models is a unification of a variety of
stochastic processes of significance for probability and statistical physics,
including percolation, Ising, and Potts models; in addition, their study has
impact on the theory of certain random combinatorial structures, and of
electrical networks. Much (but not all) of the physical theory of Ising/Potts
models is best implemented in the context of the random-cluster representation.
This systematic summary of random-cluster models includes accounts of the
fundamental methods and inequalities, the uniqueness and specification of
infinite-volume measures, the existence and nature of the phase transition, and
the structure of the subcritical and supercritical phases. The theory for
two-dimensional lattices is better developed than for three and more
dimensions. There is a rich collection of open problems, including some of
substantial significance for the general area of disordered systems, and these
are highlighted when encountered. Amongst the major open questions, there is
the problem of ascertaining the exact nature of the phase transition for
general values of the cluster-weighting factor q, and the problem of proving
that the critical random-cluster model in two dimensions, with 1\le q\le 4,
converges when re-scaled to a stochastic L\"owner evolution. Overall the
emphasis is upon the random-cluster model for its own sake, rather than upon
its applications to Ising and Potts systems.
g.r.grimmett@statslab.cam.ac.uk
1973. A LAW OF LARGE NUMBERS FOR RANDOM WALKS IN RANDOM MIXING ENVIRONMENTS
Francis Comets, Ofer Zeitouni
We prove a law of large numbers for a class of multidimensional random walks
in random environments where the environment satisfies appropriate mixing
conditions, which hold when the environment is a weak mixing field in the sense
of Dobrushin and Shlosman. Our result holds if the mixing rate balances moments
of some random times depending on the path. It applies in the non-nestling
case, but we also provide examples of nestling walks that satisfy our
assumptions. The derivation is based on an adaptation, using coupling, of the
regeneration argument of Sznitman-Zerner.
zeitouni@ee.technion.ac.il
1974. SYMMETRIC ORNSTEIN-UHLENBECK SEMIGROUPS AND THEIR GENERATORS
A. Chojnowska-Michalik and B. Goldys
We provide necessary and sufficient conditions for a Hilbert space-valued
Ornstein-Uhlenbeck process to be reversible with respect to its invariant
measure $\mu$. For a reversible process the domain of its generator in $L^p(\mu
)$ is characterized in terms of appropriate Sobolev spaces thus extending the
Meyer equivalence of norms to any symmetric Ornstein-Uhlenbeck operator. We
provide also a formula for the size of the spectral gap of the generator. Those
results are applied to study the Ornstein-Uhlenbeck process in a chaotic
environment. Necessary and sufficient conditions for a transition semigroup
$(R_t)$ to be compact, Hilbert-Schmidt and strong Feller are given in terms of
the coefficients of the Ornstein-Uhlenbeck operator. We show also that the
existence of spectral gap implies a smoothing property of $R_t$ and provide an
estimate for the (appropriately defined) gradient of $R_t\phi$. Finally, in the
Hilbert-Schmidt case, we show that for any $\phi\in L^p(\mu)$ the function
$R_t\phi$ is an (almost) classical solution of a version of the Kolmogorov
equation.
beng@maths.unsw.edu.au
1975. A NEW FACTORIZATION PROPERTY OF THE SELFDECOMPOSABLE PROBABILITY
MEASURES
Aleksander M. Iksanov, Zbigniew J. Jurek and Bertram M. Schreiber
We prove that the convolution of a selfdecomposable distribution with its
background driving law is again selfdecomposable if and only if the background
driving law is s-selfdecomposable. We will refer to this as the factorization
property of a selfdecomposable distribution; let $L^f$ denote the set of all
these distributions. The algebraic structure and various characterizations of
$L^f$ are studied. Some examples are discussed, the most interesting one being
given by the L\'evy stochastic area integral. A filtration of the class $L^f$
into decreasing subsemigroups $L^{f}_n, n\ge 1,$ is given.
iksan@unicyb.kiev.ua
1976. AN ENLIGHTENING COUNTEREXAMPLE
Michael Hardy
We give a visually appealing counterexample to the proposition that unbiased
estimators are better than biased estimators.
hardy@math.mit.edu
1977. INFERENCE BY CONVERSION
Jani Lahtinen
We are discussing a modeling technique based on the idea to generate data
sequences with a number of suggested models. These sequences are transformed,
or converted, into an observed data sequence by a suitable function, or a
program. The motivation for doing so is in cases where the likelihood of
observed data is hard to compute, which is circumvented with an indirect
approximation by trying to replicate the data. It is shown that this approach
produces desirable metrics on the models of interest and a consistent method
for model selection, at least in some cases.
wolf@lce.hut.fi
1978. TRANSIENCE OF PERCOLATION CLUSTERS ON WEDGES
Omer Angel, Itai Benjamini, Noam Berger, Yuval Peres
We study random walks on supercritical percolation clusters on wedges in
$\Z^3$, and show that the infinite percolation cluster is (a.s.) transient
whenever the wedge is transient. This solves a question raised by O. Haggstrom
and E. Mossel. We also show that for convex gauge functions satisfying a mild
regularity condition, the existence of a finite energy flow on $Z^2$ is
equivalent to the (a.s.) existence of a finite energy flow on the supercritical
percolation cluster. This solves a question of C. Hoffman.
noam@stat.berkeley.edu
1979. SHARP METASTABILITY THRESHOLD FOR TWO-DIMENSIONAL BOOTSTRAP PERCOLATION
Alexander E. Holroyd
In the bootstrap percolation model, sites in an $L$ by $L$ square are
initially independently declared active with probability $p$. At each time
step, an inactive site becomes active if at least two of its four neighbours
are active. We study the behaviour as $p \to 0$ and $L \to \infty$
simultaneously of the probability $I(L,p)$ that the entire square is eventually
active. We prove that $I(L,p) \to 1$ if $\liminf p \log L > \lambda$, and
$I(L,p) \to 0$ if $\limsup p \log L < \lambda$, where $\lambda = \pi^2/18$. We
prove the same behaviour, with the same threshold $\lambda$, for the
probability $J(L,p)$ that a site is active by time $L$ in the process on the
infinite lattice. The same results hold for the so-called modified bootstrap
percolation model, but with threshold $\lambda' = \pi^2/6$. The existence of
the thresholds $\lambda,\lambda'$ settles a conjecture of Aizenman and
Lebowitz, while the determination of their values corrects numerical
predictions of Adler, Stauffer and Aharony.
holroyd@math.ucla.edu
1980. NONPARAMETRIC VOLATILITY DENSITY ESTIMATION FOR DISCRETE TIME MODELS
Bert van Es, Peter Spreij, Harry van Zanten
We consider discrete time models for asset prices with a stationary
volatility process. We aim at estimating the multivariate density of this
process at a set of consecutive time instants. A Fourier type deconvolution
kernel density estimator based on the logarithm of the squared process is
proposed to estimate the volatility density. Expansions of the bias and bounds
on the variance are derived.
spreij@science.uva.nl
1981. ON RANDOM WALKS AND DIFFUSIONS RELATED TO PARRONDO'S GAMES
R. Pyke
In a series of papers, G. Harmer and D. Abbott study the behavior of random
walks associated with games introduced in 1997 by J. M. R. Parrondo. These
games illustrate an apparent paradox that random and deterministic mixtures of
losing games may produce winning games. In this paper, classical cyclic random
walks on the additive group of integers modulo $m$, a given integer, are used
in a straightforward way to derive the strong law limits of a general class of
games that contains the Parrondo games. We then consider the question of when
random mixtures of fair games related to these walks may result in winning
games. Although the context for these problems is elementary, there remain open
questions. An extension of the structure of these walks to a class of shift
diffusions is also presented, leading to the fact that a random mixture of two
fair shift diffusions may be transient to $+\infty$.
dabbott@eleceng.adelaide.edu.au
1982. ON PARRONDO'S PARADOX: HOW TO CONSTRUCT UNFAIR GAMES BY COMPOSING FAIR
GAMES
E.S. Key, M. Klosek, D. Abbott
We construct games of chance from simpler games of chance. We show that it
may happen that the simpler games of chance are fair or unfavourable to a
player andyet the new combined game is favourable -- this is a
counter-intuitive phenomenoknown as Parrondo's paradox. We observe that all of
the games in question are random walks in periodic environments (RWPE) when
viewed on the proper time scale. Consequently, we use RWPE techniques to derive
conditions under which Parrondo's paradox occurs.
dabbott@eleceng.adelaide.edu.au
1983. THE LAW OF LARGE NUMBERS FOR RANDOM WALKS IN A MIXING RANDOM ENVIRONMENT
Firas Rassoul-Agha
The point of view of the particle is an approach that has proven very
powerful in the study of many models of random motions in random media. We
provide a new use of this approach to prove the law of large numbers in the
case of one or higher-dimensional, finite range, transient random walks in
mixing random environments. One of the advantages of this method over what has
been used so far is that it is not restricted to i.i.d. environments.
rassoul@cims.nyu.edu
1984. FAST ALGORITHMS OF BAYESIAN SEGMENTATION OF IMAGES
B.A. Zzalesky
The network flow optimization approach is offered for Bayesian segmentation
of gray-scale and color images. It is supposed image pixels are characterized
by a feature function taking finite number of arbitrary rational values (it can
be either intensity values or other characteristics of images). The clusters of
homogeneous pixels are described by labels with values in another set of
rational numbers. They are assumed to be dependent and distributed according to
either the exponential or the Gaussian Gibbs law. Instead traditionally used
local neighborhoods of nearest pixels the completely connected graph of
dependence of all pixels is employed for the Gibbs prior distributions.
The methods developed reduce the problem of segmentation to the problem of
determination of the minimum cut of an appropriate netw
zalesky@mpen.bas-net.by
1985. DISCRETE POLYNUCLEAR GROWTH AND DETERMINANTAL PROCESSES
Kurt Johansson
We consider a discrete polynuclear growth (PNG) process and prove a functioal
limit theorem for its convergrence to the Airy process. This generalizes
previous results by Pr"ahofer and Spohn. The result enables us to express the
GOE largest eigenvalue (Tracy-Widom) distribution in terms of the Airy process.
We also show some results and give a conjecture about the transversal
fluctuations in a point to line last passage percolation problem.
kurtj@math.kth.se
1986. CONVERGENCE OF THE POINCARE CONSTANT
Oliver Johnson
The Poincare constant R(Y) of a random variable Y relates the L2 norm of a
function g and its derivative g'. Since R(Y) - Var(Y) is positive, with
equality if and only if Y is normal, it can be seen as a distance from the
normal distribution. In this paper we establish a best possible rate of
convergence of this distance in the Central Limit Theorem. Furthermore, we show
that R(Y) is finite for discrete mixtures of normals, allowing us to add rates
to the proof of the Central Limit Theorem in the sense of relative entropy.
o.johnson@statslab.cam.ac.uk
1987. PHASE TRANSITION AND CRITICAL BEHAVIOR IN A MODEL OF ORGANIZED
CRITICALITY
Marek Biskup, Philippe Blanchard, Lincoln Chayes, Daniel Gandolfo and
Tyll Krueger
We study a model of ``organized'' criticality, where a single avalanche
propagates through an \textit{a priori} static (i.e., organized) sandpile
configuration. The latter is chosen according to an i.i.d. distribution from a
Borel probability measure $\rho$ on $[0,1]$. The avalanche dynamics is driven
by a standard toppling rule, however, we simplify the geometry by placing the
problem on a directed, rooted tree. As our main result, we characterize which
$\rho$ are critical in the sense that they do not admit an infinite avalanche
but exhibit a power-law decay of avalanche sizes. Our analysis reveals close
connections to directed site-percolation, both in the characterization of
criticality and in the values of the critical exponents.
biskup@math.ucla.edu
1988. SWENDSEN-WANG DYNAMICS ON Z^D FOR DISORDERED NON FERROMAGNETIC SYSTEMS
Emilio De Santis
We study the Swendsen-Wang dynamics for disordered non ferromagnetic Ising
models on cubic subsets of the hypercubic lattice Z^d and we show that for all
small values of the temperature parameter T the dynamics has a slow relaxation
to equilibrium (it is torpid). Looking into this dynamics from the point of
view of the Markov chains theory we can prove that the spectral radius goes to
one when the size of the system goes to infinity. This means that, if we want
to use the Swendsen-Wang dynamics for a computer simulation, we have a slow
convergence to the stationary measure in low temperature. Also it is a good
example of a non-local dynamics that relaxes slowly to the equilibrium measure.
desantis@mat.uniroma1.it
1989. MOCK-GAUSSIAN BEHAVIOUR FOR LINEAR STATISTICS OF CLASSICAL COMPACT
GROUPS
C.P. Hughes and Z. Rudnick
We consider the scaling limit of linear statistics for eigenphases of a
matrix taken from one of the classical compact groups. We compute their moments
and find that the first few moments are Gaussian, whereas the limiting
distribution is not. The precise number of Gaussian moments depends upon the
particular statistic considered.
rudnick@math.tau.ac.il
1990. UNIQUENESS FOR REFLECTING BROWNIAN MOTION
IN LIP DOMAINS
R. Bass, K. Burdzy and Z.-Q. Chen
A lip domain is a Lipschitz domain where the Lipschitz
constant is strictly less than one.
We prove strong existence and pathwise uniqueness
for the solution to the Skorokhod equation
$$dX_t=dW_t+ \bn(X_s) dL_s,$$
in planar lip domains, where $W_t$ is a Brownian motion,
$\bn$ is the inward pointing unit normal
vector, and $L_t$ is a local time on the boundary
which satisfies some additional regularity conditions.
Counterexamples are given for some Lipschitz (but not lip)
three dimensional domains.
bass@math.uconn.edu burdzy@math.washington.edu zchen@math.washington.edu
- To see a preprint or other
information provided by the author
click here.
- Or
here.
- Or
here.
1991. SHARP METASTABILITY THRESHOLD FOR TWO-DIMENSIONAL BOOTSTRAP PERCOLATION
Alexander E. Holroyd
In the bootstrap percolation model, sites in an $L$ by $L$
square are initially independently declared active with
probability $p$. At each time step, an inactive site
becomes active if at least two of its four neighbours are
active. We study the behaviour as $p \rightarrow 0$ and
$L \rightarrow \infty$ simultaneously of the probability
$I(L,p)$ that the entire square is eventually active. We
prove that $I(L,p) \rightarrow 1$ if $\liminf p\log L >
\lambda$, and $I(L,p) \rightarrow 0$ if $\limsup p\log L <
\lambda$, where $\lambda = \pi^2/18$. We prove the same
behaviour, with the same threshold $\lambda$, for the
probability $J(L,p)$ that a site is active by time $L$
in the process on the infinite lattice. The same results
hold for the so-called modified bootstrap percolation model,
but with threshold $\lambda' = \pi^2/6$. The existence of
the thresholds $\lambda,\lambda'$ settles a conjecture of
Aizenman and Lebowitz, while the determination of their
values corrects numerical predictions of Adler, Stauffer
and Aharony.
holroyd@math.ucla.edu
- To see a preprint or other
information provided by the author
click here.
1992. MARKOV CHAINS WITH SELF-INTERACTIONS
Pierre Del Moral and Laurent Miclo
In this article we study a class of time self-interacting
``Markov'' chain models. We propose a novel theoretical
basis based on measure valued processes and semigroup
technics to analyze their asymptotic behavior as the time
parameter tends to infinity.
We exhibit different types of decays to equilibrium
depending on the level of interaction.
We illustrate these results in a variety of examples
including Gaussian or Poisson self-interacting models.
We analyze the long time behavior of a new class of
evolutionary self-interacting chain models. These
genetic type algorithms can also be regarded as
reinforced stochastic explorations of an environment
with obstacles related to a potential function.
delmoral@cict.fr
- To see a preprint or other
information provided by the author
click here.
1993. ON THE CONVERGENCE OF CHAINS WITH TIME EMPIRICAL
SELF-INTERACTIONS
Pierre Del Moral and Laurent Miclo
We consider stochastic chains on abstract measurable
spaces whose evolution at any given time depend on the
present position and on the occupation measure created by
the path up to this instant. This generalization of
reinforced random walks enables us to impose conditions
insuring $Lp$ or a.s. convergence of the empirical measures
toward some fixed point of a probability-valued dynamical
system. We present two sets of hypotheses based on weak
contraction properties, leading to two different proofs,
but in both situations the rates of convergence are
optimal in the examined level of generality.
delmoral@cict.fr
- To see a preprint or other
information provided by the author
click here.
1994. GENEALOGICAL MODELS IN RARE EVENT ANALYSIS
Frederic Cérou, Pierre Del Moral, Francois Le Gland and Pascal Lézaud
We present in this article a genetic type interacting
particle systems algorithm and a genealogical model
for estimating a class of rare events arising in physics
and network analysis. We represent the distribution of a
Markov process hitting a rare target in terms of a
Feynman-Kac model in path space.
We show how these branching particle models described
in previous works can be used to estimate the probability
of the corresponding rare events as well as the distribution
of the process in this regime.
delmoral@cict.fr
- To see a preprint or other
information provided by the author
click here.
1995. A NOTE ON LAPLACE-VARADHAN'S INTEGRAL LEMMA
Pierre Del Moral and Tim Zajic
We propose a complement to an integral lemma of
Laplace-Varadhan arising in large deviations literature.
We examine a situation in which the state space may
depend on the rate of deviations.
In the final part of this work we use this framework
to discuss large deviations principles for a class of
mean field interacting jump processes and a class of
mean field interacting non linear diffusions.
delmoral@cict.fr
- To see a preprint or other
information provided by the author
click here.
1996. ABOUT SUPERCONTRACTIONS OF MARKOV KERNELS
Pierre Del Moral, Michel Ledoux and Laurent Miclo
We study Lipschitz's contraction properties of general
Markovian kernels seen as operators on spaces of
probabilities endowed with entropy-like distances.
Universal quantitative bounds on the associated ergodic
constants are deduced from Dobrushin's coefficient and
strong contraction properties in Orlicz's spaces for
relative densities are proved under restrictive mixing
assumptions. Next we obtain contraction estimates in
the entropy sense around an arbitrary probability by
introducing a particular Dirichlet form and the
corresponding modified logarithmic Sobolev inequalities.
The interest of these bounds will be illustred by
inhomogeneous Gaussian examples, emphasizing the
irrelevence of the existence of an invariant measure
assumption.
delmoral@cict.fr
- To see a preprint or other
information provided by the author
click here.
1997. ON THE SPECTRUM OF MARKOV SEMIGROUPS VIA LARGE DEVIATIONS
Irina Ignatiouk-Robert
The essential spectral radius of a sub-Markovian process is defined
as the infimum of the spectral radius of all local perturbations of
the process. It is shown that, under some general assumptions, the
essential spectral radius can be expressed in terms of the sample path
large deviation rate function associated with the rescaled process.
Irina.Ignatiouk@math.u-cergy.fr
1998. HARDY INEQUALITY FOR CENSORED STABLE PROCESSES
Zhen-Qing Chen an Renming Song
For $\alpha$ between 0 and 2, a censored $\alpha$-stable
process in an open set $D\subset R^n$ is a process
obtained from a symmetric $\alpha$-stable L\'evy process
in $R^n$ by restricting its L\'evy measure to $D$.
In this paper, a Hardy inequality is established for
censored stable processes on a large class of bounded
domains including bounded Lipschitz domains.
zchen@math.washington.edu, rsong@math.uiuc.edu
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information provided by the author
click here.
- Or
here.
1999. GREEN FUNCTION ESTIMATE FOR CENSORED STABLE PROCESSES
Zhen-Qing Chen and Panki Kim
Sharp two-sided estimates for Green functions of
censored $\alpha$-stable process $Y$ in a bounded
$C^{1,1}$ open set $D$ are obtained, where $\alpha$
is a constant strictly between 1 and 2. It is shown that
the Martin boundary and minimal Martin boundary of $Y$
can all be identified with the Euclidean boundary of $D$.
Sharp two-sided estimates for the Martin kernel of $Y$ are
also derived.
zchen@math.washington.edu, pkim@math.washington.edu
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information provided by the author
click here.
2000. ANNEALED FEYNMAN-KAC MODELS
Pierre Del Moral and Laurent Miclo
We analyze the concentration properties of an annealed
Feynman-Kac model in distribution space. We characterize
the concentration regions in terms of a variational problem
involving a competition between the potential function and
the mutation kernel. When the temperature parameter is
evanescent with time and under appropriate hypotheses,
the probability mass tends to concentrate on regions with
minimal potential values. We give a precise description
of these areas using non-linear semi-group contractions
and large deviations techniques. We illustrate this
annealed model with two physical interpretations related
respectively to Markov motions in absorbing media and
interacting measure valued processes.
delmoral@cict.fr
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information provided by the author
click here.
2001. ANALYTIC CHARACTERIZATION OF CONDITIONAL GAUGEABILITY
FOR NON-LOCAL FEYNMAN-KAC TRANSFORMS
Zhen-Qing Chen
An analytic characterization of gaugeability and
conditional gaugeability is given for non-local
(or discontinuous)Feynman-Kac transforms of general
symmetric Markov processes. This analytic
characterization is very useful in determining
whether a process perturbed by a potential is gaugeable
or conditionally gaugeable in concrete cases.
zchen@math.washington.edu
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information provided by the author
click here.
2002. ABSOLUTE CONTINUITY OF SYMMETRIC MARKOV PROCESSES
Z.-Q. Chen, P.J. Fitzsimmons, M. Takeda, J. Ying and T.-S. Zhang
We study Girsanov's theorem in the context of symmetriMarkov
processes, extending earlier work of Fukushima-Takeda
and Fitzsimmons on Girsanov transformations of `gradient type'.
We investigate the most general Girsanov transformation
leading to another symmetric Markov process. This investigation
requires an extension of the forward-backward martingale method
of Lyons-Zheng, to cover the case of processes with jumps.
zchen@math.washington.edu, pfitz@euclid.ucsd.edu, takeda@math.tohoku.ac.jp,
jying@fudan.edu.cn, tzhang@maths.man.ac.uk
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information provided by the author
click here.
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here.
2003. NON-SYMMETRIC PERTURBATIONS OF SYMMETRIC DIRICHLET FORMS
Patrick J. Fitzsimmons and Kazuhiro Kuwae
We provide a path-space integral representation of the
semigroup associated with the quadratic form obtained by
lower order perturbation of a symmetric local Dirichlet form.
The representation is a combination of Feynman-Kac and
Girsanov formulas, and extends previously known results in
the framework of symmetric diffusion processes through the
use of the Hardy class of smooth measures, which contains the
Kato class of smooth measures.
pfitzsim@ucsd.edu kuwae@yokohama-cu.ac.jp
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information provided by the author
click here.
2004. THE BURGERS SUPERPROCESS
Guillaume Bonnet and Robert J. Adler
We define the Burgers superprocess to be the solution of the
stochastic partial differential equation
$$
\frac{\partial }{\partial t} u(t,x) =
\Delta u(t,x) \ -\ \lambda u(t,x) \nabla u(t,x)\ + \
\gamma \sqrt{u(t,x)} \ W(dt,dx),
$$
where $t\geq 0$, $x\in\R$, and $W$ is space-time white noise. Taking
$\gamma=0$ gives the classic Burgers equation, an important,
non-linear, partial differential equation. Taking $\lambda=0$ gives a
(Brownian) superprocess, an important, measure valued, stochastic process.
We discuss (and prove) existence of a solution to this equation, and
discuss (but cannot prove) uniqueness of the solution.
gbonnet@email.unc.edu robert@ieadler.technion.ac.il
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information provided by the author
click here.
2005. MAJORIZING KERNELS AND STOCHASTIC CASCADES WITH APPLICATIONS
TO INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
Rabi Bhattacharya, Larry Chen, Scott Dobson, Ronald B. Guenther,
Mina Ossiander, Enrique Thomann, Edward C. Waymire
A general method is developed to obtain conditions on initial data
and forcing terms for the global existence of unique regular solutions
to incompressible 3d Navier-Stokes equations.The basic idea generalizes
a probabilistic approach introduced by LeJan and Sznitman (1997) to obtain
weak solutions whose Fourier transform may be represented by an expected
value of a stochastic cascade. A functional analytic framework is also
developed which partially connects stochastic iterations and certain
Picard iterates. Some local existence and uniqueness results are also obtained
by contractive mapping conditions on the Picard iteration.
bhattach@indiana.edu chen@math.orst.edu dobsons@math.orst.edu
guenth@math.orst.edu ossiand@math.orst.edu thomann@math.orst.edu waymire@math.orst.edu
- To see a preprint or other
information provided by the author
click here.
2006. STRONG FELLER PROPERTIES FOR DISTORTED BROWNIAN MOTION
AND APPLICATIONS TO FINITE PARTICLE SYSTEMS WITH SINGULAR INTERACTIONS
Sergio Albeverio, Yuri Kondratiev, Michael Roeckner
We prove strong Feller properties for a class of distorted Brownian
motions on $\RR^d$. We also construct a weak solution to the corresponding
stochastic differential equation starting from any point in $\{\rho \not= 0\}$
and staying in $\{\rho \not= 0\}$ before possibly going out of any ball in $\RR^d$.
Here $\rho$ is the Lebesgue density of the symmetrizing measure $\mu$.
Our condition on the logarithmic derivative $\frac{\nabla \rho}{\rho}$ is
that it should be locally in $L^{d+\epsilon}$, but only with respect to the
symmetrizing measure $\mu = \rho \; dx$, not necessarily Lebesgue measure $dx$.
This allows applications to singular situations.
In particular, finite particle systems with two body interactions with
infinitely strong repulsion can be treated by our results.
Among other things it is shown that they never meet no matter what their
starting configuration was. Another application treats diffusions in random
media.
Dedicated to Len Gross on the occasion of his 65th birthday.
albeverio@uni-bonn.de kondrat@mathematik.uni-bielefeld.de roeckner@mathematik.uni-bielefeld.de
2007. HARNACK AND FUNCTIONAL INEQUALITIES FOR GENERALIZED MEHLER SEMIGROUPS
M. Roeckner, F.Y. Wang
Harnack inequalities are established for a class of
generalized Mehler semigroups, which in particular imply
upper bound estimates of the transition density. Moreover,
Poincare and log-Sobolev inequalities are proved in terms
of estimates for the square field operators. Furthermore,
under a condition, well-known as the Gaussian case, we prove
that generalized Mehler semigroups are strong Feller.
The results are illustrated by concrete examples.
In particular, we show that a generalized Mehler semigroup
with an $\alpha$-stable part is not hyperbounded but
exponentially ergodic, and that the log-Sobolev constant
obtained by our method in the special Gaussian case can be
sharper than the one following from the usual curvature
condition.
Moreover, a Harnack inequality is established for the
generalized Mehler semigroup associated with the Dirichlet
heat semigroup on (0,1). We also prove that this semigroup
is not hyperbounded.
roeckner@mathematik.uni-bielefeld.de
wangfy@bnu.edu.cn
