Probability Abstracts 43
This document contains abstracts 939-977.
They have been mailed on January 30, 1998.
939. RANDOM SPANNING TREES OF CAYLEY GRAPHS
AND AN ASSOCIATED COMPACTIFICATION OF SEMIGROUPS
Steven N. Evans
A sequential construction of a random spanning
tree for the Cayley graph of a finitely generated,
countably infinite subsemigroup
of a group is considered. At each stage, the spanning
tree is approximated by a finite tree
rooted at the identity. The approximation
at the next stage is obtained by adding edges to all the vertices
not in the current tree
that can be obtained from existing vertices via multiplication
by a random walk step with values in the generating set.
This construction leads to a compactification of the
semigroup in which a sequence of semigroup elements
that is not eventually constant is convergent
if the random geodesic through the spanning tree that joins the origin
to the $n^{\mathrm th}$ element of the sequence
converges in distribution as $n \rightarrow \infty$.
The compactification is identified in a number of examples.
Also, it is shown that
asymptotically the height (respectively, the size) of
the sequence of approximating trees grows linearly
(respectively, exponentially).
evans@stat.Berkeley.EDU
- To see a preprint or other
information provided by the author
click here.
940. MARKOV MARGINAL PROBLEMS AND THEIR APPLICATIONS TO MARKOV
OPTIMAL CONTROL
Toshio Mikami
In this paper we discussed the problems that can be stated, roughly, as
follows.
Fix $T>0$.
Let $X^u (t)$ be a $R$-valed semimartingale which satisfies the following:
for $0\le t\le T$,
$$dX^u (t)=u(t)dt+\sigma (t,X^u (t))dW(t).$$
Find $b^n (t,x)$ and $\sigma^n (t,x)$
$((t,x)\in [0,T]\times R)$ for which the following has a weak solution
$$X^n (t)=X^n (0)+\int_0^t b^n (s,X(s))ds+ \int_0^t\sigma^n (s,X(s))dW(s),
\quad (t\in [0,T])$$
and
$$P(X^u (t)\in dx)\sim P(X^n (t)\in dx)\quad \hbox{ for }t\in [0,T].$$
$$P(u(t)\in dx)\sim P(b^n (t,X^n (t))\in dx)\quad\hbox{ for }t\in (0,T].$$
as $n\to\infty$.
As an application we consider Markov optimal control problems.
mikami@math.sci.hokudai.ac.jp (plain-TeX file, PS file and hardcopy available)
941. STRONG UNIQUENESS FOR A CLASS OF INFINITE DIMENSIONAL DIRICHLET
OPERATORS AND APPLICATIONS TO STOCHASTIC QUANTIZATION
Vitali Liskevich, Michael R"ockner
Strong and Markov uniqueness problems in $L^2$ for Dirichlet operators
on rigged Hilbert spaces are studied. An analytic approach based on
a--priori estimates is used. The extension of the problem to the
$L^p$-setting is discussed. As a direct application essential
self--adjointness and strong uniqueness in $L^p$ is proved for the
generator (with initial domain the bounded smooth cylinder functions)
of the stochastic quantization process for Euclidean quantum field
theory in finite volume $\Lambda \subset \R^2$.
V.Liskevich@bristol.ac.uk roeckner@mathematik.uni-bielefeld.de
942. GENERALIZED MEHLER SEMIGROUPS: THE NON--GAUSSIAN CASE
Marco Fuhrmann, Michael R"ockner
We study generalized Mehler semigroups, introduced in \cite{BRS}, with
special emphasis on the non--Gaussian case. We review and simplify the
method of construction. In the general (non--Gaussian) case we
construct an associated cadlag Markov process in an appropriate state
space obtained as a solution of a stochastic equation which can be
solved ''$\omega$ by $\omega$''. We also show tightness of the
associated $(r,p)$-capacities. Invariant measures, time regularity and
a definition of the generator are also studied.
marfuh@mate.polimi.it roeckner@mathematik.uni-bielefeld.de
943. STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY LEVY SPACE-TIME
WHITE NOISE
David Applebaum and Jiang-Lun Wu
The purpose of this paper is to investigate the problem of solving the
following parabolic stochastic partial differential equation (SPDE)
$$ (\frac{\delta}{dt} - \frac{\delta^{2}}{\delta x^{2}}) u(t,x)
= a(t,x, u(t,x)) + b(t,x,u(t,x)) F_{t,x} $$
with suitable boundary and initial conditions (for all $(t,x) \in [0,
\infty) \times [O,L]$ where $F_{t,x}$ is so-called Levy space-time white
noise. The probelm is interpreted as a stochastic integral equation of jump
type involving evolution kernels. We obtain the existence and uniqueness in
the $L^{2}$ sense
and show that it gives rise to a stochastic flow (in time).
dba@maths.ntu.ac uk Jiang-Lun.Wu@ruhr-uni-bochum.de (Latex file available)
944. COMPOUND POISSON PROCESSES AND LEVY PROCESSES IN GROUPS AND
SYMMETRIC SPACES
David Applebaum
A sample path description of compound Poisson processes on groups is given
and applied to represent Levy processes on connected Lie groups as almost
sure limits of sequences of Brownian motions with drift interlaced with
random jumps. We obtain spherically symmetric Levy processes in Riemannian
symmetric spaces of the form $M = G \setminus K$ where $G$ is a semisimple
Lie group and $K$ is a a compact subgroup by projection of symmetric
horizontal processes in $G$ and give a straightforward proof of Gangolli's
Levy-Khintchine formula for their spherical transform. Finally, we show that
such processes can be realised in Fock space in terms of creation,
conservation and annihilation processes. They appear as generators of a new
class of factorisable representationsof the current group $C(R^{+},G)$. In
the case where $G$ is of non-compact type, these Fock space processes are
indexed by the roots of $G$ and the natural action of the Weyl group induces
a (second quantised) unitary equivalence between them.
dba@maths.ntu.ac.uk Latex file available.
945. SPEEED OF $\overline d$-CONVERGENCE FOR MARKOV APPROXIMATIONS OF
CHAINS WITH COMPLETE CONNECTIONS. A COUPLING APPROACH
Xavier Bressaud, Roberto Fern\'andez and Antonio Galves
We compute the speed of convergence of the canonical Markov
approximation of a chain with complete connections with summable
decays. We show that in the $\overline{d}$-topology the approximation
converges at least at a rate proportional to these decays.
This is proven by explicitly constructing a coupling between
the chain and each range-$k$ approximation.
bressaud@ime.usp.br rf@ime.usp.br galves@ime.usp.br
946. THE LOCAL TIME OF SIMPLE RANDOM WALK IN RANDOM
ENVIRONMENT
Yueyun Hu and Zhan Shi
We study the upper and lower limits of the local time,
at any fixed level, of Sinai's simple random walk in
random environment and Brox's diffusion process with
Brownian potential.
hu@proba.jussieu.fr, shi@ccr.jussieu.fr (Plain) TeX file available
947. THE LIMITS OF SINAI'S SIMPLE RANDOM WALK IN
RANDOM ENVIRONMENT
Yueyun Hu and Zhan Shi
Several integral criteria are presented which completely
characterize all the possible Levy classes of Sinai's
simple random walk in random environment and Brox's
diffusion process with Brownian potential.
hu@proba.jussieu.fr, shi@ccr.jussieu.fr (Plain) TeX file available
948. RATES OF CONVERGENCE OF DIFFUSIONS WITH DRIFTED
BROWNIAN POTENTIALS
Yueyun Hu, Zhan Shi and Marc Yor
We obtain all the convergence rates for a diffusion
process whose random potential is Brownian motion with
a constant drift. The limiting stable distributions are
related to Cauchy's principal values of Brownian local
times.
hu@proba.jussieu.fr, shi@ccr.jussieu.fr, secret@proba.jussieu.fr
(Plain) TeX file available
949. A LOCAL TIME CURIOSITY IN RANDOM ENVIRONMENT
Zhan Shi
The maximum local times of Sinai's random walk in random
environment and Brox's diffusion with Brownian potential
have rather different forms of asymptotics.
shi@ccr.jussieu.fr (Plain) TeX file available
950. REFINED GIBBS CONDITIONING PRINCIPLE FOR CERTAIN INFINITE
DIMENSIONAL STATISTICS
A. Dembo and J. Kuelbs
Let X_1, ..., X_n, be independent, identically distributed random
observations taking values in a Polish space S, and T:S-->E a
statistic on S with values in a separable Banach space E.
We examine the limit law of (X_1,...,X_k) conditional on
n^{-1} sum^n_{i=1} T(X_i) being in an open convex subset D of E.
In the limit n-->infinity, the conditional laws converge to a k-fold product
probability Q^k, where Q is determined by the Gibbs conditioning principle.
Our results describe the allowed dependence of k=k(n) on n in terms
of explicit geometric conditions related to smoothness of the boundary of
D at a dominating point.
amir@stat.stanford.edu
- To see a preprint or other
information provided by the author
click here.
951. MONOTONICITY OF UNIQUENESS FOR PERCOLATION ON CAYLEY GRAPHS:
ALL INFINITE CLUSTERS ARE BORN SIMULTANEOUSLY
Olle H\"aggstr\"om and Yuval Peres
Consider site or bond percolation with retention parameter $p$ on an
infinite Cayley graph. In response to questions raised by Grimmett
and Newman (1990) and Benjamini and Schramm (1996), we show that the
property of having (almost surely) a unique infinite open cluster is
increasing in $p$. Moreover, in the standard coupling of the
percolation models for all parameters, a.s.\ for all $p_2>p_1>p_c$,
each infinite $p_2$-cluster contains an infinite $p_1$-cluster;
this yields an extension of Alexander's (1995) ``simultaneous
uniqueness'' theorem. As a corollary, we obtain that the probability
$\theta_v(p)$ that a given vertex $v$ belongs to an infinite cluster
is a continuous function of $p$ throughout the supercritical phase
$p>p_c$. We also show that when there are multiple infinite clusters,
a.s. each of them has uncountably many ends and ``cluster repulsion''
holds. All our results extend to quasi-transitive infinite graphs
with a unimodular automorphism group.
olleh@math.chalmers.se peres@stat.berkeley.edu
- To see a preprint or other
information provided by the author
click here.
952. UNIFORM SPANNING FORESTS
Itai Benjamini, Russell Lyons, Yuval Peres, and Oded Schramm
We study uniform spanning forest measures in infinite graphs, which are
weak limits of uniform spanning tree measures from finite subgraphs.
These limits can be taken with free or wired boundary conditions.
Pemantle (1991) proved that the free and wired spanning forests coincide
in $\Z^d$ and that they give a single tree iff $d<5$. In the present
work, we extend Pemantle's alternative to general graphs and exhibit
further connections of uniform spanning forests to random walks,
potential theory, invariant percolation, and amenability. The uniform
spanning forest model is related to random cluster models in statistical
physics, but, because of the preceding connections, its analysis can be
carried further. Among our results are the following:
* The free spanning forest and wired spanning forest in a graph $G$
coincide iff all harmonic Dirichlet functions on $G$ are constant.
* The tail $\sigma$-field of the wired spanning forest and the free
spanning forest is trivial on any graph.
* In any Cayley graph which is not a finite extension of $\Z$, all
component trees of the wired spanning forest have one end; this
is new in $\Z^d$ for $d>4$.
* In any tree, and in any graph with spectral radius less than $1$,
a.s. all components of the wired spanning forest are recurrent.
* The basic topology of the free and the wired uniform spanning
forest measures on lattices in hyperbolic space is determined.
* A Cayley graph is amenable iff for all
$\epsilon>0$, the union of the wired spanning forest
and Bernoulli percolation with parameter $\epsilon$ is connected.
* Harmonic measure from infinity is shown to exist on any recurrent
proper planar graph with finite co-degrees.
We also present fourteen open problems and conjectures.
itai@wisdom.weizmann.ac.il rdlyons@indiana.edu
peres@stat.berkeley.edu schramm@wisdom.weizmann.ac.il
953. COUPLING THE TOTALLY ASYMMETRIC SIMPLE EXCLUSION
PROCESS WITH A MOVING INTERFACE
Timo Seppalainen
This paper is an expanded version of the author's lecture
at the I Escola Brasileira de Probabilidade at IMPA this
past August. First we explain how scaling limits for the
one-dimensional totally asymmetric simple exclusion process
(TASEP) can be derived through a coupling of the process with
a moving interface. Secondly we prove new results about the
large deviations of the interface model and the exclusion
process. We derive the exact rate function for lower tail
deviations of the interface, and for lower tail deviations
of a tagged particle in the TASEP. The only assumption
required on the initial distribution of the TASEP is that
lower tail rate functions exist. We also show how deviations
from the hydrodynamic limit that force particles to accelerate
can decay exponentially in $n^2$ instead of the usual rate of
exponential in $n$.
seppalai@iastate.edu (AMS-TeX file, ps file, and hardcopy available)
954. SELFDECOMPOSABILITY: AN EXCEPTION OR A RULE?
Zbigniew J. Jurek
A characteristic function / Fourier transform f is said to be
SELFDECOMPOSABLE (or class L distribution) if for each 0 < c < 1 there
exists another characteristic function g (depending on c) such that
f(t) = f(ct)g(t) , for all real t .
It is known that these are exactly those characteristic functions that are
limits of normalized partial sums of independent but not necesserily
identically distributed random variables. (Thus all stable distributions
are in class L!).
In the paper we first give some conditions that allow to identify class L
distributions among those which are infinitely divisible . Then we list
examples of distributions that are in L . It includes gamma and logarithms
of gamma random varialbes, inverse Gaussian, Barrndorff-Nielsen generalized
hyperbolic distributions, log-normal, Student t-distribution,
Fisher F- distributions, etc.
At the same time we present procedures and properties that exclude
distributions from being selfdecomposable ones. In particular, compund
Poisson and geometric compound ( waiting time for the first success ) are
not in class L.
The paper concludes with a construction of an innovation process for
some autoregression of order one , and a criterion for existence of p-moments
of selfdecomposable random variables.
zjjurek@math.uni.wroc.pl (preprint available)
955. A PROPOSED PROBABILITY DISTRIBUTION FUNCTION
FOR THE STATE OF ZERO INFORMATION
Sam Allen
Probability is the science of drawing tentative conclusions based
on incomplete information. Classically it has been held that a
state of zero information corresponds to a uniform PDF over 0..1,
expressing the premise that in the absence of any information
about a particular phenomenon or event all probabilities are
equally likely. Since this distribution has a mean of 1/2, Laplace
and his followers concluded that an event about which nothing
is known has a probability of occurring of 1/2. This premise leads
to a number of paradoxes that have been hotly debated for 200 years.
It is proposed instead that in the absence of any information about
an event no probability of its occurrence should be given and that
its PDF consists of two half-height Dirac Delta functions, one at
p=0 and the other at p=1. Given the information that the event has
occurred once and has failed to occur once, this transforms into
the uniform PDF over 0..1.
samba@earthdome.com
- To see a preprint or other
information provided by the author
click here.
956. LARGE DEVIATIONS FROM THE CIRCULAR LAW
G. Ben Arous and O. Zeitouni
We prove a full large deviations principle, in the scale $N^2$,
for the empirical measure of the eigenvalues of an $N\times N$
(nonsymmetric) matrix composed of i.i.d. zero mean random variables with
variance $N^{-1}$. The (good) rate function which governs
this rate function possesses as unique minimizer
the circular law, providing an alternative proof of convergence to
the latter. The techniques are related to recent work by
Guionnet and Ben Arous, who treat the symmetric case. A crucial
role is played by precise determinant computations due to Edelman.
zeitouni@ee.technion.ac.il
957. THE SECOND LOWEST EXTREMAL INVARIANT MEASURE OF THE CONTACT PROCESS
M. Salzano, R. H. Schonmann
We study the ergodic behavior of the contact process on infinite
connected graphs of bounded degree. We show that the fundamental
notion of complete convergence is not as well behaved as it was
thought to be. In particular there are graphs for which complete
convergence holds in any number of separated intervals of values
of the infection parameter and fails for the other values of this
parameter. We then introduce a basic invariant probability measure
related to the recurrence properties of the process, and an associated
notion of convergence that we call ``partial convergence''. This notion
is shown to be better behaved than complete convergence, and to hold in
certain cases in which complete convergence fails. Relations between
partial and complete convergence are presented, as well as tools
to verify when these properties hold. For homogeneous graphs we show that
whenever recurrence takes place (i.e., whenever local survival occurs)
there are exactly two extremal invariant measures.
rhs@math.ucla.edu (electronic copy available)
958. LACK OF MONOTONICITY IN FERROMAGNETIC ISING MODEL PHASE DIAGRAMS
R. H. Schonmann, N. I. Tanaka
We study patterns of the phase diagram of ferromagnetic
Ising models on graphs under an external magnetic field.
We provide an example of a tree with only two types of
vertices on which for a range of values of the external field
there is a unique Gibbs distribution at low enough and at high
enough temperatures, while at intermediate temperatures there
is phase coexistence (in other words, a reentrance transition
takes place).
rhs@math.ucla.edu (electronic copy available)
959. A NEW PROOF THAT FOR THE CONTACT PROCESS ON HOMOGENEOUS TREES
LOCAL SURVIVAL IMPLIES COMPLETE CONVERGENCE
M. Salzano, R. H. Schonmann
We provide a new proof, substantially simpler than Zhang's
original one, that for the contact process on homogeneous
trees local survival implies complete convergence.
rhs@math.ucla.edu (electronic copy available)
960. WULFF DROPLETS AND THE METASTABLE RELAXATION OF KINETIC ISING MODELS
R. H. Schonmann, S. B. Shlosman
We consider the kinetic Ising models (Glauber dynamics)
corresponding to the infinite volume Ising model in
dimension 2 with nearest neighbor ferromagnetic interaction
and under a positive external magnetic field $h$. Minimal
conditions on the flip rates are assumed, so that all the common
choices are being considered. We study the relaxation towards
equilibrium when the system is at an arbitrary subcritical
temperature $T$ and the evolution is started from a distribution
which is stochastically lower than the ($-$)-phase. We show that
as $h \to 0$ the relaxation time blows up as $\exp(\lambda_c(T)/h)$,
with $\lambda_c(T) = w(T)^2/(12 T m^*(T))$. Here $m^*(T)$ is the
spontaneous magnetization and $w(T)$ is the integrated surface
tension of the Wulff body of unit volume. Moreover,
for $0 < \lambda < \lambda_c$, the state of the process at time
$\exp(\lambda/h)$ is shown to be close, when $h$ is small,
to the ($-$)-phase. The difference between this state and the
($-$)-phase can be described in terms of an asymptotic expansion
in powers of the external field, which can be interpreted as stating
that this state is in a sense a smooth continuation of the Gibbs
distributions with small negative $h$. This expansion can
be interpreted as describing a set of $\Cal C^\infty$ continuations
in $h$ of the family of Gibbs distributions with the negative magnetic
fields into the region of positive fields.
rhs@math.ucla.edu (electronic copy available)
961. DOBRUSHIN-KOTECK\'Y-SHLOSMAN THEOREM UP TO THE CRITICAL TEMPERATURE
D.Ioffe, R. H. Schonmann
We develop a non-perturbative version of the
Dobrushin-Koteck\'{y}-Shlosman theory of phase separation
in the canonical 2D Ising ensemble. The results are valid
for all temperatures below critical.
rhs@math.ucla.edu (electronic copy available)
962. THE TRIANGLE CONDITION FOR CONTACT PROCESSES ON HOMOGENEOUS TREES
R. H. Schonmann
We complete work of C. C. Wu, by showing that for contact
processes on homogeneous trees with degree at least 3 the
triangle condition is satisfied below the second critical
point. In particular it holds at the first critical point
which therefore has mean-field critical exponents.
rhs@math.ucla.edu (electronic copy available)
963. THE SECOND LOWEST EXTREMAL INVARIANT MEASURE OF THE CONTACT PROCESS II
M. Salzano, R. H. Schonmann
We continue the investigation of the behavior of the contact
process on infinite connected graphs of bounded degree. Some
questions left open in Salzano and Schonmann (preprint 1996)
concerning the notions of complete convergence, partial
convergence, and the criterion $r=s$ are answered.
The continuity properties of the survival probability and the
recurrence probability are studied. These order parameters are
found to have a richer behavior than expected, with the possibility
of the survival probability being discontinuous at or above the
threshold for survival. A condition which guaranties the continuity
of the survival probability above the survival point is introduced
and exploited. The recurrence probability is shown to always be
left-continuous above the recurrence point, and a necessary and sufficient
condition for its right-continuity is introduced and exploited.
It is shown that for homogeneous graphs the survival probability can
only be discontinuous at the survival point, and the recurrence
probability can only be discontinuous at the recurrence point.
For graphs which are obtained by joining a finite number of
severed homogeneous trees by means of a finite number of vertices
and edges, the survival point, the recurrence point and the
discontinuity points of the survival and recurrence probabilities
are located.
rhs@math.ucla.edu (electronic copy available)
964. A NOTE ON PERCOLATION IN A VORONOI COMPETITION--GROWTH MODEL
E.Kira, E.J.Neves, R. H. Schonmann
We study a model in which two entities (e.g., plant species,
political ideas, ...) compete for space on a plane, starting
from randomly distributed seeds, and growing deterministically
at possibly different rates. An entity which forms an infinite
cluster is considered to dominate over the other. We analyze the
occurence of such a form of domination in situations in which one
entity starts from a much larger density of seeds than the other one,
but the latter one grows at a much faster rate than the former one.
The model studied here generalizes the problem of Voronoi percolation.
rhs@math.ucla.edu (electronic copy available)
965. STABILITY OF INFINITE CLUSTERS IN SUPERCRITICAL PERCOLATION
R. H. Schonmann
A recent theorem by H\"aggstr\"om and Peres concerning independent
percolation is extended to all the quasi-transitive graphs.
This theorem states that if $0 < p_1 < p_2 \leq 1$ and percolation
occurs at level $p_1$, then every infinite cluster at level
$p_2$ contains some infinite cluster at level $p_1$. Consequences
are the continuity of the percolation probability above the percolation
threshold and the monotonicity of the uniqueness of the infinite
cluster, i.e., if at level $p_1$ there is a unique infinite cluster
then the same holds at level $p_2$.
These results are further generalized to graphs with a ``uniform
percolation'' property. The threshold for uniqueness of the infinite
cluster is characterized in terms of connectivities between large
balls.
rhs@math.ucla.edu (electronic copy available)
966. EVENTUAL INTERSECTION FOR SEQUENCES OF LEVY PROCESSES
Steven N. Evans and Yuval Peres
Consider the events $\{F_n \cap \bigcup_{k=1}^{n-1} F_k = \emptyset\}$,
$n =1,2, \ldots$, where $(F_n)_{n=1}^\infty$ is an i.i.d. sequence of
stationary random subsets of a compact group.
A plausible conjecture is that these events will not occur infinitely often
with positive probability if
$F_i$ and $F_j$ intersect with positive probability for $i \ne j$.
We present a counterexample to show that this
condition is not sufficient, and give one that is.
The sufficient condition always holds when
$F_n = \{X_t^n : 0 \le t \le T\}$ is the range of
a L\'evy process $X^n$ on the $d$-dimensional torus
with uniformly distributed initial position
and $P\{\exists 0 \le s, t \le T : X_s^i = X_t^j \} > 0$ for $i \ne j$.
We also establish an analogous result for the sequence of
graphs $\{(t,X_t^n) : 0 \le t \le T\}$.
evans@stat.Berkeley.EDU, peres@stat.Berkeley.EDU
- To see a preprint or other
information provided by the author
click here.
967. PERFECT SIMULATION IN STOCHASTIC GEOMETRY
W.S. Kendall and E. Thoennes
Simulation plays an important role in stochastic geometry and related
fields, because all but the simplest random set models tend to be
intractable to analysis. Many simulation algorithms deliver (approximate)
samples of such random set models, for example by simulating the
equilibrium distribution of a Markov chain such as a spatial
birth-and-death process. The samples usually fail to be exact because the
algorithm simulates the Markov chain for a long but finite time, and thus
convergence to equilibrium is only approximate. The seminal work by Propp
and Wilson made an important contribution to simulation by proposing a
coupling method, Coupling from the Past (CFTP), which delivers
perfect, that is to say exact, simulations of Markov chains. In this paper
we introduce this new idea of perfect simulation and illustrate it using
two common models in stochastic geometry: the dead leaves model and a
Boolean model conditioned to cover a finite set of points.
For an animated comparison (image size 128Kb) of imperfect and perfect
simulations of the dead-leaves tessellation,follow the link from
http://www.warwick.ac.uk/statsdept/Staff/WSK/abstracts.html#323
w.s.kendall@warwick.ac.uk, elke@stats.warwick.ac.uk
- To see a preprint or other
information provided by the author
click here.
- Or
here.
968. TRANSITION OPERATORS OF DIFFUSIONS REDUCE ZERO-CROSSING
Steven N. Evans and Ruth J. Williams
If $u(t,x)$ is a solution of a one-dimensional, parabolic,
second-order, linear partial differential equation (PDE), then
it is known that, under suitable conditions, the number of
zero-crossings of the function $u(t,\cdot)$ decreases (that is,
does not increase) as
time $t$ increases. Such theorems have applications
to the study of blow-up of solutions of semilinear PDE,
time dependent Sturm Liouville theory,
curve shrinking problems and control theory.
We generalise the PDE results by showing that the transition operator
of a (possibly time-inhomogenous) one-dimensional
diffusion reduces the number of zero-crossings
of a function or even, suitably interpreted, a signed measure.
Our proof is completely
probabilistic and depends in a transparent manner on little more than the
sample-path continuity of diffusion processes.
evans@stat.berkeley.edu williams@russel.ucsd.edu
- To see a preprint or other
information provided by the author
click here.
969. ON THE EXISTENCE AND NON-EXISTENCE OF FINITARY CODINGS FOR A CLASS OF
RANDOM FIELDS
J. van den Berg and J. E. Steif
We study the existence of finitary codings (also called finitary
homomorphisms or finitary factor maps) from a finite-valued i.i.d.\
process to certain random fields. For Markov random fields we show,
using ideas of Marton and Shields, that the presence of a phase transition
is an obstruction for the existence of the above coding: this yields
a large class of Bernoulli shifts for which no such coding exists.
Conversely, we show that for the stationary distribution of a monotone
exponentially ergodic probabilistic cellular automaton such a coding does
exist. The construction of the coding is partially inspired by the
Propp-Wilson algorithm for exact simulation.
In particular, combining our results with a theorem of Martinelli and
Olivieri, we obtain the fact that for the plus state for the ferromagnetic
Ising model on $\bbbz^d$, $d \geq 2$, there is (not) such a coding when
the interaction parameter is below (above) its critical value.
J.van.den.Berg@cwi.nl steif@math.chalmers.se
970. LARGE DEVIATIONS FOR SOME POISSON RANDOM INTEGRALS
Zbigniew J. Jurek and Liming Wu
In the theory of large deviations Schilder Theorem is one of the
main examples. It gives the large deviation estimates for convergence of
{tW} to zero, as t tends to zero, where W is the Brownian Motion. In our
paper we investigate analogous problem for {tN~(f)}, as t tends to zero,
where N~(f) is the integral of f with respect to the compensated Poisson
point process N~, on a measure space (E,m). We have proved that the speed
function in the LDP ( large deviation principle) is (-logt)/t and the
intensity functional I is given as the Minkowski functional of the compact
convex hull spanned by the support of a measure fm . This is for the case
of vector-valued functions f which are bounded and m-square integrable .
For unbounded real f we also established LDP with different speed and
rate. Finally we apply our results to selfdecomposable distributions given
as probability distributions of random integrals with respect to Levy
procesess ( their BDLP = background driving Levy processes).
zjjurek@math.uni.wroc.pl or Wuliming@ucfma.univ-bpclermont.fr
971. THE LIMIT BEHAVIOUR OF ELEMENTARY SYMMETRIC POLYNOMIALS OF I.I.D.
RANDOM VARIABLES WHEN THEIR ORDER TENDS TO INFINITY
Peter Major
Let $\xi_1,\xi_2,\dots$ be a sequence of i.i.d. random variables, and
consider the elementary symmetric polynomial $S^{(k)}(n)$ of order
$k=k(n)$ of the first $n$ elements $\xi_1,\dots,\xi_n$ of this sequence.
We are interested in the limit behaviour of $S^{(k)}(n)$ with an
appropriate transformation if ${k(n)\over n}\to\alpha$, $0<\alpha<1$.
Since $k(n)\to\infty$ as $n\to\infty$, the classical methods cannot be
applied in this case, and new kind of results appear. We solve the
problem under some conditions which are satisfied in the generic case.
major@math-inst.hu
- To see a preprint or other
information provided by the author
click here.
972. ON THE TAIL BEVAVIOR OF THE DISTRIBUTION FUNCTION OF MULTIPLE
STOCHASTIC INTEGRALS IN SEPARABLE METRIC SPACES
Peter Major and Lidia Rejto
Let $\bar{\mu}_{n}(\cdot)$ denote the empirical measure of a sample
of i.i.d. random variables with values on a separable metric space
$(X, {\cal X})$ with distribution $\mu$. Define the normalized empirical
meausures $\mu_n(\cdot)=\sqrt{n}(\bar\mu_n(\cdot)-\mu(\cdot))$
and consider the integrals
$$
{\cal I}(t) = \int_{X_{t}} \int_{X} \dots
\int_{X} f(u_1,\dots,u_s) \mu_n(du_1) \dots \mu_n(du_s),
$$
where $X_s \subseteq X_t$ for all $s\le t$, $X_0=\emptyset$, $X_1=X$,
$\mu(X_t)=t$ and $f$ is a bounded measurable function which disappears
on the diagonal set of the product space $X^s$.
We give a sharp upper bound on the probability
$P(\sup_{0\leq t \leq 1}|{\cal I}(t)| >x)$.
This is a generalization of the result in a paper of Major where this
estimate was proved in the special case when the
space X is the interval [0,1], and the empirical distribution of a
sequence of independent and on [0,1] uniformly distributed random
variables is considered.
major@math-inst.hu
- To see a preprint or other
information provided by the author
click here.
973. ALMOST SURE LIMIT THEOREMS
Ildar Ibragimov, Mikhail Lifshits
Let $\{z_k\}$ be a sequence of random vectors converging in law. Consider
empirical measures
$ Q_n = {1\over \ln n} \sum_{k=1}^n {1\over k} \delta_{z_k}$. The
statements
about the weak convergence of $Q_n$ (or other similar measures) with
probability one are called almost sure limit theorems. We suggest several
methods which enable to deduce a.s. limit theorems from classical limit
theorems, prove an a.s. invariance principle, investigate the convergence
of
generalized moments of empirical measures. In contrast to many previous
works,
where only normal limit law is considered, our results are valid in the
case of general limit distribution.
ibr32@pdmi.ras.ru lifts@mail.rcom.ru
974. ON STATES OF EXIT MEASURES FOR SOME SUPERDIFFUSIONS : THE
CRITICAL CASE
Romain Abraham
In a previous paper, Sheu studied the states of the exit
measure of the $(L,\beta)$-superdiffusion ($1<\beta \le 2$) from a
smooth domain of $R^d$. He proved that the critical dimension is
$d_c=(\beta +1)/(\beta -1)$. If $d<d_c$, the exit measure is absolutely
continuous with respect to the surface measure whereas, if $d>d_c$, the
exit measure is singular. Here, we study the case $d=d_c$ and $L=\Delta$
by using a path-valued process constructed via the Brownian snake
by subordination (constructed by Bertoin, Le Gall and Le Jan).
We get in that case an upper bound for the hitting probabilities
of small balls on the boundary and as a consequence obtain that
the exit measure of the $(\Delta,\beta)$-superdiffusion is still
singular in critical dimension.
abraham@math-info.univ-paris5.fr
975. SEQUENTIAL SELECTION OF AN INCREASING SEQUENCE FROM A SAMPLE
OF RANDOM SIZE
Alexander Gnedin
Following a long-standing suggestion by Samuels and Steele (1981) we
study the problem of sequential selection of an increasing sequence from
a random sample of size $N$, where $N$ is geometrically distributed. The
maximum expected length of selected subsequence is shown to be
asymptotic to $1/\sqrt{p}$, as $p\to 0.$
gnedin@math.uni-goettingen.de
976. ON THE POISSON-DIRICHLET LIMIT
Alexander Gnedin
Kingman showed that if the vector $X_N$ is distributed according to the
Dirichlet law then the vector of descending order statistics
converges, under certain conditions, to a nondegenerate
limit. Ordering of the components was motivated by the fact that any
fixed (say, second) component of
$X_N$ converges (in probability) to zero.
The following observation seems to be new: $X_N$ has a
nondegenerate limit if we identify the vector with a
random partition of the unit interval. The limiting object is then a
random open set with infinetely many component intervals which sum to
one. Convergence of this kind does not require rearranging the
components of $X_N$ and
implies existence of limit distributions for a class of functionals
which are not covered by the Kingman's approach.
The convergence of the vector of order statistics of $X_N$ to the
vector of decreasing interval lengthes is a corollary.
gnedin@math.uni-goettingen.de
977. ERGODIC CONTROL OF SEMILINEAR STOCHASTIC EQUATIONS AND
HAMILTON-JACOBI EQUATIONS
B. Goldys and B. Maslowski
In this paper we consider optimal control of
stochastic semilinear equations with linearly increasing
drift and cylindrical noise. We show existence and uniqueness (up to an
additive constant) of solutions to the stationary
Hamilton-Jacobi equation associated with the cost
functional given by the asymptotic average per unit time
cost. As a consequence we find the optimizing controls
given in a feedback form. We
prove also some new results on the transition
semigroups of semilinear diffusions acting in the spaces
of continuous functions with the weighted sup norms and
on the optimal control of semilinear equations for the
discounted cost functional.
B.Goldys@unsw.edu.au
