Probability Abstracts 50
This document contains abstracts 1229-1279.
They have been mailed on April 30, 1999.
1229. BROWNIAN MOTION, BRIDGE, EXCURSION, AND MEANDER CHARACTERIZED
BY SAMPLING AT INDEPENDENT UNIFORM TIMES
Jim Pitman
For a random process $X$ consider the random vector defined by the values
of $X$ at times $0 < U_{n,1} < ... < U_{n,n} < 1$
and the minimal values of $X$ on each of the intervals between
consecutive pairs of these times, where the $U_{n,i}$ are the order statistics
of $n$ independent uniform $(0,1)$ variables, independent of $X$.
The joint law of this random vector is explicitly described when
$X$ is a Brownian motion. Corresponding results for Brownian bridge,
excursion, and meander are deduced by appropriate conditioning.
These descriptions yield numerous new identities involving the
laws of these processes, and simplified proofs of various known results,
including Aldous's characterization of the random tree constructed by
sampling the excursion at $n$ independent uniform times,
Vervaat's transformation of Brownian bridge into Brownian excursion,
and Denisov's decomposition of the Brownian motion at the time of its minimum
into two independent Brownian meanders.
Other consequences of the sampling formulae are Brownian
representions of various special functions, including Bessel polynomials,
some hypergeometric polynomials, and the Hermite function.
Various combinatorial identities involving
random partitions and generalized Stirling numbers are also obtained.
pitman@stat.Berkeley.EDU
Technical Report No. 545, Department of Statistics, U.C. Berkeley
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1230. THE DISTRIBUTION OF LOCAL TIMES OF A BROWNIAN BRIDGE
Jim Pitman
L\'evy's approach to Brownian local times is used to give a simple derivation
of a formula of Borodin which determines the distribution of the local time
at level x up to time 1 for a Brownian bridge of length 1 from 0 to b.
A number of identities in distribution involving functionals of
the bridge are derived from this formula.
A stationarity property of the bridge local times is derived by
a simple path transformation, and related to Ray's description of the
local time process of Brownian motion stopped at an independent
exponential time.
pitman@stat.Berkeley.EDU
Technical Report No. 539, Department of Statistics, U.C. Berkeley.
To appear in S\'eminaire de Probabilit\'es XXXIII.
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1231. THE LAW OF THE MAXIMUM OF A BESSEL BRIDGE
Jim Pitman and Marc Yor
Let $M_d$ be the maximum of a standard Bessel bridge of dimension $d$.
A series formula for $P(M_d < a)$ due to Gikhman and Kiefer for
$d = 1,2, \ldots$ is shown to be valid for all real $d >0$.
Various other characterizations of the distribution of $M_d$ are given,
including formulae for its Mellin transform, which is an entire function.
The asymptotic distribution of $M_d$ as is described both as $d$ tends to
$\infty$ and as $d$ tends to $0$. Keywords: Brownian bridge, Brownian
excursion, Brownian scaling, local time, Bessel process, zeros of
Bessel functions, Riemann zeta function.
pitman@stat.Berkeley.EDU
Technical Report No. 534, Department of Statistics, U.C. Berkeley.
To appear in Electronic J. Prob.
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1232. PATH DECOMPOSITIONS OF A BROWNIAN BRIDGE RELATED TO THE RATIO
OF ITS MAXIMUM AND AMPLITUDE
Jim Pitman and Marc Yor
We give two new proofs of Csaki's formula for the law of the ratio 1-Q of
the maximum relative to the amplitude (i.e. the maximum minus minimum)
for a standard Brownian bridge.
The second of these proofs is based on an absolute continuity relation
between the law of the Brownian bridge restricted to the event (Q < v)
and the law of a process obtained by a Brownian scaling operation
after back-to back joining of two independent three-dimensional Bessel
processes, each started at v and run until it first hits 1.
Variants of this construction and some properties of the joint
law of Q and the amplitude are described.
pitman@stat.Berkeley.EDU
Technical Report No. 532, Department of Statistics, U.C. Berkeley
To appear in Studia Math. Sci. Hungar.
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1233. THICK POINTS FOR TRANSIENT SYMMETRIC STABLE PROCESSES
Amir Dembo, Yuval Peres, Jay Rosen and Ofer Zeitouni
Let T(x,r) denote the total occupation measure of the
ball of radius r centered at x for a transient symmetric
stable processes of index b<d in R^d and K(b,d)
denote the norm of the convolution with its 0-potential
density, considered as an operator on L^2(B(0,1),dx).
We prove that as r approaches 0, almost surely
$sup_{|x| < 1} T(x,r)/(r^b|log r|) \to b K(b,d)$.
Furthermore, for any 0<a<b/K(b,d), the Hausdorff
dimension of the set of ``thick points'' x for which
limsup_{r \to 0} T(x,r)/(r^b |log r|)=a,
is almost surely b-a/K(b,d); this is the correct
scaling to obtain a nondegenerate ``multifractal spectrum''
for transient stable occupation measure.
The liminf scaling of T(x,r) is quite different:
we exhibit positive, finite, non-random c(b,d), C(b,d),
such that almost surely
c(b,d) < sup_x liminf_{r \to 0} T(x,r)/r^b < C(b,d).
amir@math.stanford.edu peres@math.huji.ac.il
jrosen3@idt.net zeitouni@ee.technion.ac.il
http://www-stat.stanford.edu/~amir
1234. LONGEST INCREASING SUBSEQUENCES: FROM PATIENCE SORTING
TO THE BAIK-DEIFT-JOHANSSON THEOREM
David Aldous and Persi Diaconis
We describe a simple one-person card game, Patience Sorting.
Its analysis leads to a broad circle of ideas linking
Young tableaux with the longest increasing subsequence
of a random permutation via the Schensted correspondence.
A recent highlight of this area is the work of Baik-Deift-Johansson
which yields asymptotic probability laws via hard analysis of
Toeplitz determinants.
aldous@stat.berkeley.edu
http://www.stat.berkeley.edu/users/aldous/bibliog.html
1235. SEQUENTIAL ESTIMATION OF FUNCTIONS OF p FOR BERNOULLI TRIALS
Steven L. Hubert and Ronald Pyke
We investigate the problem of estimating a function of p, the probability
of success in a sequence of independent identically distributed Bernoulli
trials. The estimation
problems of interest here are those which become more
difficult when p is small, such as when estimating the rate, 1/p, with
squared error loss, or when estimating p with relative squared error loss.
A sequential estimation procedure is required when the specification of the
problem requires that the solution be valid for all values of p. The
formulation used is typical of those taken in the literature, but a new
approach to the asymptotic analysis of the problem is taken, in which the
asymptotics are driven by letting the parameter, p, approach zero.
Previous approaches to the study of the asymptotics have always fixed the
distributional parameters, such as p in the Bernoulli case, and allowed
some other parameter of the problem (e.g., accuracy or cost)
to vary. The emphasis in this paper is
on the estimation of powers of p with a loss function equal to squared
error loss multiplied by another power of p. Consistency and efficiency
are defined in this framework and conditions under which the procedure is
consistent and efficient are given.
pyke@math.washington.edu
1236. THE SOLIDARITY OF RECURRENT MRP'S RELATIVE TO STABLE LIMIT THEOREMS
Ronald Pyke
It is known that for a wide class of irreducible recurrent Markov Chains and
Markov Renewal processes,
the finiteness of the p-th moment of recurrence
times is finite for one state if and only if it is finite for all states.
This result also holds for a wide class of functionals of the processes
between successive occurrence times. Such results are known as solidarity
theorems. In this paper, a solidarity theorem is proved concerning the
property of `being in the domain of attraction of a stable law with
specified parameters.' A basic identity linking additive random functions
determined by two distinct states is emphasized, and used to derive
solidarity results for moments and for the WLLN.
pyke@math.washington.edu
1237. MARKOV RENEWAL MODELLING OF POISSON TRAFFIC AT INTERCECTIONS HAVING
SEPARATE TURN LANES
Rudy Gideon and Ronald Pyke
Consider a two-lane road which is intersected on one side by a single-lane
secondary road. A single car waiting on the secondary road may merge into
either the nearest or the farthest lane. It is assumed that the traffic in
each lane is independent of the other lanes and that the inter-arrival
times of cars at the intersection in their respective lanes is
exponential. The main purpose of this paper is to study the queue size on
the secondary road when the secondary road has a special right-turn lane
(or left-turn lane in some countries) which allows some cars to merge into the
nearest lane of the main road even when other cars waiting to enter the far
lane are present. The problem is approached by first setting
up a four-state Markov Renewal process to describe the traffic on the main
road. Next the merging process on the secondary road is described
as a Markov Renewal process with a random environment. The
event that the queue is empty is studied, and conditions are stated under
which this event is recurrent or transient. Finally, the quantities which
occur in the conditions for recurrence of an empty queue are derived
explicitly for a one-car right turn lane.
pyke@math.washington.edu, gideon@selway.umt.edu
1238. THE REASSEMBLING OF SHATTERED BROWNIAN SHEET
Ronald Pyke
If $D_n:n\ge 1$ is a given countable collection of Borel subsets
of the unit square $[0, 1]^2=I^2$ and Z is a standard Brownian sheet over
I^2, is it possible to reconstruct Z from the knowledge of all of the
patches $Z-c_n$ over transformed domains $\tau_n(D_n)$ for unknown
constants c_n and unknown rotation-translation transformations $\tau_n$?
We show that the answer to this question is yes under fairly natural
restrictions on the sets D_n. The main property of Brownian sheet that
leads to this possibility is that the local behaviour of Z around a point
$\bt$ actually determines $\bt$. In this sense, a Brownian sheet carries with
it its own location coordinates.
pyke@math.washington.edu
1239. THE ISING MODEL ON DILUTED GRAPHS AND STRONG AMENABILITY
Olle H\"aggstr\"om, Roberto H. Schonmann and Jeffrey E. Steif
Say that a graph has persistent transition if the Ising model on
the graph can exhibit a phase transition (nonuniqueness of Gibbs
measures) in the presence of a nonzero external field. We show that
for nonamenable graphs, for Bernoulli percolation with $p$ close
to 1, all the infinite clusters have persistent transition. On the
other hand, we show that for transitive amenable graphs, the infinite
clusters for any stationary percolation do not have persistent
transition. This extends a result of Georgii for the cubic lattice.
A geometric consequence of this latter fact is that the infinite
clusters are strongly amenable (i.e., their anchored Cheeger
constant is 0). Finally we show that the critical temperature for the
Ising model with no external field on the infinite clusters of
Bernoulli percolation with parameter $p$, on an arbitrary bounded
degree graph, is a continuous function of $p$.
olleh@math.chalmers.se rhs@math.ucla.edu steif@math.chalmers.se
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1240. PARTIAL ORDERS AND MINIMIZATION OF RECORDS
IN A SEQUENCE OF INDEPENDENT RANDOM VARIABLES
Raul Gouet and Jaime San Martin
Given independent random variables $X_1,...,X_n$, with continuous
distributions $F_1,\ldots,F_n$, we investigate the order in
which these random variables should be arranged so as to
minimize the number of upper records. We show that records
are stochastically minimized if the sequence $F_1,\ldots,F_n$
decreases with respect to a partial order, closely related
to the monotone likelihood ratio property. Also, the expected
number of records is shown to be minimal when the distributions
are comparable in terms of a one sided hazard rate ordering.
Applications to parametric models are considered.
rgouet@dim.uchile.cl jsanmart@dim.uchile.cl
1241. MULTIDIMENSIONAL SYMMETRIC STABLE PROCESSES
Zhen-Qing Chen
The purpose of this paper is to survey the recent progress
on the study of potential theoretic properties for
symmetric stable processes as well as to present some new
results on the estimates of Green function, Poisson and
Martin kernels for symmetric stable processes in bounded
$C^{1, 1}$ open sets in $R^n$ with $n\geq 2$. The new
results give explicit information on how the comparing
constants depend on parameter $\alpha$. By passing
$\alpha$ to 2, they yield a new (and unified) proof of
the Green function and Poisson kernel estimates for
Brownian motion. In addition to these new estimates, the
paper surveys recent progress in the study of notions of
harmonicity, integral representation of harmonic
functions, boundary Harnack inequality, conditional
lifetime, conditional gauge and intrinsic
ultracontractivity for symmetric stable processes.
zchen@math.washington.edu
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1242. NUMEROTATION DES RECORDS D'UN PROCESSUS DE POISSON
PONCTUEL
Jean Brossard, Christophe Leuridan
Let $X$ be a Poisson point process on $(0,\infty)$. Assume
that its intensity $\mu$ satisfies $\mu(0,\infty)=\infty)$
and $0<\mu[x,\infty)<\infty)$ for every $x>0$. We prove
that the record-breaking instants for $X$ (the times $t$
such that $X_t > X_s$ for every $s \in [0,t)$) can not be
increasingly enumerated by stopping times $T_n$ indexed
by $Z$. The proof relies on the non-existence of a Markov
process indexed by $Z$ with transition probabilities
$\Pi(x,A)= \frac {\mu(A \cap (x,\infty))} {mu(x,\infty)}$.
These results are no longer true if we replace $>$ with
$\geq$ in the definition of record-breaking instants, and
$(x,\infty)$ with $[x,\infty)$ in $\Pi(x,A)$. A necessary
and sufficient condition (in terms of atoms of $\mu$) is
given for the existence of such an enumeration.
Last, we study the problem of $Z/qZ$-enumerations (with
$q \leq 2$). We show that there is no $Z/qZ$-valued,
adapted, right-continuous process that is constant between
two record-breaking instants and has jump 1 when $X$ beats
the record. Surprisingly, there exist some independent
enlargements of the filtration in which $X$ remains
Poisson and adapted $Z/qZ$-enumerations exist.
brossard@fourier.ujf-grenoble.fr leuridan@fourier.ujf-grenoble.fr
1243. CHAINES DE MARKOV INDEXEES PAR -N : EXISTENCE ET
COMPORTEMENT
Jean Brossard, Christophe Leuridan
Let $\Pi$ be a Markov transition kernel. Does there exist
a Markov process indexed by negative integers with
transition kernel $\Pi$ ? In the paper, we give conditions
on the kernel $\Pi$ for the existence of such a process.
When such a process exists, we prove that the extremal
laws may have only two kinds of behaviour : a stationnary-
type behaviour or a deterministicly transient behaviour.
In the secund case, the time is a deterministic function
of the position before a finite random time. We give
examples illustrating these situations.
brossard@fourier.ujf-grenoble.fr leuridan@fourier.ujf-grenoble.fr
1244. A REMARK ON TSIRELSON'S STOCHASTIC DIFFERENTIAL EQUATION
M. \'Emery and W. Schachermayer
Tsirelson's stochastic differential equation is called
``celebrated and mysterious'' by Rogers and Williams [RW].
This note aims at making it a little more celebrated and
a little less mysterious.
Using a deterministic time-change, we translate the study
of Tsirelson's equation into the study of ``eternal''
Brownian motion on the circle. This allows us to show that
the filtration generated by any solution of Tsirelson's
equation is also generated by some Brownian motion (which,
however, cannot be the Brownian motion driving the equation,
because the equation has no strong solution).
[RW] --- L.C.G. Rogers and D. Williams. Diffusions, Markov
Processes, and Martingales. Volume~2: It\^o Calculus.
Wiley, 1987.
walter.schachermayer@fam.tuwien.ac.at
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1245. BROWNIAN FILTRATIONS ARE NOT STABLE UNDER EQUIVALENT TIME-CHANGES
M. \'Emery and W. Schachermayer
L. Dubins, J. Feldman, M. Smorodinsky and B. Tsirelson have
shown in [DFST~96] that a small perturbation of its probability
law can transform Brownian motion into a process whose
natural filtration is not generated by any Brownian motion
whatsoever.
Marc Yor raised the following question: Is there something
similar to the DFST-phenomenon, with a change of time instead
of a change of probability law? More precisely, does there
exist on Wiener space an absolutely continuous, strictly
increasing time-change such that the time-changed
filtration is no longer Brownian?
The present paper shows that the answer to Yor's question
is positive; moreover, as was the case with the perturbation
of measure considered in [DFST~96], the perturbation of time
can be made arbitrarily small.
[DFST~96] --- L. Dubins, J. Feldman, M. Smorodinsky \& B. Tsirelson.
Decreasing sequences of $\sigma$-fields and a measure change
for Brownian motion. Ann. Prob. 24, 882--90, 1996.
walter.schachermayer@fam.tuwien.ac.at
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1246. DOMAINS OF ATTRACTION FOR EXPONENTIAL FAMILIES
August A. Balkema, Claudia Kluppelberg and Sidney Resnick
With the distribution F of the rv X we associate
the natural exponential family of distribution
$F_\lambda $ where
$$dF_{\lambda}(x)=e^{\lambda x}dF(x)/Ee^{\lambda X}$$
for
$\lambda\in\Lambda:=\{\lambda\in R: E(\exp\{\lambdaX}
<\infty\}$}. Assume
$\lambda_\infty:=\sup \Lambda $
does not lie in $\Lambda $.
Let $\lambda \uparrow\lambda_\infty$,
then non-degenerate limit laws for the normalised
distributions $F_\lambda (a_\lambda x+b_\lambda)$
are the normal and gamma distributions. Their domains
of attractions are determined.
Applications to saddlepoint and gamma approximations
are considered.
guus@wins.uva.nl cklu@mathematik.tu-muenchen.de sid@orie.cornell.edu
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1247. STATISTICAL ESTIMATION FOR MULTIPLICATIVE CASCADES
Mina Ossiander and Ed Waymire
The probability distribution of the cascade generators in a
random multiplicative cascade represents a hidden parameter
which is reflected in the fine scale limiting behavior of
the scaling exponents of a single sample cascade realization
as a.s. constants. We identify a large class of cascade
generators uniquely determined by these scaling exponents.
For this class and using scaling exponents from single
sample realizations as a function of spatial resolution,
we provide: (i.) asymptotic consistency and (ii.) confidence
intervals for estimates of the cumulant generating function
(log Laplace transform) of the cascade generator distribution
in a neighborhood of the origin.
ossiand@math.orst.edu waymire@math.orst.edu
1248. LIMSUP RANDOM FRACTALS
Davar Khoshnevisan, Yuval Peres and Yimin Xiao
Orey and Taylor (1974) introduced sets $F(u)$ of
``fast points'' where Brownian increments are unusually
large: $F(u)$ consists of the points $t$ in $[0,1]$
where $ \limsup |B(t+h)-B(t)| / M(h)$ as $ h \to 0$
is at least $u$. Here $M(h)= \sqrt{ 2h |\log h|}$ is the
modulus of continuity. They proved that for $u$ in $(0,1]$,
the Hausdorff dimension of $F(u)$ is $1-u^2$ a.s. We prove
that for any analytic set $E$ in $[0,1]$, the supremum of
the $u$ such that $E$ intersects $F(u)$, a.s. equals the
square root of the packing dimension of $E$. We derive this
from a general result that applies to many other random
fractals defined by limsup operations. This result also yields
extensions of certain fractal functional limit laws due to
Deheuvels and Mason (1994). In particular, we prove that for
any absolutely continuous function $f$ on $[0,1]$ such that
$f(0)=0$ and the energy $\int |f'|^2$ is lower than the
packing dimension of $E$, there a.s. exists some $t \in E$ so
that $f(s)$ can be uniformly approximated in $[0,1]$ by
normalized Brownian increments $[B(t+sh)-B(t)] / M(h)$;
such uniform approximation is a.s. impossible if the energy
of $f$ is higher than the packing dimension of $E$.
davar@math.utah.edu peres@math.huji.ac.il xiao@math.utah.edu
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1249. THE MANIFOLD-VALUED DIRICHLET PROBLEM FOR SYMMETRIC
DIFFUSIONS
Jean Picard
Harmonic maps between two Riemannian manifolds $M$ and $N$
are often constructed as energy minimizing maps. This
construction is extended for the Dirichlet problem to the
case where the Riemannian energy functional on $M$ is
replaced by a more general Dirichlet form. We obtain weakly
harmonic maps and prove that these maps send the diffusion
to $N$-valued martingales. The basic tools are the
reflected Dirichlet space and the stochastic calculus for
Dirichlet processes.
picard@ucfma.univ-bpclermont.fr
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1250. SMOOTHNESS OF HARMONIC MAPS FOR HYPOELLIPTIC DIFFUSIONS
Jean Picard
Harmonic maps are viewed as maps sending a fixed diffusion
to manifold-valued martingales. Under a convexity
condition, we prove that the continuity of real-valued
harmonic functions implies the continuity of harmonic maps.
Then we prove with a probabilistic method that continuous
harmonic maps are smooth under Hormander's condition; the
proof relies on the study of martingales with values in the
tangent bundle.
picard@ucfma.univ-bpclermont.fr
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1251. LARGE DEVIATION PRINCIPLES FOR EUCLIDEAN FUNCTIONALS
AND OTHER NEARLY ADDITIVE PROCESSES
Timo Seppalainen and J. E. Yukich
We prove a large deviation principle for a process indexed
by cubes of the multidimensional integer lattice or
Euclidean space, under approximate additivity and regularity
hypotheses. The rate function is the convex dual of the
limiting logarithmic moment generating function. In some
applications the rate function can be expressed in terms of
relative entropy. The general result applies to processes in
Euclidean combinatorial optimization, statistical mechanics,
and computational geometry. Examples include the length of
the minimal tour (the traveling salesman problem), the
length of the minimal matching graph, the length of the
minimal spanning tree, the length of the k-nearest neighbors
graph, and the free energy of a short-range spin glass
model.
seppalai@iastate.edu jey0@lehigh.edu
1252. FAVOURITE SITES, FAVOURITE VALUES AND JUMPING SIZES
Endre Csaki, Pal Revesz and Zhan Shi
We determine: (a) the joint almost sure asymptotic behaviour
of the most visited site and the maximum local time of
one-dimensional simple random walk or Brownian motion; (b)
the maximal jumping size of the most visited site. These
solve two open problems of Erdos and Revesz.
csaki@math-inst.hu revesz@ci.tuwien.ac.at zhan@proba.jussieu.fr
1253. ASYMPTOTIC STABILITY OF NON LINEAR
SEMI-GROUPS OF FEYNMAN-KAC TYPE
Pierre Del Moral, Laurent Miclo
We study the asymptotic stability properties of a
class of non linear semi-group in distribution space
arising in the study of certain parabolic equation,
non linear estimation problems, biology and physics.
Here we use Dobrushin ergodic coefficient to prove
uniform bounds with respect to the total variation
norm including for temporally non homogeneous semigroups.
Our results also complement the stability properties for
McKean-Vlasov diffusions of Tamura to interacting
processes in which the interaction goes through jumps.
delmoral@cict.fr miclo@cict.fr
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1254. INTERACTING PARTICLE SYSTEM APPROXIMATIONS
OF THE KUSHNER-STRATONOVITCH EQUATION
Dan Crisan, Pierre Del Moral, Terry J. Lyons
In this paper we consider the continuous time filtering
problem and we estimate the order of convergence of an
interacting particle system scheme presented by the
authors in previous works. We will discuss how the
discrete time approximating model of the
Kushner-Stratonovitch equation and the genetic type
interacting particle system approximation combine.
We present quenched error bounds as well as mean order
converge results.
D.Crisan@statslab.cam.ac.uk delmoral@cict.fr t.lyons@ic.ac.uk
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1255. A MORAN PARTICLE SYSTEM APPROXIMATION
OF FEYNMAN-KAC FORMULAE
Pierre Del Moral, Laurent Miclo
In this paper we present a weighted sampling Moran
particles system model for the numerical solving of
a class of Feynman-Kac formulae which arises in the
study of certain parabolic differential equations,
non linear economic modelling, non linear filtering
problems and elsewhere.
Our major motivation is from advanced signal processing,
but our approach is context free. We will show that
under certain regularity conditions the resulting
interacting particles scheme converges to the considered
non linear equations.
In the setting of non linear filtering, the convergence
exponent resulting from our proof also improves recent
results on other particles interpretations of these
equations.
delmoral@cict.fr miclo@cict.fr
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1256. BRANCHING AND INTERACTING PARTICLE SYSTEM
APPROXIMATIONS OF FEYNMAN-KAC FORMULAE
WITH APPLICATIONS TO NON LINEAR FILTERING
Pierre Del Moral, Laurent Miclo
This survey paper focuses on interacting particle systems methods
for the numerical solving of a class of Feynman-Kac
formulae arising in the study of certain parabolic
differential equations, physics, non linear filtering
and elsewhere.
We have tried to give an expos'e of the mathematical theory
that it is useful in analyzing the convergence of such
particle approximating models including law of large
numbers, large deviations principles, fluctuations and
empirical process theory as well as semi-group techniques
and limit theorems for processes.
In addition, we investigate the delicate and
probably the most important problem of the long time
behavior of such interacting measure valued processes.
We will show how to connect this problem with the
asymptotic stability of the corresponding limiting process
so that to derive useful uniform convergence results
with respect to the time parameter.
Several variations and refinements of generic interacting
particle system models are also presented. In the last part
of this work we apply these results to continuous time
and discrete time filtering problems.
delmoral@cict.fr miclo@cict.fr
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1257. ENTANGLEMENT IN PERCOLATION
Geoffrey R. Grimmett and Alexander E. Holroyd
We study the existence of finite and infinite entangled graphs in the bond
percolation process in three dimensions with density $p$. After a
discussion of the relevant definitions, the entanglement critical
probabilities are defined. The size of the maximal entangled graph at the
origin is studied for small $p$, and it is shown that this graph has
radius whose tail decays at least as fast as $\exp(-\alpha n / \log n)$;
indeed, the logarithm may be replaced by any iterate of logarithm for an
appropriate positive constant $\alpha$. We explore the question of almost
sure uniqueness of the infinite maximal open entangled graph when $p$ is
large, and we establish two relevant theorems. We make several conjectures
concerning the properties of entangled graphs in percolation.
g.r.grimmett@statslab.cam.ac.uk a.e.holroyd@statslab.cam.ac.uk
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1258. ORDERED OVERLAPS IN DISORDERED MEAN-FIELD MODELS
Francis Comets and Amir Dembo
Let M(N) be a sequence of integers with M --> infinity
as N --> infinity and M=o(N).
For bounded i.i.d. r.v. $\xi_i^k$
and bounded i.i.d. r.v. $\sigma_i$, we study the large
deviation of the family of ordered scalar products
$X^k = N^{-1} \sum_{i=1}^N \sigma_i \xi_i^k, k \leq M$,
under the distribution conditioned on the
$\xi_i^k$'s. To get a full large deviation principle,
it is necessary to specify also the total norm
$(\sum_{k \leq M} (X^k)^2)^{1/2}$,
which turns to be associated with some extra Gaussian
distribution. Our results apply to disordered, mean-field
systems, including generalized Hopfield models in the
regime of a sublinear number of patterns.
We build also a class of examples where this norm is
the crucial order parameter.
comets@math.jussieu.fr amir@math.stanford.edu
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information provided by the author
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1259. ON THE COHOMOLOGY OF FLOWS OF STOCHASTIC AND RANDOM
DIFFERENTIAL EQUATIONS
Peter Imkeller and Christian Lederer
Suppose a stochastic differential equation on
$d$-dimensional Euclidean space has a global forward
flow. We show that if the Lie algebra generated by
its diffusion vector fields is finite dimensional
and solvable, then the flow is conjugate to the flow
of a non-autonomous random differential equation,
i.e. one can be transformed into the other via a
random diffeomorphism of $d$-dimensional Euclidean
space. Viewing a stochastic differential equation in
this form which appears closer to the setting of
ergodic theory, can be an advantage when dealing
with asymptotic properties of the system. To
illustrate this, we give sufficient criteria for the
existence of global random attractors in terms of
the random differential equation, which are applied
in the case of the Duffing-van der Pol oscillator
with two independent sources of noise.
imkeller@mathematik.hu-berlin.de lederer@mathematik.hu-berlin.de
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information provided by the author
click here.
1260. LOWER TAILS OF QUADRATIC FUNCTIONALS OF SYMMETRIC STABLE PROCESSES
Zhan Shi
Let $X$ be a symmetric stable process of index
$\alpha\in (0,2]$. We determine the exact asymptotics
(in the logarithmic scale) of the lower tail probability
of the integral of $X^2$ with respect to a finite measure
on $[0,1]$.
zhan@proba.jussieu.fr
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information provided by the author
click here.
1261. PERFECT METROPOLIS-HASTINGS SIMULATION OF LOCALLY STABLE POINT PROCESSES
Wilfrid S.Kendall and Jesper Moeller
In this paper we investigate the application of perfect simulation, in
particular Coupling from The Past (CFTP), to the simulation of random
point processes. We give a general formulation of the method of
dominated CFTP and apply it to the problem of perfect simulation of
general locally stable point processes as equilibrium distributions of
spatial birth-and-death processes. We then investigate discrete-time
Metropolis-Hastings samplers for point processes, and show how a
variant which samples systematically from cells can be converted into a
perfect version. An application is given to the Strauss point process.
w.s.kendall@warwick.ac.uk jm@math.auc.dk
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information provided by the author
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1262. ON THE WULFF CRYSTAL IN THE ISING MODEL
Raphael Cerf and Agoston Pisztora
We study the phase separation phenomenon in the Ising model in
dimensions $d\ge 3$. To this end we work in a large box with plus
boundary conditions and we condition the system to have an excess
amount of negative spins so that the empirical magnetization is
smaller than the spontaneous magnetization $m^*$. We confirm the
prediction of the phenomenological theory by proving that
with high probability a single droplet of the minus phase emerges
surrounded by the plus phase. Moreover, the rescaled droplet is
asymptotically close to a definite deterministic shape -- the Wulff
crystal -- which minimizes the surface free energy. In the course
of the proof we establish a surface order large deviation principle
for the magnetization. Our results are valid for temperatures $T$
below a limit of slab-thresholds $\hat T_c$ conjectured to agree
with the critical point $T_c$. Moreover, $T$ should be such that
there exist only two extremal translation invariant Gibbs states at
that temperature; a property which can fail for at most countably
many values and which is conjectured to be true for every $T$.
The proofs are based on the Fortuin-Kasteleyn representation of
the Ising model along with coarse-graining techniques. To handle
the emerging macroscopic objects we employ tools from geometric
measure theory which provide an adequate framework for the large
deviation analysis. Finally, we give a heuristic argument that for
subcritical temperatures close enough to $T_c$, the dominant minus
spin cluster of the Wulff droplet permeates the entire box and has
a strictly positive local density everywhere.
Raphael.Cerf@math.u-psud.fr pisztora@asd4.math.cmu.edu
1263. RETURNS TO ORIGIN OF A ONE-DIMENSIONAL RANDOM WALK
VISITING EACH SITE AN EVEN NUMBER OF TIMES
G.M. Cicuta, M.Contedini
The class of random walks in one dimension, returning to the origin,
restricted by the requirement that any site visited (different from the origin)
is visited an even number of times, is analyzed in the present note. We call
this class the even-visiting random walks and provide a closed expression to
evaluate them.
cicuta@parma.infn.it
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information provided by the author
click here.
1264. FOURIER-WALSH COEFFICIENTS FOR A COALESCING FLOW (DISCRETE TIME)
Boris Tsirelson
A two-dimensional array of independent random signs produces coalescing
random walks. The position of the walk, starting at the origin, after N steps
is a highly nonlinear, noise sensitive function of the signs. A typical term of
its Fourier-Walsh expansion involves the product of about square roof of N
signs.
tsirel@math.tau.ac.il
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information provided by the author
click here.
1265. FREE STOCHASTIC MEASURES VIA NONCROSSING PARTITIONS
Michael Anshelevich
We consider free multiple stochastic measures in the combinatorial framework
of the lattice of all diagonals of an $n$-dimensional space. In this free case,
one can restrict the analysis to only the noncrossing diagonals. We give
definitions of what free multiple stochastic measures are, and calculate them
for the free Poisson and free compound Poisson processes. We also derive
general combinatorial It\^{o}-type relationships between free stochastic
measures of different orders. These allow us to calculate, for example, free
Poisson-Charlier polynomials, which are the orthogonal polynomials with respect
to the free Poisson measure.
mashel@math.berkeley.edu
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information provided by the author
click here.
1266. EXTINCTION FOR TWO PARABOLIC STOCHASTIC PDE'S ON THE LATTICE
C. Mueller and E. Perkins
It is well known that, starting with finite mass, the super-Brownian motion
dies out in finite time. The goal of this article is to show that with some
additional work, one can prove finite time die-out for two types of systems of
stochastic differential equations on the lattice Z^d. Our first system involves
the heat equation on the lattice Z^d, with a nonlinear noise term u(t,x)^gamma
dB_x(t), with 1/2 <= gamma < 1. The B_x are independent Brownian motions. When
gamma = 1/2, the measure which puts mass u(t,x) at x is a super-random walk and
it is well-known that the process becomes extinct in finite time a.s.
Finite-time extinction is known to be a.s. false if gamma = 1. For 1/2 < gamma
< 1, we show finite-time die-out by breaking up the solution into pieces, and
showing that each piece dies in finite time. Our second example involves the
mutually catalytic branching system of stochastic differential equations on
Z^d, which was first studied by Dawson and Perkins. Roughly speaking, this
process consists of 2 superprocesses with the continuous time simple random
walk as the underlying spatial motion. Furthermore, each process stimulates
branching and dying in the other process. By using a somewhat different
argument, we show that, depending on the initial conditions, finite time
extinction of one type may occur with probability 0, or with probability
arbitrarily close to 1.
cmlr@troi.cc.rochester.edu
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information provided by the author
click here.
1267. SCALING LIMIT OF FOURIER-WALSH COEFFICIENTS (A FRAMEWORK)
Boris Tsirelson
Independent random signs can govern various discrete models that converge to
non-isomorphic continuous limits. Convergence of Fourier-Walsh spectra is
established under appropriate conditions.
tsirel@math.tau.ac.il
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information provided by the author
click here.
1268. SHAPE FLUCTUATIONS AND RANDOM MATRICES
Kurt Johansson
We study a certain random groeth model in two dimensions closely related to
the one-dimensional totally asymmetric exclusion process. The results show that
the shape fluctuations, appropriately scaled, converges in distribution to the
Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
kurtj@math.kth.se
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information provided by the author
click here.
1269. RANDOM MATRICES AND RANDOM PERMUTATIONS
Andrei Okounkov
We prove a version of a conjecture due to Baik, Deift, and Johansson which
says that with respect to the Plancherel measure on the set of partitions of
$n$, the 1st, 2nd, 3rd, and so on, rows behave, suitably scaled, like the 1st,
2nd, 3rd, and so on, eigenvalues of a Gaussian random Hermitian matrix as $n$
goes to infinity.
The subject turns out to have a strong connection to Kontsevich's work on
Witten conjectures.
okounkov@math.uchicago.edu
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information provided by the author
click here.
1270. DISTRIBUTIONS ON PARTITIONS, POINT PROCESSES, AND THE HYPERGEOMETRIC
KERNEL
Alexei Borodin and Grigori Olshanski
We study a 3-parametric family of stochastic point processes on the
one-dimensional lattice originated from a remarkable family of representations
of the infinite symmetric group. We prove that the correlation functions of the
processes are given by determinantal formulas with a certain kernel. The kernel
can be expressed through the Gauss hypergeometric function; we call it the
hypergeometric kernel.
In a scaling limit our processes approximate the processes describing the
decomposition of representations mentioned above into irreducibles. As we
showed before, see math.RT/9810015, the correlation functions of these limit
processes also have determinantal form with so-called Whittaker kernel. We show
that the scaling limit of the hypergeometric kernel is the Whittaker kernel.
The integral operator corresponding to the Whittaker kernel is an integrable
operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the
hypergeometric kernel can be considered as a kernel defining a `discrete
integrable operator'.
We also show that the hypergeometric kernel degenerates for certain values of
parameters to the Christoffel-Darboux kernel for Meixner orthogonal
polynomials. This fact is parallel to the degeneration of the Whittaker kernel
to the Christoffel-Darboux kernel for Laguerre polynomials.
borodine@math.upenn.edu
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information provided by the author
click here.
1271. SCALING LIMITS OF LOOP-ERASED RANDOM WALKS AND UNIFORM SPANNING TREES
Oded Schramm
The uniform spanning tree (UST) and the loop-erased random walk (LERW) are
related probabilistic processes. We consider the limits of these models on a
fine grid in the plane, as the mesh goes to zero. Although the existence of
scaling limits is still unproven, subsequential scaling limits can be defined
in various ways, and do exist. We establish some basic a.s. properties of the
subsequential scaling limits in the plane. It is proved that any LERW
subsequential scaling limit is a simple path, and that the trunk of any UST
subsequential scaling limit is a topological tree, which is dense in the plane.
The scaling limits of these processes are conjectured to be conformally
invariant in 2 dimensions. We make a precise statement of the conformal
invariance conjecture for the LERW, and show that this conjecture implies an
explicit construction of the scaling limit, as follows. Consider the Loewner
differential equation
${\partial f\over\partial t}
= z {\zeta(t)+z \over \zeta(t)-z} {\partial f\over\partial z}$
with boundary values $f(z,0)=z$, in the range $z\in\U=\{w\in\C\st |w|<1\}$,
$t\le 0$. We choose $\zeta(t):= \B(-2t)$, where $\B(t)$ is Brownian motion on
$\partial \U$ starting at a random-uniform point in $\partial \U$. Assuming the
conformal invariance of the LERW scaling limit in the plane, we prove that the
scaling limit of LERW from 0 to $\partial\U$ has the same law as that of the
path $f(\zeta(t),t)$. We believe that a variation of this process gives the
scaling limit of the boundary of macroscopic critical percolation clusters.
schramm@wisdom.weizmann.ac.il
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information provided by the author
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1272. ON THE DISTRIBUTIONS OF THE LENGTHS OF THE LONGEST INCREASING AND
DECREASING SUBSEQUENCES IN RANDOM WORDS
Craig A. Tracy and Harold Widom
We consider the distributions of the lengths of the longest weakly increasing
and strongly decreasing subsequences in words of length N from an alphabet of k
letters. We find Toeplitz determinant representations for the exponential
generating functions (on N) of these distribution functions and show that they
are expressible in terms of solutions of Painlev\'e V equations. We show
further that in the weakly increasing case the generating function gives the
distribution of the smallest eigenvalue in the k x k Laguerre random matrix
ensemble and that the distribution itself has, after centering and normalizing,
an N -> infinity limit which is equal to the distribution function for the
largest eigenvalue in the k x k Gaussian Unitary Ensemble.
tracy@itd.ucdavis.edu
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information provided by the author
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1273. ON CANTOR'S SINGULAR MOMENTS
Helmut Prodinger
In problem 10621 of the American Mathematical Monthly the question was raised
whether it is possible to explicitly compute Cantor's singular moment of order
-1. This is answered in the affirmative.
helmut@gauss.cam.wits.ac.za
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information provided by the author
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1274. BLOCKING AND PERSISTENCE IN THE ZERO-TEMPERATURE DYNAMICS OF HOMOGENEOUS
AND DISORDERED ISING MODELS
C.M. Newman, D.L. Stein
A ``persistence'' exponent theta has been extensively used to describe the
nonequilibrium dynamics of spin systems following a deep quench: for
zero-temperature homogeneous Ising models on the d-dimensional cubic lattice,
the fraction p(t) of spins not flipped by time t decays to zero like
t^[-theta(d)] for low d; for high d, p(t) may decay to p(infinity)>0, because
of ``blocking'' (but perhaps still like a power). What are the effects of
disorder or changes of lattice? We show that these can quite generally lead to
blocking (and convergence to a metastable configuration) even for low d, and
then present two examples --- one disordered and one homogeneous --- where p(t)
decays exponentially to p(infinity).
dls@cosmo.physics.arizona.edu
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information provided by the author
click here.
1275. UNIVERSALITY AND SCALING OF CORRELATIONS BETWEEN ZEROS ON COMPLEX
MANIFOLDS
Pavel Bleher, Bernard Shiffman, Steve Zelditch
We study the limit as $N\to\infty$ of the correlations between simultaneous
zeros of random sections of the powers $L^N$ of a positive holomorphic line
bundle $L$ over a compact complex manifold $M$, when distances are rescaled so
that the average density of zeros is independent of $N$. We show that the limit
correlation is independent of the line bundle and depends only on the dimension
of $M$ and the codimension of the zero sets. We also provide some explicit
formulas for pair correlations. In particular, we provide an alternate
derivation of Hannay's limit pair correlation function for SU(2) polynomials,
and we show that this correlation function holds for all compact Riemann
surfaces.
shiffman@chow.mat.jhu.edu
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information provided by the author
click here.
1276. THE EXPECTED NUMBER OF REAL ROOTS OF A MULTIHOMOGENEOUS SYSTEM OF
POLYNOMIAL EQUATIONS
Andrew McLennan
Theorem 1 is a formula expressing the mean number of real roots of a random
multihomogeneous system of polynomial equations as a multiple of the mean
absolute value of the determinant of a random matrix. Theorem 2 derives closed
form expressions for the mean in special cases that include earlier results of
Shub and Smale (for the general homogeneous system) and Rojas (for ``unmixed''
multihomogeneous systems). Theorem 3 gives upper and lower bounds for the mean
number of roots, where the lower bound is the square root of the generic number
of complex roots, as determined by Bernstein's theorem. These bounds are
derived by induction from recursive inequalities given in Theorem 4.
mclennan@atlas.socsci.umn.edu
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information provided by the author
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1277. (NON-) GIBBSIANNESS AND PHASE TRANSITIONS IN RANDOM LATTICE SPIN MODELS
C. Kuelske
We consider disordered lattice spin models with finite volume Gibbs measures
$\mu_{\L}[\eta](d\s)$. Here $\s$ denotes a lattice spin-variable and $\eta$ a
lattice random variable with product distribution $\P$ describing the disorder
of the model. We ask: When will the joint measures
$\lim_{\L\uparrow\Z^d}\P(d\eta)\mu_{\L}[\eta](d\s)$ be [non-] Gibbsian measures
on the product of spin-space and disorder-space? We obtain general criteria for
both Gibbsianness and non-Gibbsianness providing an interesting link between
phase transitions at a fixed random configuration and Gibbsianness in product
space: Loosely speaking, a phase transition can lead to non-Gibbsianness,
(only) if it can be observed on the spin-observable conjugate to the
independent disorder variables.
Our main specific example is the random field Ising model in any dimension
for which we show almost sure- [almost sure non-] Gibbsianness for the single-
[multi-] phase region. We also discuss models with disordered couplings,
including spinglasses and ferromagnets, where various mechanisms are
responsible for [non-] Gibbsianness.
kuelske@wias-berlin.de
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information provided by the author
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1278. ON THE STOCHASTIC KURAMOTO-SIVASHINSKY EQUATION
Jinqiao Duan and Vincent Ervin
In this article we study the solution of the Kuramoto-Sivashinsky equation
(for surface erosion or surface growth) on a bounded interval subject to a
random forcing term. We show that a unique solution to the equation exists for
all time and depends continuously on the initial data.
duan@math.clemson.edu
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information provided by the author
click here.
1279. ON SEPARABLE FOKKER--PLANK EQUATIONS WITH A CONSTANT DIAGONAL DIFFUSION
MATRIX
Alexander Zhalij
We classify (1+3)-dimensional Fokker-Plank equations with a constant diagonal
diffusion matrix that are solvable by the method of separation of variables. As
a result, we get all possible forms of the drift coefficients providing
separability of the corresponding Fokker-Plank equations and carry out variable
separation in the latter. It is established, in particular, that the necessary
condition for the Fokker-Plank equation to be separable is that the drift
coefficients must be linear. We also find the necessary condition for
R-separability of the Fokker-Plank equation. Furthermore, exact solutions of
the Fokker-Plank equation with separated variables are constructed.
zhaliy@imath.kiev.ua
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information provided by the author
click here.
