Probability Abstracts 32
This document contains abstracts 646-668.
They have been mailed on March 29, 1996.
646. FIRST PASSAGE TIMES FOR THRESHOLD GROWTH DYNAMICS ON Z^2
Janko Gravner and David Griffeath
In the Threshold Growth model on an integer lattice, the occupied
set grows according to a simple local rule: a site becomes occupied
iff it sees at least a threshold number of already occupied sites
in its prescribed neighborhood. In this paper, we analyze the
behavior of two-dimensional threshold growth dynamics started from a
sparse Bernoulli density of occupied sites. We explain how nucleation
of rare centers, invariant shapes, and interaction between growing
droplets influence the first passage time in the supercritical
case. We also briefly address scaling laws for the critical case.
Postscript version available: fdv3.ps (1.8 MB including figures)
by anonymous ftp to cam8.math.wisc.edu, or from the CAM8 Ftp link of
http://math.wisc.edu/~griffeat/sink.html
gravner@feller.ucdavis.edu griffeat@math.wisc.edu
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information provided by the author
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647. MULTITYPE THRESHOLD GROWTH: CONVERGENCE TO POISSON--VORONOI
TESSELLATIONS
Janko Gravner and David Griffeath
A Poisson-Voronoi tessellation (PVT) is a tiling of the Euclidean
plane in which centers of individual tiles constitute a Poisson
field and each tile comprises the locations that are closest to a
given center with respect to a prescribed norm. Many spatial
systems in which rare, randomly-distributed centers compete for
space should be well-approximated by a PVT. Examples that we can
handle rigorously include multitype threshold vote automata,
in which kappa different camps compete for voters stationed on the
two-dimensional lattice. According to the deterministic,
discrete-time update rule, a voter changes affiliation only to that
of a unique opposing camp having more than theta representatives
in the voter's neighborhood. We establish a PVT limit for such
dynamics started from completely random configurations, as the
number of camps becomes large, so that the density of initial
"pockets of consensus" tends to 0. Our methods combine nucleation
analysis, Poisson approximation, and shape theory.
Postscript version available: vd3.ps (5.2 MB including figures)
by anonymous ftp to cam8.math.wisc.edu, or from the CAM8 Ftp link of
http://math.wisc.edu/~griffeat/sink.html
gravner@feller.ucdavis.edu griffeat@math.wisc.edu
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information provided by the author
click here.
648. LYAPUNOV EXPONENTS OF STOCHASTIC FLOWS
Ming Liao
We obtain a formula for Lyapunov exponents of stochastic flows generated
by sde (stochastic differential equations) on compact manifolds. As an
application, we consider the stochastic flow generated by the following
sde on the (n-1)-dimensional sphere S^{n-1}.
dx_t = \sum Y_{j_1,j_2,...,j_m}(x_t) \circ dw_t^{j_1j_2...j_m}
where Y_{j_1,j_2,...,j_m} = x_{j_1}...x_{j_{m-1}} Y(j_m), Y(j) is the
orthogonal projection of \partial/\partial x_j to S^{n-1}, and
w_t is an n^m dimensional standard Brownian motion. The one point
motion of the stochastic flow is a Brownian motion on S^{n-1}.
Its Lyapunov exponents are given by
-[(n+m-2)/2] + [(m-1)/2](n-1-k) for k=1,2,...,n-1
We note that the stochastic flow become instable in the sense that its
top exponent is positive for large n and m.
liaomin@mail.auburn.edu (latex file available)
649. LYAPUNOV EXPONENTS OF LINEAR STOCHASTIC FUNCTIONAL DIFFERENTIAL
EQUATIONS PART II: EXAMPLES AND CASE STUDIES
Salah-Eldin A. Mohammed and Michael K. R. Scheutzow
We give several examples and examine case studies of linear stochastic
functional differential equations. The examples fall into two broad
classes: regular and singular, according to whether an underlying stochastic
semiflow exists or not. In the singular case, we obtain upper and lower
bounds on the maximal exponential growth rate $\lambda_1 (\sigma)$ of the
trajectories expressed in terms of the noise variance $\sigma$. Roughly
speaking we show that for small $\sigma$, $\lambda_1 (\sigma)$ behaves like
$-{\sigma^2}/2$, while for large $\sigma$, it grows like $\log \s$. For
several regular one-dimesional examples, it is shown that discrete Oseledec
spectra exist, and upper bounds on the top exponent $\lambda_1$ are provided.
These bounds are sharp in the sense they reduce to known estimates in the
deterministic or non-delay cases.
salah@math.siu.edu ms@math.tu-berlin.de (AMS TeX file or hard copy available)
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650. ANCESTRAL PROCESSES WITH SELECTION
Stephen M. Krone and Claudia Neuhauser
In this paper, we present the genealogy of a sample of genes
at a single locus taken from a haploid population of size $N$ which
evolves according to a Moran model with selection and mutation. The
genealogy of the sample is embedded in a graph which we call the
ancestral selection graph. This graph contains all the information
about the ancestry; it is the analogue of Kingman's coalescent process which
arises in the case with no selection. The ancestral selection graph can
be easily simulated and we outline an algorithm for simulating samples.
The main goal is to analyze the ancestral selection graph and to
compare it to Kingman's coalescent process. In the case of no mutation,
we find that the distribution of the time to the most recent common
ancestor does not depend on the selection coefficient, and hence is the
same as in the neutral case. When the mutation rate is positive, we
give a procedure for computing the probability that two individuals in
a sample are identical by descent and evaluate the first two terms
of its power series in terms of the selection coefficient. This
probability depends on both the selection coefficient and the mutation
rate and is in general different from the analogous probability in the
neutral case. We also provide a recursion formula that can be used to
approximate the probability of a given sample by simulating backwards
along the sample paths of the ancestral selection graph, a technique
developed by Griffiths and Tavar\'e .
krone@uidaho.edu
651. RAY-KNIGHT THEOREMS VIA CONTINUOUS TREES.
Romain Abraham and Laurent Mazliak
We consider here a brownian motion perturbed by its local time:
$X_t=\vert B_t\vert +\Delta ^{-1}(2\ell _t)$ where $B_t$ is
a standard brownian motion, $\ell $ is its local time at 0
and $\Delta$ is a strictly increasing $C^1$-function on $[0,+\infty [$ such
that $\Delta (0)=0$ and $\lim _{t\rightarrow +\infty}\Delta (t)=+\infty$.
Yor has extended the classical Ray-Knight theorem about the local
time process in the space variable taken at infinity. We obtain than
a generalized Bessel square process of "dimension" $\Delta$.
We give here a new proof of this result based on the continuous
tree enbedded in the process $X$.
abraham@math-info.univ-paris5.fr
mazliak@ccr.jussieu.fr
652. DISTRIBUTION OF OVERLAP PROFILES IN THE ONE-DIMENSIONAL
KAC-HOPFIELD MODEL
Anton Bovier, V\'eronique Gayrard, Pierre Picco
We study a one-dimensional version of the Hopfield model with
long, but finite range interactions below the critical temperature.
In the thermodynamic limit we obtain large deviation estimates for the
distribution of the ``local'' overlaps, the range of the interaction,
$\gamma^{-1}$, being the large parameter. We show in particular that the
local overlaps in a typical Gibbs configuration are constant and equal to one
of the mean-field equilibrium values on a scale $o(\g^{-2})$. We also
give estimates on the size of typical ``jumps''. i.e. the regions where
transitions from one equilibrium value to another take place. Contrary to the
situation in the ferromagnetic Kac-model, the structure of the
profiles is found to be governed by the quenched disorder rather than
by entropy.
bovier@wias-berlin.de gayrard@cpt.univ-mrs.fr
picco@cpt.univ-mrs.fr (hardcopy or file available upon request)
653. CATALYTIC SUPER BROWNIAN MOTION
Jean-Fran\c{c}ois Delmas
We study a general class of catalytic super-Brownian
motions. Informally, such a process $Z$ describes the evolution of a
large number
of small particles which move according to independent Brownian motions
in $R^d$ and branch only when they visit a given set $D$ (the
catalyst), which may be of zero Lebesgue measure. We first construct the
processus $Z$ as a weak limit of branching particle systems. We then
obtain detailed information about the continuity properties of
$Z$. Using excursion theory for Brownian motion in $R^d$, we prove a
representation theorem for $Z$ outside the catalyst $D$. In the special
case when $D$ has zero Lebesgue measure, this representation theorem
shows that a.s. for every $t>0$, the measure $Z_t$ is absolutely
continuous with respect to Lebesgue measure, and its density solves the
heat equation outside $D$.
delmas@cermics.enpc.fr
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654. ANALYSIS OF SIMPLE ALGORITHMS FOR DYNAMIC LOAD BALANCING
Murat Alanyali and Bruce Hajek
The principle of load balancing is examined for dynamic resource
allocation subject to certain constraints. The emphasis is on the
performance of simple allocation strategies which can be implemented
on-line. Either finite capacity constraints on resources or migration
of load can be incorporated into the setup. The load balancing problem
is formulated as a stochastic optimal control problem. Variants of a
\lq\lq Least Load Routing \rq\rq policy are shown to lead to a fluid
type limit and to be asymptotically optimal.
b-hajek@uiuc.edu
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information provided by the author
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655. ON LARGE DEVIATIONS IN LOAD SHARING NETWORKS
Murat Alanyali and Bruce Hajek
Three policies, namely Optimal Repacking, Least Load Routing, and
Bernoulli Splitting, are considered for dynamic resource allocation
in load sharing networks with standard Erlang type statistics. Large
deviations principles are established for the three policies in a
simple network of three consumer types and two resource locations,
and are used to identify the network overflow exponents. The overflow
exponents for networks with arbitrary topologies are identified for
Optimal Repacking and Bernoulli Splitting policies, and conjectured
for the Least Load Routing policy.
b-hajek@uiuc.edu
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information provided by the author
click here.
656. ON LARGE DEVIATIONS OF MARKOV PROCESSES WITH DISCONTINUOUS STATISTICS
Murat Alanyali and Bruce Hajek
This paper establishes a process-level large deviations principle for
Markov processes in the Euclidean space with a discontinuity in the
transition mechanism along a hyperplane. The transition mechanism of
the process is assumed to be continuous on one closed halfspace, and
also continuous on the complementary open halfspace. A similar result
was recently obtained by Dupuis and Ellis for lattice-valued Markov
processes satisfying a mild communication/controllability condition.
Our proof relies on the work of Blinovskii and Dobrushin, which in turn
is based on an earlier work of Dupuis and Ellis.
b-hajek@uiuc.edu
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information provided by the author
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657. HEAVY TAILS AND LONG RANGE DEPENDENCE IN ON/OFF PROCESSES
AND ASSOCIATED FLUID MODELS
David Heath, Sidney Resnick and Gennady Samorodnitsky
On/off models are common inputs for a variety of
communication network models as well as storage and inventory models.
A stationary renewal process alternating periods of activity
(active transmission
fluid buildup, etc) with periods of inactivity (silence, no transmission,
inputs off, etc) induces a stationary indicator process which indicates if
the system is active or not. Heavy tails for the on periods
induces a covariance function for the indicator process which decreases
slowly at a rate characteristic of long range dependence. This has dramatic
consequences for fluid models where fluid flows in at constant rate and
there is a constant rate of release.
Available as TR1144.ps.Z at http://www.orie.cornell.edu/trlist/trlist.html
davidh@orie.cornell.edu sid@orie.cornell.edu
gennady@orie.cornell.edu
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information provided by the author
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658. THE GAUSSIAN CORRELATION CONJECTURE FOR ELLIPSOIDS
Gideon Schechtman, Thomas Schlumprecht and Joel Zinn
The Gaussian Correlation Conjecture states that for any two symmetric,
convex sets in n-dimensional space and for any centered, Gaussian
measure on that space, the measure of the intersection is greater than
or equal to the product of the measures. In this paper we obtain
several results which substantiate this conjecture. For example, we
show there is a positive constant, c, such that the conjecture is true
if the two sets are in the Euclidean ball of radius $c\sqrt{n}$.
Further we show that if for every n the conjecture is true when the sets are
in the Euclidean ball of radius $\sqrt{n}$, then it is true in general.
Our most concrete result is that the conjecture is true if the two
sets are (arbitrary) centered ellipsoids.
jzinn@math.tamu.edu
659. DENSITY IN SMALL TIME FOR LEVY PROCESSES
J. Picard
The density of real-valued L\'evy processes is studied
in small time under the assumption that the process has many small
jumps. We prove that the real line can be
divided into three subsets on which the
density is smaller and smaller: the set of points that the process can
reach with a finite number of jumps ($\Delta$-accessible points); the set
of points that the process can reach with an infinite number of jumps
(asymptotically $\Delta$-accessible points); and the set of points that the
process cannot reach by jumping ($\Delta$-inaccessible points).
Compressed PostScript file available at
ftp://ucfma.univ-bpclermont.fr/pub/PUBLI/lma/1995/18-95.ps.gz
picard@ucfma.univ-bpclermont.fr
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information provided by the author
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660. ASYMPTOTIC BEHAVIOUR OF DISCONNECTION AND NON-INTERSECTION
EXPONENTS
Wendelin Werner
We study the asymptotic behaviour of disconnection
and non-intersection exponents for planar Brownian motion
when the number of considered paths tends to infinity.
In particular, if $\eta_n$ (respectively $\xi (n, p)$) denotes the
disconnection exponent for $n$ paths (respectively the
non-intersection exponent for $n$ paths versus $p$ paths),
then we show that
$\lim_{n \rightarrow \infty} {\eta_n /n} = 1/2$ and that
for fixed $a>0$ and $b>0$,
$\lim_{n \rightarrow \infty} \xi ([na],[nb])/n = (\sqrt {a} + \sqrt {b})^2/2.$
wwerner@dmi.ens.fr
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information provided by the author
click here.
661. THE LOCAL TIME OF ITERATED BROWNIAN MOTION
Endre Cs\'aki, Mikl\'os Cs\"org\H{o}, Ant\'onia F\"oldes and P\'al
R\'ev\'esz
The local time process $L^*(x,t);\ x\in R,\ t\geq 0$ of the iterated
process $W_1(|W_2(t)|)$ is defined and its properties are studied.
Here $W_1$ and $W_2$ are independent Wiener processes. It is shown
that $L^*$ is jointly continuous in $x$ and $t$ and upper and lower
functions are given for $L^*(0,t)$ and $\sup_x L^*(x,t)$ as
$t\to\infty$. These results however are not sharp enough for
concluding an exact LIL. Strong invariance theorems are also given
for $L^*$ approximating the local time of the iterated random walk.
csaki@novell.math-inst.hu (LaTeX file available)
662. STRASSEN THEOREMS FOR A CLASS OF ITERATED PROCESSES
Endre Cs\'aki, Ant\'onia F\"oldes and P\'al R\'ev\'esz
A general Strassen theorem is proved for a stochastic process
satisfying certain conditions and it is applied for the iterated
process $W(L(t))$, where $W$ is a standard Wiener process and $L$ is
a local time of a symmetric L\'evy process, independent from $W$. It
is shown in particular that if $L$ is a Wiener local time, then the
Strassen class of
$${W(L(xt))\over 2^{5/4}3^{-3/4}t^{1/4}(\log\log t)^{3/4}}$$
is the class of functions $f(x)$, absolutely continuous with respect
to the Lebesgue measure, $f(0)=0$ and $\int_0^1 |f'(x)|^{4/3}\,
dx\leq 1.$ This is a "direct" Strassen class for an iterated process
as opposed to "composite" Strassen class of the form $f=g(h(x)),\quad
\int_0^1 (|g'(x)|^2 +|h'(x)|^2)\, dx\leq 1$ proved previously for
iterated processes. In obtaining direct Strassen class, it is
essential that the inner process $L$ is nondecreasing.
csaki@novell.math-inst.hu (LaTeX file available)
663. THE CONVEX HULL OF A SELF-SIMILAR MEASURE
Irene Hueter
Let $ X_1, X_2, \ldots, $ be i.i.d. random points in $ I\!\!R^2 $
with distribution $ \nu ,$ and let $ N_n $ denote the number
of points spanning the convex hull of $ X_1, X_2 , \ldots X_n . $
We obtain the asymptotic growth rate of $ E(N_n) $ under the
assumption that $ \nu $ is a certain self-similar measure on the
unit disk. This extends some well-known results in the absolutely
continuous case to the singular case.
Moreover, we find a limiting expression for the probability of
the `cap region', i.e. the union of caps of fixed probability,
which has been introduced by B\'{a}r\'{a}nyi and Larman (1988)
to measure the speed at which the convex hull approaches the
boundary of the support. Our main tool consists in a geometric
application of the renewal theorem. Exactly the same approach
can be adopted to prove the analogous results in $ I\!\!R^d. $
hueter@math.ufl.edu
664. CONICAL MAXIMA ON CONVEX REGIONS
Irene Hueter
We consider a sequence of i.i.d. random points $ X_1, X_2,
\ldots, $ from an absolutely continuous distribution on
a convex region in the Euclidean space. The purpose of this
note is to give an easy explanation for a curious `gap behaviour'
of the expected number of conical maxima $M_n $ and to present
an easy method to compute lower and upper bounds of the expectation.
We calculate these for the uniform distribution on a polygon.
hueter@math.ufl.edu
665. A SAMPLE PATH LARGE DEVIATION PRINCIPLE FOR $L^2$-MARTINGALE MEASURE
Boualem Djehiche and Ingemar Kaj
Gaussian White Noise, super-Brownian motion and the diffusion-limit
Fleming-Viot process are examples of such infinite-dimensional Markov
processes with continuous paths and $L^2$-martingale measures we study
in this work as regards to their sample path large deviation
probabilitites and their associated large deviation rate functions in
the limit of small perturbations. We present a unified approach based
on Girsanov transform techniques. We derive the rate function as a
Lagrangian functional and, as an alternative representation, via some
generalized derivatives in a 'Cameron-Martin space'.
boualem@math.kth.se ingemar.kaj@math.uu.se (Hard copy and Tex file available)
666. LOCALIZATION TRANSITION FOR A POLYMER NEAR AN INTERFACE
Erwin Bolthausen and Frank den Hollander
Consider the directed process $(i,S_i)_{i \geq 0}$ where the second component
is simple random walk on $\Z$ ($S_0=0$). Define a transformed path measure by
weighting each $n$-step path with a factor $\exp[\lambda \sum_{1 \leq i \leq n}
(\omega_i + h){\rm sign}(S_i)]$. Here, $(\omega_i)_{i \geq 1}$ is an
i.i.d.\ sequence of random variables taking values $\pm 1$ with
probability $1/2$ (acting as a random medium), while $\lambda \in [0,\infty)$
and $h \in [0,1)$ are parameters. The weight factor has a tendency to pull the
path towards the horizontal, because it favors the combinations
a phase transition occurs between localized and delocalized behavior
(in the vertical direction). We derive several properties of this curve,
in particular, its behavior for $\lambda \downarrow 0$. To obtain this
behavior, we prove that as $\lambda, h \downarrow 0$ the free energy
scales to its Brownian motion analogue.
eb@amath.unizh.ch denholla@sci.kun.nl (LaTeX file available)
667. PERFECT SIMULATION FOR THE AREA-INTERACTION POINT PROCESS.
Wilfrid S. Kendall
The ideas of Propp and Wilson (``Exact sampling with coupled Markov chains
and applications to statistical mechanics'', to appear in Random Structures
and Algorithms, 1996) for exact simulation of Markov chains are extended to
deal with perfect simulation of attractive area-interaction point processes
in bounded windows. A simple modification of the basic algorithm is
described which provides perfect simulation of the non-monotonic case of
the repulsive area-interaction point process. Results from simulations
using a C computer program are reported; these confirm the practicality of
this approach in both attractive and repulsive cases. The paper concludes
by mentioning other point processes which can be simulated perfectly in
this way, and by speculating on useful future directions of research.
w.s.kendall@warwick.ac.uk
668. TRACE ON THE BOUNDARY FOR SOLUTIONS OF NONLINEAR DIFFERENTIAL
EQUATIONS
E. B. Dynkin and S. E. Kuznetsov
Let L be a second order elliptic differential operator in
d-dimensional Euclidean space with no zero order terms and let
E be a bounded domain with smooth boundary E'. We say that a
function h is L-harmonic if Lh=0 in E. Every
positive L-harmonic function has a unique representation
h(x)=\int_{E'} k(x,y) \nu(dy)
where k is the Poisson kernel for L and \nu is a finite measure on
E'. We call \nu the trace of h on E'.
Our objective is to investigate positive solutions of a
nonlinear equation
L u=u^\alpha in E
for 1<\alpha\le 2 [the restriction \alpha\le 2 is imposed because our
main tool is the \alpha-superdiffusion which is not defined for
\alpha>2]. We associate with every solution u a pair (\Gamma,\nu)
where \Gamma is a closed subset of E' and \nu is a Radon measure on
O=E'\setminus \Gamma. We call (\Gamma,\nu) the trace of u on E'.
\Gamma is empty if and only if u is dominated by an L-harmonic
function. We call such solutions moderate. A moderate solution is
determined uniquely by its trace. In general, many solutions can have the
same trace. We establish necessary and sufficient conditions for
a pair (\Gamma,\nu) to be a trace and we give a probabilistic formula
for the maximal solution with a given trace.
ebd1@cornell.edu
