June 58, 2007
Probability and Stochastic Processes Symposium in honour of
Donald A. Dawson's work,
on the occasion of his 70th birthday
School of Mathematics and Statistics
Carleton University
ABSTRACTS

Ted Cox, Syracuse University
Convergence of interacting particle systems to superBrownian
motion
This talk will survey some past and current work with Ed Perkins
and Rick Durrent on a particular interacting particle system. We
show that nearly critical scaled systems converge to superBrownian
motion, and then use this and additional arguments to obtain information
about the particle system from the limiting superBrownian motion.
Colleen D. Cutler, University of Waterloo
Repeat Sampling of Extreme Observations with Error: Regression
to the Mean and Asymptotic Error Distributions
The phenomenon of regression to the mean was described by Sir Francis
Galton in a series of prestigious works in the 19th century. This
phenomenon refers to the fact that, in the presence of measurement
error, a second independent measurement (or repeat sampling) of
an extreme value typically produces a value less extreme than the
first. This inward shift from the tails toward the mean occurs even
when there has been no intervention or change in the underlying
population and error distributions. A typical example used here
is that of blood pressure: patients who score high on a first measurement
generally score lower (on average) on a second measurement, even
without treatment.
The arguments used to explain this regression effect typically
appeal to the classical Gaussian model. We describe a framework
for studying repeating sampling effects in the tails of arbitrary
distributions, and identify three distinct aymptotic forms of "regression
to the mean'', of which only one is the classical Gaussian form.
Examples will be given.
Plenary Talk
Don Dawson, Carleton University
Reflections on probability and stochastic processes 1957—2007
The first part of the lecture will consist of some personal reflections
on probability and stochastic processes around 1960, a look at a
few aspects of the amazing development of the subject over the past
50 years and some comments on current challenges. The second part
will be a report on some ongoing work on branching processes, excursions,
emergence of rare mutants, and catalytic cycles and networks.
Shui Feng, McMaster University
Limiting Theorems Associated With PoissonDirichlet Distribution
PoissonDirichlet distribution arises in many subjects. The single
paremeter $\theta$ is the scaled population mutation rate in the
contexts of population genetics. Limiting theorems such as fluctuations,
moderate deviations and large deviations will be presented for the
limiting procedure of $\theta$ approaching infinity.
Jürgen Gärtner, Fachbereich
Mathematik, Technische Universitat Berlin
Germany
On the parabolic Anderson model driven by catalytic exclusion
and voter dynamics
This is joint work with Frank den Hollander and Gregory Maillard.
We consider the parabolic Anderson equation $\partial u/\partial
t=\kappa\Delta u + \xi u$ on $\mathbb Z^d$ driven by a catalytic
potential $\xi$. We assume that $\xi$ is modeled either by a symmetric
exclusion process in Bernoulli equilibrium or by a symmetric voter
dynamics with Bernoulli or equilibrium initial distribution. In
this talk the focus is on the comparision of the annealed Lyapunov
exponents $\lambda_p=\lim_{t\to\infty} \log \langle u(t,0)^p \rangle^{1/p}$
for $p=1,2,\dots$. It will be shown that these Lyapunov exponents
display an interesting dependence on the dimension $d$ and the diffusion
constant $\kappa$ indicating an asymptotically intermittent or nonintermittent
spatial structure of the solution.
Peter Glynn, Stanford University
Heavy Tails, Sharp Thresholds, and Stochastic Modeling
In many stochastic models arising in applied probability, it is
known that the tail behavior of the governing random variables plays
a key role in the rare event behavior of the system. On the other
hand, the typical behavior of the system is often influenced primarily
by the center of the distribution of the governing random variables
(for example, moments). In this talk, we will discuss the temporal
and spatial scales at which the tail behavior manifests itself for
several stylized queueing models. In some settings, a sharp threshold
can be identified at which the moments determine system behavior
below the threshold, and the tail structure above the threshold.
We will also discuss the implications of these results for stochastic
modeling of queueing systems. These ideas give a partial justification
for the use of heavytailed distributions even in settings in which
physical constraints imply that the random variables must be bounded.
This represents joint work with Jose Blanchet and Mariana OlveraCravioto.
Luis G. Gorostiza, Centro de Investigacion
y de Estudios Avanzados, Mexico
SelfSimilar Stable Processes arising from HighDensity Occupation
Times of Particle Systems
For the branching particle system in dimension d with symmetric
alphastable Levy motion, (1 + beta)branching, and initial homogeneous
Poisson configuration (i.e., Lebesgue intensity measure), occupation
time fluctuation limits exist only in dimensions d > alpha/beta,
since the system becomes locally extinct if d =< alpha/beta.
By increasing the initial density of particles, the local extinction
in dimensions d =<
alpha/beta is compensated, and limit results are obtained for every
d in three cases: low dimensions, d < alpha(1+beta)/beta, critical
dimension, d = alpha(1+beta)/beta, and high dimensions, d > alpha(1+beta)/beta.
In low dimensions, a selfsimilar longrange dependence stable process
(which is nonLevy) appears in the limit. In the Gaussain case (beta
= 1) the low dimension results split into three different cases:
d < alpha, d = alpha, and alpha < d < 2alpha. If the initial
Poisson configuration has finite intensity measure, again high density
compensates extinction, and the occupation time limits are qualitatively
different from those of the homogeneous Poisson case. Analogous
highdensity occupation time results are obtained for the particle
system without branching. For low dimension, an extension of fractional
Brownian motion appears.
Andreas Greven, Mathematisches Institut,
FriedrichAlexanderUniversitat
Evolving genealogical trees
We consider stochastic processes with values in trees, which arise
from population models. The appropriate state space will be discussed
and wellposed martingale problems are formulated for the treevalued
FlemingViot diffusion. This diffusion arises as limit of the genealogies
of the Moran model. As equilibrium we identify the tree arising
from the Kingman coalescent. Finally we consider the functional
of the process corresponding to the distribution of the length of
finite sampled subtrees. The last comments concern the challenge
to incorporate selection, mutation and recombination as well as
spatial aspects.
Gail Ivanoff, University of Ottawa
Poisson Limits for Empirical Point Processes
We present a simple and unified approach to studying weak Poisson
limits for scaled empirical point processes on ddimensional Euclidean
space. This method is based on the theory of multiparameter martingales
and requires only the straightforward computation of socalled *compensators
to identify the appropriate limit. The compensator method is not
only dimensionfree (immediately extending results from the univariate
to the multivariate case), but also allows one to handle the joint
local behaviour of the process at multiple time points with ease.
This method extends previous results in several directions. We
obtain limits at points where the underlying (multivariate) density
may be zero, but has regular variation. Our results are both multivariate
and multidimensional. Applications include weak limits for nearestneighbour
estimates of joint densities at several points simultaneously and
new extreme value limits for multivariate copulas.
This is joint work with Andre Dabrowski and Rafal Kulik.
Michael A. Kouritzin, University
of Alberta
Microstructure Filtering in Finance
Our work is based upon recent financial models for stocks and options
that include both tradingbehaviour microstructure and valuebased
macrostructure. We will discuss filteringbased methods for testing
for such things stochastic volatility and longrange dependence
in tickbytick price data. We will also discuss use for inferring
model parameters, including the difficulttoestimate Hurst parameter,
from tickbytick data. In the process, we will expose new financial
models, robust filter equations and particle filter algorithms.
The talk is based upon joint work with Jianhui Huang and Yong Zeng.
Reg Kulperger, University of Western
Ontario
Sufficient Conditions for Ergodicity for a Stochastic Competing
Species Model
A classic model of preypredator behaviour is the VolterraLotka
process. Extensions have been made to competing species models,
to
two or more species.
In this talk we consider a stochastic generalization of these models
by adding to the growth rates independent Brownian motions. These
generalizations may be unstable in the sense they tend to the boundary.
We obtain sufficient conditions for these to be ergodic.
Since these conditions are obtained using simple Markov inequalities
based on appropriate test functions they are relatively easy to
apply to multiple species as well as some nonlinear forms of the
growth rate functions.
Thomas G. Kurtz, University of Wisconsin
 Madison
Poisson representations of measurevalued processes
Measurevalued diffusions and measurevalued solutions of stochastic
partial differential equations can be represented in terms of the
Cox measures of particle systems that are conditionally Poisson
at each time t. The representations are useful for characterizing
the processes, establishing limit theorems, and analyzing the behavior
of the measurevalued processes. Examples will be given and some
of the useful methodology will be described.
Neal Madras, York University
Polymers and Percolation on Hyperbolic Graphs
We first introduce some simple models of statistical mechanics:
lattice polymers (including selfavoiding walks and lattice animals)
and percolation. The asymptotic combinatorial and probabilistic
properties of these models in Euclidean lattices have been of considerable
interest in statistical physics, and provide many challenging, easily
stated problems to mathematicians. As an interesting theoretical
variant, we consider these models on "nonEuclidean lattices",
specifically graphs that correspond to regular tilings of the hyperbolic
plane (or 3space). One example is the infinite planar graph in
which every face is a triangle and eight triangles meet at every
vertex. To a physicist, such graphs display "infinite dimensional"
characteristics. Thus, our models should behave as they would in
highdimensional Euclidean space, or, more simply, on an infinite
regular tree. In physics terminology, the models should exhibit
"mean field behaviour". We have made progress towards
understanding these problems, but some open questions remain. This
talk is based on joint work with C. Chris Wu.
Peter March, The Ohio State University
and National Science Foundation
Some models of polymer dynamics
I'll review some of the classic models, due to Rouse and KratkyPorod,
of a polymer in dilute solution. These simple models express the
balance elastic restoring forces between monomers and thermal forces
of the solvent, while ignoring all other physical effects, via linear
stochastic partial differential equations. One can try to include
relevant physical effects, such as volume exclusion, hydrodynamic
interaction, confinement, etc., leading to nonlinear perturbations
and boundary conditions of the basic linear spde's. I'll report
on some results, due to Scott McKinley, Seung Lee, and Wei Xiong,
and point out a number of interesting open problems.
Peter Mörters, University of
Bath
Localisation of mass in random media
We look at a model of mass transport in an iid random potential,
the parabolic Anderson model, and discuss how the random potential
leads to the localisation of mass in favourable islands. I report
about efforts to determine how size, shape and number of these islands
evolve in time. The talk is based on joint work with Remco van der
Hofstad (Eindhoven), Wolfgang Konig (Leipzig) and Nadia Sidorova
(Bath).
Carl Mueller, University of Rochester
Negative moments for a linear SPDE
When using Malliavin calculus, we often differentiate an equation
to obtain a linear equation for the derivative. Next, among other
things, we study the moments of the derivative. Following this motivation,
we study the negative moments of solutions of a linear SPDE, and
show that the moments are finite in some cases.
Leonid Mytnik, Technion, Haifa
Large Regularity of densities for $(\alpha,d,\beta)$superprocess
We study that the density of (a,d,ß)superprocess (0<
ß <1) for fixed times in dimension d=1. We show that this
density is Hölder continuous of order ? for every ?<a/(ß+1)1.
Moreover we show that the density is not Hölder continuous
of order ? if ?> a/(ß+1)1.
This is a joint work with Klaus Fleischmann and Vitali Wachtel.
Edwin Perkins, University of British
Columbia
On uniqueness for some singular sde's arising in branching models
We prove uniqueness in law for a class of finitedimensional sde's
which arise as limits of interacting branching particle systems
with finitely many sites. The resulting systems have degenerate
nonLipschitz continuous coefficients. Different perturbative techniques
are introduced to handle regular branching, cyclically catalytic
branching and catalytic branching networks, and also to handle different
regularity conditions on the coefficients. Motivated by a twodimensional
renormalization program of Dawson, den Hollander, Greven, Sun and
Swart, we allow the interaction mechanisms to be only continuous.
This lecture is based on joint papers with Rich Bass and Don Dawson.
Bruno Remillard, HEC Montreal
Malliavin calculus and ClarkOcone formula for functionals of
a squareintegrable Levy process
We construct a Malliavin derivative for functionals of squareintegrable
Levy processes and derive a ClarkOcone formula. The Malliavin derivative
is defined via chaos expansions involving stochastic integrals with
respect to Brownian motion and Poisson random measure. As an illustration,
the explicit martingale representation for the maximum of a Levy
process is computed. This a joint work with JeanFrancois Renaud.
Tom Salisbury, York University
Conditioned superBrownian motion
I will speak about some joint work with Deniz Sezer (York University)
on conditioning superBrownian motion in a domain by its exit measure.
We use a formalism of Dynkin's to get at conditioning on events
such as that the exit measure agrees with a specified measure on
the boundary of the domain, or that the mass of the exit measure
takes a specified value. As in earlier work with John Verzani (CUNY),
there is both an analytic formulation of the conditioned process,
and a probabilistic representation in terms of the genealogy of
mass reaching the boundary of the domain. The latter has the form
of a spatial fragmentation process.
David Sankoff, University of Ottawa
Genome evolution and random graphs
During evolution, the order of elements on chromosomes, and their
partition among the chromosomes, is scrambled by the accumulation
of rearrangement mutation events. The main kinds of genetic event
are the inversion of a chromosomal segment of arbitrary length and
the reciprocal translocation of segments of arbitrary length (prefix
and/ or sufffix exchange) between two chromosomes. Though there
are many algorithms for inferring rearrangement histories from contemporary
comparative genetic maps, there is a need for ways to statistically
validate the results. Are the characteristics of the evolutionary
history of two related genomes as inferred from an algorithmic analysis
different from the chance patterns obtained from two unrelated genomes?
Implicit in this question is the notion that the null hypothesis
for genome comparison is provided by two genomes, where the order
of elements in one is an appropriately randomized permutation of
the order in the other.
Rearrangement algorithms are generally formulated in terms of the
"breakpoint graph" induced by two genomes. Key to our
approach is the introduction of randomness into the construction
of the breakpoint graph rather than into the genomes themselves,
which facilitates the analysis without materially affecting the
results. The justification for this is a conjecture extending a
theorem of Kim and Wormald.
We discuss a number of ways of exploring these problems and present
analytic results, as well as simulations, on the distribution of
the number of arrangements inferred as a function of the number
of
randomly ordered genomic elements. Our results suggest that some
recent hypotheses about how genomes evolve are based entirely on
non significant associations.
Byron Schmuland, University of Alberta
Reversible FlemingViot processes
What forces the mutation operator of a reversible FlemingViot process
to be uniform? Our explanation is based on Handa's result that reversible
distributions must be quasiinvariant under a certain flow, making
the mutation operator satisfy a cocycle identity.
We also apply these ideas to a system of interacting FlemingViot
processes as defined and studied by Dawson, Greven, and Vaillancourt.
Gordon Slade, University of British
Columbia
Random walk on the incipient infinite cluster for oriented percolation
We consider simple random walk on the incipient infinite cluster
for the spreadout model of oriented percolation in d spatial dimensions
and one time dimension. For d > 6, we obtain bounds on exit times,
transition probabilities, and the range of the random walk, which
establish that the spectral dimension of the incipient infinite
cluster is 4/3, and thereby prove a version of the AlexanderOrbach
conjecture in this setting.
This is joint work with Martin Barlow, Antal Jarai and Takashi
Kumagai.
Wei Sun, Concordia University
On Girsanov and generalized FeynmanKac transformationsfor symmetric
Markov processes
Let $X$ be a Markovprocess, which is assumed to be associated with
a symmetricDirichlet form (\mathcal{E},\mathcal{D}(\mathcal{E}))$.
For$u\in{\mathcal{D}}({\mathcal{E}})_e$, the extended Dirichletspace,
we have Fukushima's decomposition: $\tilde u(X_{t})\tildeu(X_{0})
= M_{t}^u + N_{t}^u$, where $\tilde u$ is aquasicontinuous version
of $u$, $M_{t}^u$ the martingale part and$N_{t}^u$ the zero energy
part. In this talk, we investigate theGirsanov transform induced
by $M_{t}^u$ and the generalizedFeynmanKac transform induced by
$N_{t}^u$. For the Girsanovtransform, we present necessary and sufficient
conditions forwhich to induce a positive supermartingale and characterize
theDirichlet form associated with the Girsanov transformed process.For
the generalized FeynmanKac transform, we give a necessary andsufficient
condition for the generalized FeynmanKac semigroup($P^u_tf(x):=E_x[e^{N^u_t}f(X_t)]$)
to be strongly continuous.
Anton Wakolbinger, J.W. GoetheUniversitat
Interacting locally regulated diffusions and the Virgin Island
model
We study countable systems of interacting diffusions whose prototype
examples are locally logistic Feller branching diffusions with migration:
as long as an island's population is small, it grows supercritically,
but when the island becomes crowded, there is a strong reverting
drift which decreases the population size. For these systems, comparison
with a "mean field model" gives a sufficient criterion
for local extinction when started, say, with equal mass on each
island. In the logistic Feller case, a selfduality translates this
criterion for local extinction into one for global extinction of
the system started with a finite total mass. For more general drifts
and diffusions, such a duality is missing, and so are criteria for
global extinction. As a candidate for a comparison "in spe",
we investigate a model in which each migration leads to a virgin
island.
This model can be described through a tree of excursions, and a
criterion for global survival versus extinction can be derived.
The first part of the lecture is based on joint work with Martin
Hutzenthaler (Ann. Appl. Probab. 2007), and also the second part
is based on Martin's PhD thesis.
FengYu Wang, Beijing Normal
Harnack inequality for Markov semigroups and applications
A coupling method is introduced to derive Harnack inequality
for Markov semigroups. Concrete examples as well as applications
to estimates on the transition density and various contractivity
properties are also presented.
Hao Wang, University of Oregon
A Basic Interacting Model and Beyond
In this talk, we will give a simple survey of the development of
a class of interacting superprocesses starting from a basic interacting
branching model based on the effort of several authors. Then, more
details will be discussed about the recent progress on the
generalization of the model.
Jie Xiong, University of Tennessee
Local extinction for superprocesses in random environments
We consider a superprocess in a random environment represented by
a random measure which is white in time and colored in space with
correlation kernel $g(x,y)$. Suppose that $g(x,y)$ decays at a rate
of $xy^{\al}$, $0\le\al\le 2$, as $xy\to\infty$. We show
that the
process, starting from Lebesgue measure, suffers longterm local
extinction. If $0\le\al<2$, then it even suffers finite time
local extinction. This property is in contrast with the classical
superBrownian motion which has a nontrivial limit when the spatial
dimension is higher than 2. We also show that in dimensions $d=1,2$
superprocess in random environment suffers local extinction for
any bounded function $g$. This talk is based on a paper jointly
with Mytnik.
Xiaowen Zhou, Concordia University
Large A zeroone law of almost sure local extinction for superBrownian
motion
In this talk we consider a ddimensional ($1+\beta$)superBrownian
motion $(X_t)$ starting at Lebesgue measure on $R^d$. For each time
$t>0$ let $B_t$ be a closed ball in $R^d$ with center the origin
and radius $g(t)$, where $g$ is a nondecreasing and right continuous
function. Let $T$ be the last time when the superBrownian motion
$(X_t)$ charges $(B_t)$. We say that $(X_t)$ suffers almost sure
local extinction with respect to $(B_t)$ if $T<\infty$. We are
going to present two integral tests concerning the above mentioned
almost sure local extinction behavior. For $d\beta<2$ we are
going to show that the probability $\bP\{T=\infty\}$ is either $0$
or $1$ depending on whether the value of integral $\int_1^\infty
g(t)^d t^{1{1\over \beta}} dt$ is finite or not.
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