April 18, 2014

June 5-8, 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


Ted Cox, Syracuse University
Convergence of interacting particle systems to super-Brownian 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 super-Brownian motion, and then use this and additional arguments to obtain information about the particle system from the limiting super-Brownian 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 Poisson-Dirichlet Distribution
Poisson-Dirichlet 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

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 non-intermittent 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 heavy-tailed 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 Olvera-Cravioto.

Luis G. Gorostiza, Centro de Investigacion y de Estudios Avanzados, Mexico
Self-Similar Stable Processes arising from High-Density Occupation Times of Particle Systems
For the branching particle system in dimension d with symmetric alpha-stable 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 self-similar long-range dependence stable process (which is non-Levy) 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 high-density 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, Friedrich-Alexander-Universitat
Evolving genealogical trees
We consider stochastic processes with values in trees, which arise from population models. The appropriate state space will be discussed and well-posed martingale problems are formulated for the tree-valued Fleming-Viot 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 d-dimensional Euclidean space. This method is based on the theory of multiparameter martingales and requires only the straightforward computation of so-called *-compensators to identify the appropriate limit. The compensator method is not only dimension-free (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 nearest-neighbour 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 trading-behaviour microstructure and value-based macrostructure. We will discuss filtering-based methods for testing for such things stochastic volatility and long-range dependence in tick-by-tick price data. We will also discuss use for inferring model parameters, including the difficult-to-estimate Hurst parameter, from tick-by-tick 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 prey-predator behaviour is the Volterra-Lotka 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 measure-valued processes
Measure-valued diffusions and measure-valued 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 measure-valued 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 self-avoiding 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 "non-Euclidean lattices", specifically graphs that correspond to regular tilings of the hyperbolic plane (or 3-space). 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 high-dimensional 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 Kratky-Porod, 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 non-linear 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

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 finite-dimensional sde's which arise as limits of interacting branching particle systems with finitely many sites. The resulting systems have degenerate non-Lipschitz 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 two-dimensional 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 Clark-Ocone formula for functionals of a square-integrable Levy process
We construct a Malliavin derivative for functionals of square-integrable Levy processes and derive a Clark-Ocone 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 Jean-Francois Renaud.

Tom Salisbury, York University
Conditioned super-Brownian motion
I will speak about some joint work with Deniz Sezer (York University) on conditioning super-Brownian 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 Fleming-Viot processes
What forces the mutation operator of a reversible Fleming-Viot process to be uniform? Our explanation is based on Handa's result that reversible distributions must be quasi-invariant under a certain flow, making the mutation operator satisfy a cocycle identity.

We also apply these ideas to a system of interacting Fleming-Viot 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 spread-out 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 Alexander--Orbach
conjecture in this setting.

This is joint work with Martin Barlow, Antal Jarai and Takashi Kumagai.

Wei Sun, Concordia University
On Girsanov and generalized Feynman-Kac 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 aquasi-continuous 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 generalizedFeynman-Kac 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 Feynman-Kac transform, we give a necessary andsufficient condition for the generalized Feynman-Kac semigroup($P^u_tf(x):=E_x[e^{N^u_t}f(X_t)]$) to be strongly continuous.

Anton Wakolbinger, J.W. Goethe-Universitat
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 self-duality 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.

Feng-Yu 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 $|x-y|^{-\al}$, $0\le\al\le 2$, as $|x-y|\to\infty$. 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 super-Brownian motion which has a non-trivial 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 zero-one law of almost sure local extinction for super-Brownian motion
In this talk we consider a d-dimensional ($1+\beta$)-super-Brownian 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 super-Brownian 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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