THEMATIC PROGRAMS

April 16, 2014

Program in Probability and Its Applications
Seminar Series


August 19, 1998
Mean-field lattice trees
Remco van der Hofstad, Delft University of Technology

Abstract: TBA


August 25, 1998
Strict concavity of the Brownian motion intersection exponent
Greg Lawler, Duke University

Abstract: TBA


September 9th, 1998
A Pattern Theorem for Lattice Clusters
Neal Madras, York University

Abstract
In 1963, Kesten proved a Pattern Theorem for self-avoiding walks, which says that any finite sequence of steps that can occur in the middle of a long self-avoiding walk must in fact occur pretty often on almost all self-avoiding walks. This result has had many applications in the theory of self-avoiding walks, including several ratio limit theorems. This talk will describe an analogous theorem for lattice trees and lattice animals. A weighted version of this theorem also applies to polymer collapse and percolation models. Applications and open questions will be discussed.


September 16th, 1998
Moment asymptotics for the Anderson model
Wolfgang Koenig, Technische Universität Berlin / York University

Abstract: TBA


September 23th, 1998
Self-diffusion for Brownian Motions with Local Interaction
Ilie Grigorescu, McMaster University / The Fields Institute

Abstract: TBA


September 23rd, 1998
Infinite Systems of Diffusions in Population Biology
Jan Swart, University of Nijmegen

Abstract: TBA


September 23rd, 1998
Ruelle's probability cascades and an abstract cavity method
Erwin Bolthausen, University of Zurich

Abstract
The so called generalized random energy model (GREM for short) has been introduced by Derrida as a very simple model in spin glass theory. Some of its remarkable properties in the limit had been discussed by Ruelle. In a joint paper with Alain-Sol Sznitman, it was found, that there is a very simple and appealing clustering process which governs some of the crucial properties of this model. This clustering process also allows to formulate a continuous version of the GREM without the necessity of cumbersome limiting procedures. Furthermore, one can give an abstract version of the so called cavity method which plays a crucial role in the nonrigorous discussion of the Sherrington-Kirkpatrick model.


October 13th, 1998
On a Conjecture of B. Jorgensen and A.D. Wentzell:
from Extreme Stable Laws to Tweedie Exponential Dispersion Models

Vladimir Vinogradov, University of Northern British Columbia

Abstract
The class of Tweedie exponential dispersion models includes such well-known continuous distributions as the normal and gamma, the purely discrete sclaled Poisson distribution, as well as the class of mixed compound Poisson-Gamma distributions which have positive mass at zero, but are otherwise continuous. The remaining Tweedie models are derived by exponential tilting of extreme stable distributions. This quite heterogeneous class of distributions was introduced in statistics by Tweedie (1984),mainly because of the simple form of their unit variance function: V(m) = m^p.

It is known (see, e.g., Jorgensen (1997)) that Tweedie models possess scaling properties similar to those of the stable laws and also that they emerge as weak limits of appropriately scaled natural exponential families. Domains of attraction to Tweedie models are described in Jorgensen, Martinez and Tsao (1994) and Jorgensen, Martinez, Vinogradov (1998). At the same time, Jorgensen (1997, pp. 150-151) and Wentzell (1998,personal communication) independently conjectured that the classical theorems on weak convergence to stable laws and those on weak convergence to Tweedie models should be related.

In this lecture, it will be shown how the theorems on weak convergence to Tweedie models with index p>2 can be derived from those on weak convergence to the positive stable laws.


October 14th, 1998
Large Deviation methods in Risk Theory
Anders Martin-Lof, Stockholm University

Abstract
It will be shown that many well known estimates of e.g. ruin probabilities for compound Poisson processes can in a natural way be derived using L.D. techniques including second order refinements. The L.D framework provides an interesting description of how to allocate buffer capital to two systems in "thermodynamic equilibrium".


November 4th, 1998
Random walks in a field of traps: from simple random walk on Z^d to any Markov chain
Stanislav Evguenievich Volkov, York University/Fields Institute

Abstract
Start with a S.R.W. Designate some points of Z^d (d>2) to be "traps" so that if a particle hits one of them it stays in it forever. The traps are placed at random independtently from each other. We are interested in identifying the maximal critical density of traps for which the random walk is still transient (that is, it eventually goes to infinity). Two cases are considered:

A) annealed, when the set of traps is updated at each unit of time;
Q) quenched, when the set of traps is fixed once and for all.
We will show that the solutions to these two cases coincide.

Next we consider an arbitrary Markov chain on any space S and show the equivalence of the annealed and quenched problems under uniform boundedness of Greens function. Moreover, assuming some geometry on the space S, a relative spherical symmetry of the density q(x) of the field of traps implies a necessary and sufficient condition for the transience. This condition consists in the finiteness of the sum, sum_{x \in S} g(x_0,x)q(x).

The talk is based on two papers: one joint with F. den Hollander and M.Menshikov, the other with R. Pemantle.


November 4th, 1998
Large deviations for observation and related processes
George L. O'Brien, York University

Abstract
Let $(X_n, n \geq 1)$ be an i.i.d.\ sequence of positive random variables with distribution function $H$. Let $\Phi_H := \{ (n,X_n),\;n \geq 1\}$ be the associated observation process. We view $\Phi_H$ as a measure on $E := [0,\infty) \times (0,\infty]$ where $\Phi_H(A)$ is the number of points of $\Phi_H$ which lie in $A$. A family $(V_s, s > 0)$ of transformations is defined on $E$ in such a way that for suitable $H$ the distributions of $(V_s\Phi_H, s > 0)$ satisfy a large deviation principle and that a related Strassen-type law of the iterated logarithm also holds. Some related large deviation principles and loglog laws are then derived for extreme values and partial sums processes. Similar results are proved for $\Phi_H$ replaced by certain planar Poisson processes, with parallel applications to extremal processes and spectrally positive Levy processes.


November 18th, 1998
Duality for a stochastic p.d.e. with density dependent branching noise.
Siva Athreya, Fields Institute (Joint work with Roger Tribe)

Abstract
We establish a duality relation for the moments of the one dimensional stochastic partial differential equation
\[ u_t = \Delta u + \sqrt{\sigma(u)} \dot W_{t,x}, \]
where $\dot W_{t,x}$ is space-time white noise and $\sigma$ is a real analytic function satisfying certain growth and regularity conditions. The dual process is a system of Brownian particles with an interactive branching mechanism. In certain cases the duality relation implies weak uniqueness for the s.p.d.e. We will also present ongoing work in applying the same methodology towards uniqueness issues in finite dimensional stochastic differential equations.

(Joint work with Roger Tribe)


November 18th, 1998
The lace expansion and oriented percolation.
Eric Derbez, McMaster University / Fields Institute

Abstract
The lace expansion (originated by Brydges and Spencer) has been used to tackle many problems in statistical mechanics. This talk will consist of an exposition of some of the results in oriented percolation and the way in which the lace expansion can be set up to handle this model (as per work in progress with R v d Hofsdat and G Slade). If overhead projectors permit, Java demonstrations may be used to illustrate the salient points.


November 25th, 1998
Labyrinths
Stanislav Volkov, York University / the Fields Institute

Abstract
First, consider the two-dimensional lattice Z^2. Let each point of it be a "tunnel", a "\" mirror, a "/" mirror or a "normal point" independently from the others. Once a realization of this random field is fixed, a particle performing a random walk on Z^2 given this realization obeys the following laws: If a point the walker hits is a tunnel, it goes straight through it; a mirror, it reflects from it like a beam of light would; a normal point, it behaves like a simple RW, i.e. choses one of the four possible neighbors with equal probabilities.

This model is called a "random walk in a random labyrinth" and was introduced by Grimmett, Menshikov and V. in 1996.

The labyrinth is called {\em recurrent} if the RW on it is recurrent, and transient otherwise. The labyrinth is called {\em localised} if the number of points which a walker can visit is finite.

Assuming that the density of normal points is non-zero, we show 1) in the case of Z^2, a labyrinth is recurrent a.s. and 2) under which conditions it is non-localized with positive probability. For a general model on Z^d, d>2, we present sufficient conditions ensuring that a labyrinth is transient with positive probability.


December 9th, 1998
Metastability and Droplet Growth
Frank den Hollander, University of Nijmegen

Abstract
In this talk we describe {\it metastability and droplet growth} for the two-dimensional lattice gas with Kawasaki dynamics at low temperature and low density. Particles perform simple exclusion on $\mathbb Z^2$, and inside a {\it finite} but arbitrary box $\Lambda_0$ there is a binding energy $U$ between neighboring occupied sites. The initial configuration is chosen such that $\Lambda_0$ is empty while outside $\Lambda_0$ particles are placed randomly with density $\rho = e^{-\beta\Delta}$ for some $\Delta \in (U,2U)$, where $\beta$ is the inverse temperature. Since in equilibrium $\Lambda_0$ wants to be fully occupied when $\beta$ is large, the dynamics will tend to fill $\Lambda_0$ with particles. However, since the particle density $\rho$ is small this will take a long time to happen. We investigate how the transition from empty to full takes place. In particular, we identify the size and shape of the {\it critical droplet}, the time of its creation, and the typical trajectory prior to its creation, in the limit as $\beta \to \infty$. The choice $\Delta \in (U,2U)$ corresponds to the situation where the gas is supersaturated and the critical droplet has side length $>1$.

Because particles are {\it conserved} under Kawasaki dynamics, the analysis of metastability and droplet growth is more difficult than for Ising spins under Glauber dynamics. The key point is to show that at low density the gas outside $\Lambda_0$ can be treated as a reservoir that creates and annihilates particles randomly at the boundary of $\Lambda_0$. Once this has been achieved, the problem reduces to local metastable behavior on $\Lambda_0$, and standard techniques from non-conservative dynamics can be applied. Inside $\Lambda_0$ the dynamics is still conservative, but this can be handled via local geometric arguments.


December 9th, 1998
Large Time Asymptotics of the Nonlinear Filter
Rami Atar, York University / the Fields Institute

Abstract
The nonlinear filtering equation describes the conditional law of a Markov process $\{x_t\}$ at time $t$ conditioned on the path $y_s \int_0^s g(x_u)du + w_s$ for $s\in[0,t]$. Here $g$ is a fixed function and $\{w_t\}$ is a Brownian motion independent of $\{x_t\}$. We study the exponential decay rate of the variation distance between solutions that correspond to different initial conditions. The question is related to Lyapunov exponents, Birkhoff's contraction and large deviations.


January 6th, 1999
Conditioning Super Brownian Motion
Tom Salisbury,York University

Abstract
I'll describe some joint work with John Verzani, on conditioning super Brownian motion (SBM) via its exit measure. In high dimensions, we condition on the support of the exit measure hitting a given finite collection of points, and represent this process in terms of a tree-like backbone throwing off mass. The goal is to give an explicit dynamic description of this tree. Along the way, I'll review LeGall's construction of the super Brownian excursion.


January 19th, 1999
Asymptotic Behavior of Solutions of Parabolic SPDE
Min Kang, York University / The Fields Institute

Abstract
We investigate the weak convergence as time goes to infinity of the solutions of parabolic stochastic partial differential equations with Dirichlet boundary conditions. We first prove a result for the linear case, which is easy since the solution is Gaussian. Then we get a result for non-linear drift case by proving a new comparison theorem in SDDE, stochastic difference differential equations.


January 20th, 1999
Liouville property and a conjecture of de Giorgi
Martin Barlow, University of British Columbia

Abstract
In 1978 De Giogi conjectured that if $u(x)$, $x=(x_1, \dots, x_n) \in R^n$ is a global bounded solution of the non-linear PDE $$ \Delta u + u - u^3 =0, \eqno(1)$$ which is monotone in the $x_1$ direction, then $u$ is essentially one-dimensional. Recently Ghoussoub and Gui have proved this conjecture for $n=2$.

It turns out that this problem can be reformulated in terms of a Liouville property for a divergence form operator $$ {\cal L} = \nabla \sigma^2 \nabla. \eqno(2)$$ Here $\sigma(x)$, $x \in R^n$ is a positive (but not uniformly positive) function, defined in terms of the solution $u$. This problem can be studied using probabilistic techniques.

In this talk I will describe recent work (with R.F. Bass and C. Gui) in which we prove a Liouville theorem for some operators of the form (2). As a consequence, we can prove the conjecture under some additional hypotheses.


January 20th, 1999
Generalizing the Martingale Central Limit Theorem.
Dean Slonowsky, Fields Institute

Abstract
For each $n \geq 1$, let $X_n = \{ X_n(t) : t \in [0,1] \}$ be an $L_2$ martingale in $D([0,1])$. Roughly speaking, the martingale central limit theorem states that if
\begin{itemize}
\item the jumps in the sample paths of $X_n$ become negligible (in a certain sense) as $n$ gets large and
\item the predictable (or optional) quadratic variations of the $X_n$ converge to a continuous, increasing and deterministic limit $h : [0,1] \rightarrow [0,{\infty})$ as $n$ goes to infinity
\end{itemize}
then the $X_n$ converge weakly in $D([0,1])$ to a Brownian motion $B_h$ which is ``stretched-out'' by $h$ (cf.\ Helland 82).

In this talk, we generalize the above result to the situtation in which the latter condition holds for a sequence of {\em{general quadratic variations}}---processes which resemble predictable/optional quadratic variation but are not necessarily adapted. An application to the convergence of set-indexed strong martingales will be briefly discussed.


February 2nd, 1999
Graph Colouring with the Probabilistic Method
Michael Molloy, University of Toronto

Abstract
The probabilistic method is a powerful technique, pioneered by Erdos,by which one proves the existence of a combinatorial object by showing that an attempt to produce such an object randomly will succeed with positive probability. Often this argument will lead to an efficient algorithm to construct the desired object. In this talk, we will survey several applications of this technique to graph colouring. We will see how to prove that under certain conditions, a graph has a colouring meeting some desireable properties, by showing that a simple randomised procedure to construct such a colouring will succeed with positive probability. Much of the work described here is joint with Bruce Reed.


February 10th, 1999
A Microscopic Model For Porous Medium Equation
Shui Feng, McMaster University

Abstract
We introduce a stochastic lattice model that leads to porous medium equation under diffusive scaling limit. This is a joint work with Ian Iscoe and Timo Seppalainen.


February 17th, 1999
Diffusions and heat kernel analysis on an infinite dimensional group.
Masha Gordina, McMaster University

Abstract
First I'll construct the heat kernel measure $\mu _t$ on an infinite dimensional complex group $G$ using a diffusion in a Hilbert space. Using properties of this diffusion I can prove that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, $\mathcal{H}L^2(G, \mu _t)$, is one of two spaces of holomorphic functions I consider. Also I construct a subgroup $G_{CM}$ of $G$ which is an analog of the Cameron-Martin subspace. I'll describe an isometry from the first space to another space of holomorphic functions $\mathcal{H}L^2(G_{CM})$. The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from $\mathcal{H}L^2(G_{CM})$ into the Hilbert space associated with a certain Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups. I'll explain how my results on the Cameron-Martin subgroup are related to the work of I. Shigekawa and H. Sugita on the complex Wiener space.


February 17th, 1999
Steiner distances in random trees
Amram Meir, York University

Abstract
Let G be a connected graph, S a subset of its vertices . The Steiner distance d(S,G) of S in G is the minimum number of edges in any connected subgraph of G that contains S. The total Steiner k-distance D(k;G) (k=2,3,...) is the sum of d(S,G) over all subsets S of k vertices in G. (When k=2, the quantity is known as Wiener index of G). The purpose of this talk is to present asymptotic results (for large n): (i) For the expected value M(k;n) of d(S,T) over all randomly chosen subsets of k nodes in trees T with n nodes, which belong to certain (infinite) families F of random rooted trees. (ii) For p(k;n,m), the probability that d(S,T)=m-1, when |S|=k fixed, |T|=n and T belongs to a family F. (joint work with L. Clark and J.W. Moon)


February 24th, 1999
Trapping and saturation in a random media
Alejandro F. Ramírez, École Polytechnique Fédérale de Lausanne

Abstract
We consider a model of diffusion in a random media originated from the nuclear waste management industry. In each site of the cubic lattice $Z^{d}$ there is an obstacle with a positive integer valued random capacity. On the other hand, on the origin and independently of the distribution of the obstacles, independent random walks are introduced so that at time $t$, $N(t)$ random walks have been born, and so that they survive until touching an obstacle. When this happens the capacity of the touched obstacle decreases by one. An obstacle of 0 capacity disappears. For low values of $N(t)$ we prove that with probability 1, the saturated obstacles correspond to a sphere of volume $N(t)$. With the help of this fact, and using the second version of the method of enlargement of obstacles of Sznitman, we prove that there are three injection regimes, depending on $N(t)$ and the dimension. In particular estimates on the asymptotic behaviour of the principal Dirichlet eigenvalue of the discrete Laplacian operator on a box of side $t$ with sites removed with positive probability, each one independent of the others, is required. This is a joint work with Gerard Ben Arous.


February 24th, 1999
Vertex-Reinforced Random Walks: What's New?
Stanislav Volkov, York University/Fields Institute

Abstract
Vertex-Reinforced Random Walk (VRRW) is a non-Markovian random process which "prefers" to visit the states it has visited before. The questions about VRRW are classical examples of problems which are very easy to formulate but extremely hard to solve. We consider VRRW on a very general class of infinite graphs and show that, with a positive probability, VRRW on them visits only finitely many points. We conjecture that on all graphs of bounded degree this takes place a.s. (this has only been shown for the case of trees). The proof is based on identifying a "trapping" subgraph on which VRRW can get stuck and showing how this actually happens (and what the limiting empirical occupational measure is). An interesting finding is that this limiting measure is dependent on whether the graph contains triangles or not. These results represent my on-going research and generalize those obtained by Pemantle and Volkov (1998) for $\Z^1$, though using a different technique.


March 3rd, 1999
Some results for long range percolation and maniscale models
Mikhail Menchikov, Moscow State University

Abstract
We consider two-dimensional Bernoulli long range percolation model. We prove that if we have a percolation for this model, we can find a positive N such that if we cut all edges of size more then N, anyway percolation will be. We demonstrate a conterexample for the case of dependent percolation model. This example closely connected with discrete and continuous multiscale models.


March 3rd, 1999
Skew Convolution Semigroups and Immigration Processes
Zenghu Li, Beijing Normal University

Abstract
The immigration phenomena associated with branching models have been investigated quite extensively in the past decades. In the measure-valued setting, immigration processes have been studied with different motivations by Dawson, Dynkin, Evans, Gorostiza, Ivanoff, Shiga and others. We focus on a special type of immigration associated with DW superprocesses formulated by skew convolution semigroups. Transition semigroups of the immigration processes can be characterized in terms of infinitely divisible probability entrance laws and their trajectory structures can be investigated by using Kuznetsov processes. The immigration processes provide a number of new limit theorems. In particular, the fluctuation limits of them lead to some infinite dimensional Ornstein-Uhlenbeck processes described by the branching mechanisms of the superprocesses. Other types of limit theorems can be obtained by randomizing the skew convolution semigroup.


March 17th, 1999
Vertex Reinforced Random Walks: the techniques.
Stas Volkov, Fields Institute/York University

Abstract
This talk is intended for people who attended my first talk on VRRW in February AS WELL AS for those who did not have a chance to come; I will try my best to make it interesting for both. Vetrex-Reinforced Random Walk (VRRW) on a graph (e.g. $Z^2$) is a nearest-neighbor walk that goes to a vertex with a probability proportional to the number of times this vertex has been visited before. Similar models for {\em edges} has been posed/studied by Diaconis, Davis, Durrett, Selke, Toth, Pemantle and others. It has been shown by Pemantle and Volkov that on $Z^1$ VRRW visits only finitely many vertices a.s. (and just 5 with a positive probability). Recently I proved that the similar statements hold for many other graphs (lattices, trees, etc.) In this talk I will outline the results with the stress on HOW they can be obtained and what are the possible approaches for this kind of problems.


March 24th, 1999
How does the Vervaat process really behave?
Ricardas Zitikis, Laboratory for Research in Statistics and Probability, Carleton University

Abstract
It is well-known that the appropriately normalized Vervaat process asymptotically behaves like one half times the squared empirical process. During the talk we shall give a complete description of the strong and weak asymptotic behaviour of the Lp-distance between these two processes. In particular, the law of the iterated logarithm, the ``other'' law of the iterated logarithm, weak convergence, etc. for the aforementioned distance will be discussed in detail.


March 24th, 1999
An Introduction to Set-Indexed Strong Submartingales.
Dean Slonowsky,Fields Institute

Abstract
In their 1994 paper, G. Ivanoff and E. Merzbach introduced the notion of a set-indexed strong submartingale, a generalization of the planar strong submartingales defined in the 1975 paper of R. Cairoli and J. Walsh.

After motivating their definition through simple examples, we will present several results in the theory of set-indexed strong submartingales including:

- A Doob-Meyer-type decomposition.
- The definition and existence of quadratic variation for square integrable set-indexed strong martingales.
- The role of set-indexed strong martingales in a Levy-type characterization of set-indexed Brownian motion.

Time permitting, the framework for strong martingale central limit theorems will be discussed.


March 31st, 1999
On the long-term behaviour of superprocesses with a general branching mechanism
Yongjin Wang, Nankai University

Abstract
The persistence criteria for branching systems and superprocesses have been established through a variety of ways since Dawson [1977]. We here give an analytic approach to this argument, in which a simple proof of the criteria is available. By extending our result to a larger class of superprocesses with a general branching mechanism, we then get a complete characterization of the long-term behaviour of the superprocesses on the tempered measure space as well as on the finite measure space. Finally, using Dynkin's stopped historical superprocesses, we also show that the extinction and non-extinction of superprocesses at large time are heavily dependent on the branching mechanisms.


March 31st, 1999
Elements of free probability
Alexandru Nica, University of Waterloo

Abstract
The name ``free probability'' does not refer to an attitude in the practice of probability, but to a theory initiated by D. Voiculescu about 15 years ago, with motivation from problems on free products of von Neumann algebras. The subject has evolved since then into a kind of parallel to basic probability theory, where tensor products and independent random variables are replaced by free products and free (non-commuting) random variables. The talk will avoid the von Neumann algebra side of the theory, and will focus on the latter parallelism (free-ness for random variables, as a non-commutative analogue for the concept of independence). For instance -- there exist free analogues of the Gaussian and of the Poisson distributions; the free analogue of the Gaussian is the semicircle law of Wigner, and appears as limit in the free central limit theorem of Voiculescu. A nice way to prove the free CLT goes by using the free counterpart of the notion of characteristic function, which is called ``the R-transform'' (of a probability measure on the real line). The list of free analogues could certainly be continued (e.g. free analogues of infinitely divisible distributions and of the Levy-Khinchin theorem, free stable laws, processes with free increments) and is a subject of current research. A notable feature of free probability is its connection to the theory of random matrices, which is already signaled by the occurrence of the Wigner's law in the free CLT, but also shows up in various other places.


April 7th, 1999
p-variation, integration and stock price modelling
Rimas Norvaisa, Institute of Mathematics and Informatics, Lithuania

Abstract
The p-variation is a generalization of the total variation of a function by replacing the absolute value of the increment in its definition by the p-th power of this increment. Then a.a. sample paths of a BM have bounded p-variation for each p >2. The usefulness of p-variation property for the Riemann-Steiljes integral was shown by L.C. Young (1936). We plan to discuss how the p-variation could be used in stochastic calculus and stock price modelling.


April 7th, 1999
Random spanning trees of Cayley graphs and compactifications of groups
Steven Evans, Department of Statistics, U.C. Berkeley

Abstract
We consider a natural algorithm for producing a random spanning tree for the Cayley graph of a finitely generated, countably infinite group. Associated with such a tree there is a (deterministic) compactification of the group: a sequence of group elements that is not eventually constant is convergent if the random geodesic through the spanning tree that joins the identity to the nth element of the sequence converges in distribution as n goes to infinity. We identify this compactification in a number of simple examples and present some open problems.


April 14th, 1999
Assymptotic properties of integral functionals of geometric stochastic processes.
Mikolos Csorgo, Carleton University

Abstract
This talk is based on a forthcoming paper by E. Csaki, M. Csorgo, A. Foldes and P. Revesz, where we study strong asymptotic properties of two types of integral fuctionals of geometric stochastic processes. These integral functionals are of interest in financial modeling, yielding various Asian type option pricings via appropriate selection of the processes in their respective integrands. We show that, under fairly general conditions on the latter processes, the log of the integral functionals themselves asymptotically behave like appropriate sup functionals of the processes in the exponents of their respective integrands. In addition to presenting these theorems in their general form, we will also illustrate them via geometric Brownian and geometric fractional Brownian motions.


April 14th, 1999
Forward and backward integrals revisited.
Terry Lyons, Imperial College

Abstract
In 1998 Lyons and Zheng (Asterisque,157-158) observed that the Stratonovich integral of a one form against a reversible and stationary diffusion process makes sense even if the one form is square integrable. Interestingly the resulting process is not a semi-martingale. However, it is a difference of a forward and backward martingale. The technique has lead to a number of useful estimates and is considerably more powerful than an ItF4 approach which would require significant smoothness.

A major failing of the techniques mentioned above, involved the need to start the process with the stationary measure. Integrating the form along the sample path of the process conditioned to start at a pointmight well not make sense. (In 1990 Proc. Royal Soc. Edinburgh Sect. A 115) it was shown by Zheng and Lyons that the integral existed if the one form was bounded. Stoica and Lyons (Annals of Prob 1999) proved that if the one form was in the natural conjectural Lebesgue space (coming from the Sobolev embedding theorem). Results about the behaviour at infinity were also given.

In this talk I will survey the basic results and, if time permits, I will give applications.


April 28th, 1999
Optimal approximation of stochastic differential equations by adaptive step-size control
Norbert Hofmann, University of Erlangen

Abstract
The talk will discuss the strong approximation of stochastic differential equations with respect to the global error in the L2-norm as well as in the uniform norm. For equations with additive noise we present sharp lower and upper bounds for the minimal error in the class of arbitrary methods which use a fixed number of observations of the driving Brownian motion. We introduce Euler schemes with adaptive step-size control which perform asymptotically optimal.


May 5th, 1999
Layout problems on random geometric graphs
Mathew Penrose, University of Durham

Abstract
Given a graph, layout problems are concerned with choosing an ordering of its vertices so that adjacent vertices lie close together in the ordering. For example, the aim might be to minimize the sum over all edges of the separation of the edge's endpoints in the ordering. We investigate problems of this type for graphs generated randomly, by percolation or by continuum analogs. Such graphs have been used by computer scientists for comparing heuristics for these problems. If time permits we may also discuss some connectivity properties of these graphs.


May 13th, 1999
Level sets of additive Levy processes
Davar Khoshnevisan, University of Utah

Abstract
I will talk about recurrence issues for level sets of some multiparameter Levy processes. This approach unifies and explains some earlier existing results in Markov processes, and points to a notion of regeneration which makes sense in any dimension. This is on-going work with Yimin Xiao.


May 19th, 1999
Quasi-potential of the Fleming-Viot process with neutral mutation and selection
Kenji Handa, Saga Universty

Abstract
In 1998, Dawson and Feng established large deviation principles for the Fleming-Viot processes with neutral mutation and selection, and the corresponding reversible measures as the sampling rate tends to 0. In this talk an identity between quasi-potential defined through the dynamical rate function and the static rate function is discussed.


May 26th, 1999
Slice sampler Markov chains
Jeffrey S. Rosenthal, University of Toronto

Abstract
The slice sampler is a recent example of a Markov chain Monte Carlo (MCMC) algorithm, designed to approximately sample from a given probability distribution. In this talk, we will review recent work (joint with G.O. Roberts) on the convergence properties of this algorithm. A particular variant, the polar slice sampler, will be shown to have strikingly good convergence rate for a large class of distributions.


June 2nd, 1999
Relaxation to equilibrium for interacting random walks
Jeremy Quastel, University of Toronto

Abstract
We will survey results and methods for hydrodynamic limits and decay estimates for different models of interacting conservative systems and discuss open problems in the field.


June 9th, 1999
State Dependent Multitype Spatial Branching
Don Dawson, The Fields Institute

Abstract:
This talk will describe joint work with Andreas Greven on spatial infinite type branching systems indexed by a countable group, for example $Z^d$ or the hierarchical group. The space of types is [0,1] and the state of the system at a given site is a measure on [0,1]. The spatial components of the system interact via migration. Instead of the classical independence assumption on the evolution of different families of the branching population, we introduce interaction between the families through a state dependent branching rate of individuals and state dependent mean offspring of individuals but for most results we restrict attention to the critical case. One objective is to establish that the large scale structure of surviving types is related to the immortal clan of super-Brownian motion.