# Past Probability Seminars Fall 2002

## UW Math Probability Seminar Fall 2002

Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted.

Organized by [index.html Timo Seppäläinen ]

### Schedule and Abstracts

|| Tuesday, September 3, 2:25 PM, Van Vleck B223 || || Note unusual time and place. || || * Hermann Thorisson,* University of Iceland || || * Coupling* ||

Coupling means the joint construction of two or more random elements. The aim is usually to establish some distributional relation between the individual elements. But the aim can also be the reverse: to turn a distributional relation into a pointwise relation. Examples are the turning of stochastic domination into pointwise domination, weak convergence into pointwise convergence, and liminf convergence of densities into pointwise convergence where the random elements actually hit the limit. This deepens our understanding of the distributional relation itself, may enable us to establish previously hard-to-prove facts by simple pointwise arguments, and often leads to unexpected new results.

This talk starts off with the above examples and then moves to stochastic processes (exact coupling, shift-coupling, and epsilon-couplings). Applications to Markov Processes, Regenerative Processes and in Palm Theory will be indicated. The view is then extended to random fields with applications in Palm Theory, and finally to random elements under a topological transformation group which opens up many new possibilities for applications.

Reference: Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, NY.

|| Thursday, September 5 || || * Hermann Thorisson,* University of Iceland || || * Point-Stationarity* ||

Let N^o be the Palm version of a stationary Poisson process N in R^d, that is, N^o has the same distribution as N + \delta_0. Consider the following problem: when d > 1, is there some non-randomized way of shifting the origin of N^o from the point at the origin to another point T so that the distribution of N^o does not change?

This is clearly possible when d = 1, since then the intervals between points are i.i.d. exponential and remain so when the origin is shifted to the nth point on the right (or on the left) of the point at the origin. And when d > 1, it is shown in Thorisson (2000) that such a T - with P(T ^� 0) arbitrarily close to 1, - exists if external randomization is allowed. But is there a strictly non-zero non-randomized T?

We shall show that the answer is yes. There is actually a sequence (T_n : n \in Z) of such points, and for d = 2 and d = 3 this sequence strings up the points of N^o.

If we go beyond the Poisson case, a more general problem concerns the concept of "point-stationarity". Intuitively, point-stationarity means that the behaviour of a point process N^o relative to a given point of the process is independent of the point selected as origin. Formally, this concept is defined in Thorisson (2000) to be distributional invariance under bijective point-shifts "against any independent stationary background" and shown to be the characterizing property of the Palm version N^o of any stationary point process N in R^d. A natural question is whether the definition of "point-stationarity" can be reduced to distributional invariance under non-randomized bijective point-shifts. An approach to this problem will be outlined.

Reference: Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, NY.

|| Thursday, September 12 || || * Steven Lalley,* University of Chicago || || * Random walks on infinite free products and infinite algebraic systems of generating functions * || || [lalley.pdf Abstract] ||

|| Thursday, September 19 || || * Ananda Weerasinghe,* Iowa State University || || * Optimal strategies for a class of singular stochastic problems * ||

Consider a real valued Ito process which is controlled through a dynamic and adaptive choice of drift and diffusion coefficients and by an added bounded variation process. In this talk we consider the problem of choosing optimal controls to minimize the infinite horizon discounted cost which consists of two parts: a running cost which penalizes the state of the process for being away from the origin, and a cost for using the bounded variation process. We derive an optimal strategy and show that the value function is twice continuously differentiable. We also establish Abelian limit relationships between the value functions of the discounted cost problem and the corresponding ergodic control problem.

|| Thursday, September 26 || || * Maury Bramson,* University of Minnesota || || * Stationary Measures for One Dimensional Exclusion Processes * ||

The exclusion processes constitute one of the main families of interacting particle systems. Particles move about according to a given random walk kernel, except that no more than one particle is permitted at each site. Even in one dimension, the theory is incomplete when the random walk kernel is asymmetric. Here, we discuss recent results on the equilibrium measures for such kernels.

|| Thursday, October 3 || || * Alexander Holroyd, * University of British Columbia and UC Berkeley || || * Two-dimensional bootstrap percolation * ||

Bootstrap percolation is a simple cellular automaton model. Sites in an L by L square are initially independently occupied with probability p. At each time step, an unoccupied site becomes occupied if it has at least two occupied neighbours. We study the behaviour as p tends to 0 and L tends to infinity simultaneously of the probability I(L,p) that the entire square is eventually occupied. We prove that I tends to 1 if liminf p log L > lambda, and I tends to 0 if limsup p log L < lambda, where lambda = pi^2/18. The existence of lambda settles a conjecture of Aizenman and Lebowitz, while the determination of the value corrects numerical predictions of Adler, Stauffer and Aharony.

|| Saturday-Sunday, October 12-13 || || 2002 Fall Central Section Meeting of the AMS ||

This conference takes place at the UW Madison campus. There is a special session on probability that meets on both conference days. Click above to see the program.

|| Thursday, October 17 || || No seminar on account of the Midwest Probability Colloquium in Evanston, Illinois. ||

|| Monday, October 21, 2:25 PM, Van Vleck B329 || || Note unusual time and place. || || * Wenbo Li,* University of Delaware || || * Large and Moderate Deviations for Intersection Local Times * ||

There are various motivations for the study of deviations for intersection local times generated by, respectively, independent Brownian local times and independent local times of symmetric random walks. In this talk, we will present results on mixed (self and independent) intersections. Our approach relies on a Feynman-Kac type large deviation for Brownian occupation time, certain localization techniques from Donsker-Varadhan (1975) and Mansmann (1991), and some general methods developed along the line of probability in Banach space. Our treatment in the case of random walks also involves rescaling, spectral representation and invariance principle. This talk is based on a joint work with Xia Chen.

|| Thursday, October 31 || || * * || || * * ||

|| Thursday, November 7 || || *Jim Kuelbs, * UW Madison || || * Moderate Deviation Probabilities for Open Convex Sets: Non-logarithmic Behavior * ||

Precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite and infinite dimensional settings. Our results are based on the existence of dominating points for these sets, a related representation formula, and asymptotics for the integral term in this formula.

|| Thursday, November 14 || || *Ofer Zeitouni, * University of Minnesota and Technion || || * Coagulation-fragmentation processes, random transpositions, and the Poisson Dirichlet law * ||

Consider the Markov chain on the space of (countable) partitions of the interval [0,1], obtained first by size biased sampling twice (allowing repetitions) and then merging the parts (if the sampled parts are distinct) or splitting the part uniformly (if the same part was sampled twice). Vershik conjectured that the Poisson-Dirichlet law is the unique invariant measure for this Markov chain. Our proof of Vershik's conjecture uses a combination of probabilistic, combinatoric, and representation-theoretic arguments. (Joint work with P. Diaconis, E. Mayer-Wolf and M. Zerner.)

|| Thursday, November 21 || || *William Sandholm, * Department of Economics, UW Madison || || * Evolution in games with randomly disturbed payoffs * ||

We study evolutionary dynamics in games whose payoffs are described by probability distributions. To begin our analysis, we establish global convergence of the perturbed best response dynamics in three classes of games: stable games, potential games, and supermodular games. We then use these results to prove global convergence in a model of deterministic evolution in Bayesian games, and consider an application of this framework to a general model of traffic networks. Finally, we use our initial analysis to study a model of stochastic evolution. We apply tools from stochastic approximation theory to prove finite and infinite horizon convergence results, and use techniques from large deviation theory to provide a detailed characterization of long run behavior in potential games under the logit choice rule. (Joint work with Josef Hofbauer, U. of Vienna.)

|| Thursday, November 28 || || Thanksgiving Break - No Seminar ||

|| Thursday, December 5 || || * Timo Seppäläinen, * UW Madison || || * Increasing path models and interacting particle systems * ||

I will discuss some models of increasing paths on the square lattice and interacting particle systems connected with them.