# Past Probability Seminars Spring 2009

## Probability Seminar Spring 2009

The Probability Seminar meets Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. Organized by Tom Kurtz

### Schedule

- January 29*

- *

- Rohini Kumar, *University of Wisconsin-Madison

- Current fluctuations for independent random walks in multiple dimensions*

I will talk about current fluctuations in a system of asymmetric random walks in multiple dimensions. The current process of interest is defined as a distribution valued process. Scaling time by _n_ and space by _n^1/2^_ gives current fluctuations of order _n^d^__^/2^_ where _d_ is the space dimension. The scaling limit of the normalized current process is a distribution valued Gaussian process with given covariance.

- February 5 *

- Jeremy Quastel*, University of Toronto

- Effect of noise on traveling fronts*

Reaction-diffusion equations are used widely as models in population biology, genetics, chemistry, combustion theory etc., and traveling fronts are one of their most important qualitative features. However they are really approximations to more basic models involving discreteness (particles) and randomness. I will discuss how random perturbations affect the front speed in one of the simplest models.

- February 12*

- *

- Timo Seppäläinen, *University of Wisconsin-Madison

- *

- Fluctuation bounds for a class of asymmetric zero range processes*

We show that the variance of the particle current across a characteristic is of order t^{2/3} for a class of totally asymmetric zero range processes with concave jump rates. This conforms with the expected universal behavior of asymmetric particle systems with concave flux. The bound for the current follows from superdiffusive moment bounds for a second class particle. The proof relies on coupling constructions. (Joint work with M\'arton Bal\'azs and J\'ulia Komj\'athy, Budapest.)

- February 19 *

- *

- Benedek Valko*, University of Wisconsin-Madison

- The TASEP speed process*

In a multi-type totally asymmetric simple exclusion process (TASEP) on the line, each site of Z is occupied by a particle labeled with a number and two neighboring particles are interchanged at rate one if their labels are in increasing order. We consider the process with the initial configuration where each particle is labeled by its position. It is known that in this case a.s. each particle has an asymptotic speed which is distributed uniformly on [-1,1]. We study the joint distribution of these speeds: the TASEP speed process. We prove that the TASEP speed process is stationary with respect to the multi-type TASEP dynamics. Consequently, every ergodic stationary measure is given as a projection of the speed process measure. By relating this form to the known stationary measures for multi-type TASEPs with finitely many types we compute several marginals of the speed process, including the joint density of two and three consecutive speeds. One striking property of the distribution is that two speeds are equal with positive probability and for any given particle there are infinitely many others with the same speed. (This is joint with G. Amir and O. Angel)

- *

- February 26*

- *

- Jim Rawlings, *University of Wisconsin-Madison**

- *

- State estimation: Moving horizon estimation and particle filtering*

- *

This seminar provides an overview of currently available methods for state estimation of linear, constrained and nonlinear dynamic systems. The seminar begins with a brief overview of the Kalman filter, which is the optimal estimator for a linear dynamic system subject to independent, normally distributed disturbances. Next, alternatives for treating nonlinear and constrained dynamic systems are discussed. Two complementary methods are presented in some detail: moving horizon estimation, which is based on optimization, and particle filtering, which is based on sampling. The advantages and disadvantages of these two approaches are presented. Topics for new research are suggested that address combining the best features of moving horizon estimators and particle filters.

- *

- March 5 No Seminar*

- *

- March 12*

- *

- Nicos Georgiou, *University of Wisconsin-Madison

- Soft edge results for longest increasing paths on the planar lattice*

For two-dimensional last-passage time models of weakly increasing paths several results have been found about the hard edge (close to the axis). For strictly increasing paths of Bernoulli(p) marked sites,

questions about the hard edge are addressed using the SLLN. The SLLN also applies when we want to study the behavior of these paths in a rectangle of dimensions n/p and n as n becomes large. We are interested in the maximal cardinality and fluctuations of marked sites on a strictly increasing path in a slightly smaller rectangle (but still of the same order of magnitude) where we are unable to use SLLN.

- *

- March 19 No Seminar, Spring Break*

- *

- March 26*

- *

- Luc Rey-Bellet,* University of Massachusetts-Amherst

- *

- Large deviations for the periodic Lorentz gas and nonuniformly hyperbolic dynamical systems.***

- *

The periodic Lorentz gas (or Sinai billiard) is a mechanical system consisting of a point mass bouncing elastically off a periodic array of strictly convex smooth obstacles in the plane. This is a chaotic dynamical system and can be seen as a "deterministic random walk". A famous result of Sinai, Bunimovich and Chernov is the ergodicity (and mixing) of this dynamical system as well the central limit theorem. Around ten years ago Lai-Sang Young proved the long-standing conjecture that the correlations decay exponentially fast for this system. Building on the techniques used in this proof we prove a large deviation principle for ergodic averages of suitable observables. Our results apply not only to billiards but also to various nonuniformly hyperbolic dynamical systems such has quadratic maps and Henon maps and so on...

This is a joint work with Lai-Sang Young.

- *

- April 2***

- *

- Tom Kurtz, *University of Wisconsin-Madison

- Prophetic constructions of branching and related processes*

A collection of well-known population models (branching processes, branching Markov processes, branching processes in random environments, etc.) is constructed in a manner that associates with each individual in the population a characteristic called a level. If the levels are known to an observer, then a great deal is known about the future behavior of individuals (e.g., the exact time of death). If the levels are not known, then the models evolve as the observer would expect from their classical descriptions. The constructions enable straight forward proofs of a variety of known and not-so-well-known results including limit theorems, conditioning arguments, and derivation of properties of genealogies.

- *

- April 9 No Seminar*

- *

- April 16*

- *

- Jon Peterson, *University of Wisconsin-Madison

- Current fluctuations for a system of independent one-dimensional RWRE*

In this talk I will consider a system of an infinite number of independent random walkers all moving in a common random environment. As a first step in describing the long-term behavior of this system I will prove a hydrodynamic limit which essentially says that all the particles move with a constant (deterministic) speed. The deviations from this deterministic limit are studied by what is known as the current process.

The current is the net flow of particles seen by an observer traveling at the deterministic speed given by the hydrodynamic limit. For classical random walks (i.e. non-random environment) the fluctuations of the current are of the order n^{1/4}. In this model, however, the main contribution to the current process comes only from the environment and has fluctuations of order n^{1/2}. If the current process is centered by the quenched mean, then the fluctuations are of order n^{1/4} but are still different from the case of classical random walks. Our proof relies on the quenched and annealed central limit theorems for one-dimensional RWRE. (Joint work with Timo Seppalainen).

- *

- April 23*

- *

- Scott McKinley, *Duke University

- Diffusion in Soft Matter*

Stochastic models for diffusion of Brownian particles in soft matter (viscoelastic media) play a central role in polymer dynamics and rheology, microrheology, and medical science. A sufficiently robust class of stochastic processes is required to capture the range of observed anomalous diffusive behavior, in particular transient power law scaling of the mean-squared displacement (MSD) of tracked particles. We consider the Generalized Langevin Equation characterized by a Prony series approximation to the relaxation kernel, and study in particular this system in its zero mass limit. Such a study reveals a robust class of models which exhibit transient anomalous diffusion while remaining amenable to rigorous analysis. (Joint work with Greg Forest, UNC-Chapel Hill and Lingxing Yao, Utah)

- *

- April 30*

- *

- Nancy Garcia,* Universidade Estadual de Campinas – UNICAMP

- Explicit solution of the Monge-Kantorovich problem for chains of infinite order*

In modern probabilistic terms the Monge-Kantorovich problem can be presented as follows. Suppose that P and Q are two Borel probability measures given on a Polish space. Given a product-measurable non-negative cost function c, find \mu^* between P and Q that minimizes the expected cost. In this work, we address the question of finding an explicit solution for laws of infinite memory chains on a finite alphabet A. The construction works provided that the transition probabilities of the two chains are continuous and lose memory fast enough.

- *

- May 7*

- *

- Bill Sandholm, *University of Wisconsin-Madison

- Stochastic Evolutionary Game Theory: Overview and Recent Results*

Population games provide a general model of strategic interactions among large numbers of agents; network congestion, multilateral externalities, and natural selection are among their many applications. Behavior in these games is most naturally modeled as a stochastic dynamic adjustment processes. One begins with a particular game and a model of how individual agents make decisions. When the number of agents is large enough and the time horizon of interest not too long, the evolution of aggregate behavior is well approximated by solutions to the mean dynamic, an ordinary differential equation describing the expected increments of the underlying stochastic process. If one is interested in behavior over very long time spans, one studies the stochastic evolutionary processes directly, using its stationary distribution as the basis for predictions; using the large deviations methods of Freidlin and Wentzell, one can obtain unique predictions of infinite horizon behavior even when the mean dynamic admits multiple stable equilibria. In this talk, I will explain the main models of stochastic evolutionary game theory, present some recent results, and indicate directions for future research.

See http://www.ssc.wisc.edu/~whs/research/egt.pdf for a survey.