# Past Probability Seminars Fall 2006

## UW Math Probability Seminar Fall 2006

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

Organized by Jason Swanson

### Schedule and Abstracts

|| Thursday, September 7 || || * Malwina Luczak, * The London School of Economics and Political Science || || * Laws of large numbers for epidemic models with countably many types* ||

In modelling parasitic diseases, it is natural to distinguish hosts according to the number of parasites that they carry, leading to a countably infinite type space. Proving the analogue of the deterministic equations used in models with finitely many types has previously either been done case by case, using some special structure, or else not attempted. In this paper, we prove a general theorem of this sort, and complement it with a rate of convergence in the l_1-norm. This is joint work with Andrew Barbour.

|| Thursday, September 14 || || * Thomas G. Kurtz, * University of Wisconsin-Madison || || * Particle representations and uniqueness for SPDEs* ||

Stochastic partial differential equations arise naturally as limits of finite particle systems given by systems of ordinary stochastic differential equations. In many such cases, the solution of the limiting SPDE can be represented using an infinite system of stochastic differential equations. The solution of the infinite system has either exchangeability properties or is conditionally Poisson. The measure-valued solution of the SPDE is then given in terms of the de Finetti measure of the exchangeable system or the Cox measures of the conditionally Poisson system. A class of such models will be described that includes the standard equations of nonlinear filtering theory. Conditions will be given under which uniqueness of the system of stochastic ordinary differential equations implies uniqueness for the corresponding SPDE.

|| Thursday, September 21 || || * Rohini Kumar, * University of Wisconsin-Madison || || * Current across characteristics for independent random walks* ||

The hydrodynamic limit of particle distribution in a random walk model is the solution of a transport PDE. This tells us that the initial density profile gets transported rigidly along the characteristics of the transport equation. Fluctuations from the hydrodynamic limit on the central limit scale are the initial fluctuations transported rigidly along the characteristic. Fluctuations of order n^{1/4} are given by the net current across charateristics at the microscopic level. We study this net current across characteristics over time and space; the spatial parameter is scaled by n^{1/2}. We see that the scaled net current across characteristics converges to a two-parameter Gaussian process.

|| Thursday, September 28 || || * Timo Sepp�l�inen, * University of Wisconsin-Madison || || * Current fluctuations in the asymmetric simple exclusion process* ||

This talk describes work on the fluctuations of the current in the asymmetric simple exclusion process, ranging from Ferrari-Fontes 1994 through recent joint work with Marton Balazs.

|| Thursday, October 5 || || * Nathana�l Berestycki, * University of British Columbia || || * Coalescents, Trees, Lookdowns and Oysters* ||

Lambda-Coalescents, or coalescents with multiple collisions, were introduced in 1999 by Pitman and by Sagitov. They have recently generated a lot of interest among probabilists and geneticists alike, because they arise in a number of population models as a way to describe the genealogical relationships in a sample (in particular when we cannot neglect the effect of selection or large reproductive success). After briefly recalling this connection, I will report on recent results that I have obtained with collaborators J. Berestycki, V. Limic and J. Schweinsberg, concerning the small-time asymptotics of Lambda-coalescents. At the heart of this approach is a new connection between Lambda-coalescents, continuous random trees, lookdown systems, and Fleming-Viot processes. This connection now holds in almost complete generality, subject only to a small assumption on the regularity of the measure Lambda. Time-permitting, I will discuss some of the implications of this result.

|| Thursday, October 12 || || * Kevin Buhr, * University of Wisconsin-Madison || || * A Lookdown Construction of Arratia Flow* ||

In his 1979 thesis and an uncompleted 1981 manuscript, Arratia described a coalescing Brownian flow on the real line. Intuitively, Brownian motions start at every space-time point and evolve independently until they meet and coalesce. Arratia's unpublished results are often cited, and remarkably an entirely new construction recently appeared in Annals of Probability (Fontes et al., 2004). In this talk, I'll give a simple, straightforward, and concrete construction using a particle lookdown system. This approach is in the spirit of Arratia's original direct but incomplete 1979 construction and seems truer to the intuitive description of the flow. It also allows for nearly transparent proofs of certain flow properties and suggests some new connections between Arratia flow and genetic particle models.

|| Thursday, October 19 || || * No seminar because of* || || * MIDWEST PROBABILITY COLLOQUIUM* ||

|| Thursday, October 26 || || * Davar Khoshnevisan, * University of Utah || || * Dynamical Percolation on General Trees* ||

H�ggstr�m, Peres, and Steif (1997) have introduced a dynamical version of percolation on a graph G. When G is a tree they derived a necessary and sufficient condition for percolation to exist at some time t. In the case that G is a spherically symmetric tree, Peres and Steif (1998) derived a necessary and sufficient condition for percolation to exist at some time t in a given target set D. The main result of the present talk is a necessary and sufficient condition for the existence of percolation, at some time t in D, in the case that the underlying tree is not necessary spherically symmetric. This answers a question of Yuval Peres (personal communication). We present also calculations of the Hausdorff dimension of exceptional times of percolation.

|| Thursday, November 2 || || * Julien Dubedat, * Courant Institute of Mathematical Sciences || || * Schramm-Loewner Evolutions on Riemann surfaces* ||

Introduced by Oded Schramm in 1999, Schramm-Loewner Evolutions (SLE) are random simple paths connecting two boundary points of a planar, simply connected domain. SLE has proved an efficient way to describe the scaling limit of critical 2d systems such as percolation. Defining SLEs corresponding to more complicated configurations, such as simply connected domains with several marked points or open Riemann surfaces, involves several difficulties. We will discuss "interactions" of several SLE strands and some constructions of SLE on open Riemann surfaces, in relation with zeta-regularization and Virasoro algebra representations.

|| Thursday, November 9 || || * Robert Nowak, * University of Wisconsin-Madison || || * Genomic Network Tomography* ||

Living cells respond to environmental changes via a sequence of intracellular protein-protein interactions, communicating the extracellular conditions to the nucleus where they ultimately lead to the production of proteins necessary for fundamental cellular operations. Biologists have discovered or conjectured certain signaling pathways, but our knowledge of cellular communications is still very incomplete. Due to the intrinsic difficulties associated with intracellular measurement, we consider the problem of inferring the structure of a cellular signaling network from co-occurrence data: observations in the form of lists indicating which proteins occur in each signaling pathway without revealing their order within the pathway. We call this problem Genomic Network Tomography (GNT), since it bears a strong resemblance, both physically and mathematically, to tomographic imaging and related network tomography problems. Without pathway order information, every permutation of the proteins leads to a different feasible network, resulting in combinatorial explosion of the feasible set. However, the physical principles underlying cellular signaling networks suggest that not all feasible solutions are equally likely. Proteins that co-occur in many pathways are probably more closely connected. Building on this intuition, we model path co-occurrence data as randomly shuffled samples of a random walk on the network. We derive a polynomial-time Monte Carlo network inference algorithm based on this model and, via novel concentration inequalities for importance sampling estimators, prove that the algorithm converges to a Maximum Likelihood solution of the GNT problem with very high probability.

|| Thursday, November 16 || || * David Anderson, * Duke University || || * Biochemical Reaction Systems and External Excitations* ||

There are two different natural contexts in which stochastic dynamics arises in the study of biochemical reaction networks. In the first, the stochastic chemical dynamics arises from the randomness inherent in the formation and breaking of chemical bonds. This is the randomness present in the Gillespie algorithm. If one scales up the volume and number of molecules while keeping the initial concentrations constant, then this intrinsic stochasticity becomes negligible on the scale of concentrations and the dynamical system reduces to a collection of deterministic, coupled differential equations. The second type of stochasticity, which is the focus of this talk, arises naturally in this scaling limit. In this second context, one wants to investigate the response to external excitation of a biochemical system (whose dynamics are well described by differential equations). Here the randomness is a tool used to study the out-of-equilibrium dynamics of the biochemical system and the object of study is a set of differential equations that is forced by a continuous stochastic process in one or a small number of components. In this talk, I will describe how we take two distinct approaches in our study of stochastically forced biochemical systems. In the first, we study how fluctuations propagate through relatively simple systems and study the effect of network topology on the emergent properties of the reaction system. In the second, we apply fluctuations to in silico representations of specific biological networks.

|| Thursday, November 23 || || * No seminar because of* || || * THANKSGIVING* ||

|| Note unusual day. || || Tuesday, November 28 in 901 Van Vleck Hall at 2:25 PM. || || * Brian Rider, * University of Colorado at Boulder || || * The old Riccati map in random Schroedinger and random matrix theory* ||

The classical Riccati substitution maps a solution of a one-dimensional Schroedinger equation into that of a first order non-linear ("Riccati") equation. This simple fact has long been used to study the spectral bulk of Schroedinger operators with random potential. By viewing the Riccati map as a change of measure, we explain some more recent results on the statistics of the spectral edge for an entire family of random Schroedinger operators. In a related direction, we prove that the celebrated Tracy-Widom laws of random matrix theory (describing the scaling limit of the largest eigenvalue in certain Hermitian ensembles) coincide with those of the minimal eigenvalue of a particular Schroedinger operator with white noise potential. Then, using Riccati once more, we provide an entirely new characterization of these laws in terms of the hitting distributions of a one-dimensional diffusion.

|| Thursday, November 30 || || * David Nualart, * University of Kansas || || * Central limit theorem for multiple stochastic integrals and applications* ||

The aim of this talk is to present several equivalent necessary and sufficient conditions for the convergence to a normal distribution of a sequence of multiple stochastic integrals with variance one. One of these conditions is the convergence of the moments of fourth order, and another one is expressed in terms of the Malliavin derivatives of the sequence. We will discuss applications of this result to the convergence of the renormalized self-intersection local time of the fractional Brownian motion, and also to the convergence of the p-variation of stochastic integrals with respect to the fractional Brownian motion.

|| Thursday, December 7 || || * Jan Hannig, * Colorado State University || || * Statistical Model for Tracking with Applications* ||

We propose a new tracking model that allows for birth, death, splitting and merging of targets. Targets are also allowed to go undetected for several frames. The splitting and merging of targets is a novel addition for a statistically based tracking model. This addition is essential for the tracking of storms, which is the motivation for this work. The utility of this tracking method extends well beyond the tracking of storms. It can be valuable in other tracking applications that have splitting or merging, such as vortexes, radar/sonar signals, or groups of people. The method assumes that the location of a target behaves like a Gaussian Process when it is observable. A Markov Chain model decides when the birth, death, splitting, or merging of targets takes place. The tracking estimate is achieved by an algorithm that finds the tracks that maximize the conditional density of the unknown variables given the data. The problem of how to quantify the confidence in a tracking estimate is addressed as well. Finally, some sufficient conditions for consistency of this tracking estimate are presented and an almost sure convergence of the tracking estimate to the true path is proved. The practical suitability of this method is then demonstrated on simulated and real data.

Based on a joint work with Thomas C.M. Lee and Curt B. Storlie.

|| Thursday, December 14 || || * Nathana�l Berestycki, * University of British Columbia || || * Hydrodynamic limits of spatially structured coalescents* ||

We are motivated by a question arising in population genetics, and try to describe the effect of migratory fluxes and spatial structure on the genealogy of a population. This leads to the study of systems of particles performing simple random walk on a given graph, and where particles coalescence according to a certain mechanism (typically, Kingman's coalescent) when they are on the same site. We obtain various asymptotic results for this process, at both small and large time scales, which are of intrerest to population genetics. We will also discuss some related conjectures.