# Past Probability Seminars Spring 2007

## UW Math Probability Seminar Spring 2007

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

Organized by Jason Swanson

### Schedule and Abstracts

|| Thursday, January 25 || || *Marek Biskup, * University of California, Los Angeles || || *Phase coexistence and scaling limit for gradient fields with non-convex interactions* ||

A gradient field is a family of random variables (phi_x) indexed by, say, sites of the hypercubic lattice whose joint law -- with respect to the product Lebesgue measure -- is exponential in the sum of V (phi_y - phi_x) over all nearest neighbor pairs . A prime example is the Gaussian free field where V is quadratic. This is the only case amenable to explicit calculations; it has been a challenge for mathematicians to develop tools that apply in larger generality. One fundamental result of the last decade -- due to Funaki and Spohn -- is that the infinite volume ergodic measures for strictly convex potentials V are completely characterized by the tilt. Another result -- due to Naddaf and Spencer -- states that such measures tend to (continuum) Gaussian free field in a natural scaling limit. I will discuss an example of non-convex V where there are two distinct ergodic measures with non-zero tilt and show that these measures still tend to the (continuum) Gaussian free field in the scaling limit. The latter part will use connections to random walk in random environment. Based on papers written jointly with R. Koteck\'y and H. Spohn, respectively.

|| Thursday, February 1 || || *Jason Swanson, * University of Wisconsin-Madison || || *Stochastic integration with respect to a quartic variation process* ||

Brownian motion (BM) is used to model a wide array of stochastic phenomena in a variety of scientific disciplines. Typically, this is done by using BM as a driving term in a stochastic differential equation (SDE). We are able to define and study these SDEs using Ito's stochastic calculus. Similarly, stochastic partial differential equations (SPDEs) are often used to model stochastic phenomena. In this talk, we consider a very simple example of a stochastic heat equation. The solution to this SPDE, when regarded as a process indexed by time, has a nontrivial 4-variation. It follows that we cannot use the traditional methods of the Ito calculus to define an SDE driven by this process.

In this talk, I will describe work in progress toward constructing a stochastic integral with respect to this process and a corresponding Ito-like change-of-variables formula. The integral being constructed is a limit of discrete Riemann sums. It turns out that the process we are considering has a very close relationship to a certain "flavor" of fractional Brownian motion (FBM). The quest for a calculus for FBM has led researchers in several different directions and there is a large body of literature on the topic. I will discuss some of the connections between our integral and an analogous approach for FBM.

Part of this project is joint work with Chris Burdzy.

|| Thursday, February 8 || || *Xin Qi, * University of Wisconsin-Madison || || *Functional convergence of spatial birth and death processes* ||

In Garcia and Kurtz (2006), Spatial birth and death processes are obtained as solutions of a system of stochastic equations. In this talk, first we extend the multivariate spatial central limit theorem in Penrose(2005) to a general case. Then for any bounded and integrable function f(x) on R^d (with respect to Lebesgue measure), we apply the theorem to the family of processes which is obtained by integrals of f(x) with respect to the centered and scaled spatial birth and death process. We show that this family converges weakly to some Gaussian process. By Mitoma(1983), under an appropriate topology, the centered and scaled spatial birth and death process will converge weakly to a measure-valued process.

|| Thursday, February 15 || || *Zhengxiao Wu, * University of Wisconsin-Madison || || *A Filtering Approach to Abnormal Cluster Identification* ||

A series of events $X_1,X_2,\ldots$ occur at times $\tau_1,\tau_2,\ldots$. Each event is either normal or abnormal. We model the observations as a marked point process with a randomly initiated and growing cluster which represents the abnormal events. Our goal is to compute the conditional probability that an observed event is abnormal in real time. Employing filtering techniques, we derive versions of the Zakai and Kushner-Stratonovich equations in our setting. This framework is applied in earthquake occurence modelling. Such filtering model performs well in declustering.

|| Thursday, February 22 || || ** || || ** ||

|| Thursday, March 1 || || *Hye-Won Kang, * University of Wisconsin-Madison || || *Multiscale method in heat shock model* ||

A reaction network contains multiple reactions and chemical species. The number of molecules can be modelled stochastically using continuous time Markov processes. Since the abundance of the number of each molecule is different, we apply multiscale method. We scale the number of molecules and the reaction rates with the same parameter. By averaging and law of large number, fast and slow components are separated. We can reduce the dimensionality of complex models.

|| Wednesday, March 7 in B239 Van Vleck Hall at 4:00 PM || || *MATH DEPARTMENT COLLOQUIUM* || || *Greg Lawler, * University of Chicago || || *Random walks: simple and self-avoiding* ||

I will present some problems about different types of random walks on integer lattices. Some of these walks are well understood (simple random walk) while others are still wide open problems (self-avoiding walks). Even for simple random walk there are hard, unsolved problems. My talk will emphasize the role of dimension in the study of these problems --- both the dimension of the random walker and the dimension in which the walker lives.

This talk is for a general audience --- no previous knowledge of random walks will be assumed.

|| Thursday, March 8 || || *Greg Lawler, * University of Chicago || || *Partition function view of the Schramm-Loewner evolution* ||

The Schramm-Loewner evolution (SLE) was introduced by Oded Schramm as a probability measure on curves arising as scaling limits of models in statistical physics. I will explain why it is often convenient to extend the definition to be a finite, non-probability measure on domains whose total mass is given by a normalized "partition function". Simple applications of Girsanov's theorem then give the drift in the driving function. I will discuss a number of examples including radial, two-sided radial, and multiple SLEs. I will start by giving a brief introduction to SLE.

|| Friday, March 9 in B239 Van Vleck Hall at 4:00 PM || || *DISTINGUISHED LECTURE SERIES* || || *Greg Lawler, * University of Chicago || || *Conformal invariance and two-dimensional statistical physics* ||

A number of lattice models in two-dimensional statistical physics are conjectured to exhibit conformal invariance in the scaling limit at criticality. In this talk, I will explain what the previous sentence means at least in the case of three examples: simple random walk, self-avoiding walk, and loop-erased random walk. I will describe the limit objects (Schramm-Loewner Evolution (SLE), the Brownian loop soup, and the normalized partition functions) and show how conformal invariance can be used to calculate quantities ("critical exponents") for the models. I will also describe why (in some sense) there is only a one-parameter family of conformally invariant limits. In conformal field theory, this family is parametrized by central charge.

This talk is for a general mathematical audience. No knowledge of statistical physics will be assumed.

|| Thursday, March 15 || || *M�rton Bal�zs, * Budapest University of Technology and Economics || || *Order of current variance in the simple exclusion process* ||

I will talk about _t_^2/3^-order scaling of the current variance in the simple exclusion process. After a brief introduction I will try to explain some purely probabilistic ideas that lead to this exotic scaling. The talk can be considered as a partial continuation of [probsem-F06.html Timo's September 28 talk], but is intended to be understandable by itself.

|| Thursday, March 22 || || *David Wilson, * Microsoft Research || || *Boundary Partitions in Trees and Dimers* ||

We study groves on planar graphs, which are forests in which every tree contains one or more of a special set of vertices on the outer face, referred to as nodes. Each grove partitions the set of nodes. When a random grove is selected, we show how to compute the various partition probabilities as functions of the electrical properties of the graph when viewed as a resistor network. We prove that for any partition sigma, Pr[grove has type sigma] / Pr[grove is a tree] is a dyadic-coefficient polynomial in the pairwise resistances between the nodes, and Pr[grove has type sigma] / Pr[grove has maximal number of trees] is an integer-coefficient polynomial in the entries of the Dirichlet-to-Neumann matrix. We give analogous integer-coefficient polynomial formulas for the pairings of chains in the double-dimer model. These polynomials include Pfaffian formulas for tripartite pairings for groves and double dimer configurations, which generalize the determinant formulas given by Curtis, Ingerman, and Morrow, and by Fomin, for bipartite pairings. We show that the distribution of pairings of contour lines in the Gaussian free field with certain natural boundary conditions is identical to the distribution of pairings in the scaling limit of the double-dimer model. These partition probabilities are relevant to multichordal SLE_2, SLE_4, and SLE_8. Joint work with Richard Kenyon.

|| Thursday, March 29 || || *Jim Kuelbs, * University of Wisconsin-Madison || || *Path Properties of Multigenerational Samples from Branching Processes* ||

|| Thursday, April 5 || || *No seminar because of* || || *SPRING BREAK* ||

|| Thursday, April 12 || || *Muruhan Rathinam, * University of Maryland, Baltimore County || || *Tau-leaping methods in stochastic chemical kinetics* ||

Intracellular gene regulatory mechanisms involve small numbers of large molecules and are essentially discrete and stochastic in nature. An appropriate dynamic model involves a discrete state (lattice of non-negative integer vectors) and continuous time Markov process. Exact Montecarlo simulation of sample paths of such processes, though simple, is often prohibitively expensive for systems with several molecular species and several reaction channels. In this talk we describe an ongoing project which aims to develop methods and rigorous mathematical theory for simulating these systems in an efficient way by leaping over several events at a time. We address certain important issues such as stiffness (i.e. the existence of vastly different time scales) as well as the preservation of the nonnegative integer values of the states. More broadly these schemes would be applicable to other situations such as the dynamics of market microstructures and traffic models where the detailed models are discrete and stochastic.

|| Friday, April 13 -- Sunday, April 15 || || *GRADUATE STUDENT CONFERENCE IN PROBABILITY* || || http://www.math.wisc.edu/~hkang/student_probability_conference.html || || ||

|| Monday, April 16 in 901 Van Vleck Hall at 2:25 PM || || *JOINT LIE THEORY/COMBINATORICS/PROBABILITY SEMINAR* || || *Lauren Williams, * Harvard University || || *From total positivity on the Grassmanian to the asymmetric exclusion process* ||

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. Additionally, particles may enter from the left with probability alpha and exit from the right with probability beta. In this paper we prove a close connection between the PASEP model and the combinatorics of permutation tableaux. (These tableaux come indirectly from the totally nonnegative part of the Grassmannian, via work of Postnikov, and were studied in a paper of Steingrimsson and the second author.) Namely, we prove that in the long time limit, the probability that the PASEP model is in a particular configuration tau is essentially the generating function for permutation tableaux of shape lambda(tau) enumerated according to three statistics. The proof of this result uses a result of Derrida et al on the matrix ansatz for the PASEP.

|| Thursday, April 19 || || *George Yin, * Wayne State University || || *Switching diffusion processes* ||

In this talk, we report some of our recent work on switching diffusion processes in which continuous dynamics and discrete events coexist. We focus on asymptotic properties of such processes. First, motivational examples arising from finance, singular perturbed Markovian systems, and manufacturing will be mentioned. Then we recall the notion of recurrence and regularity. After necessary and sufficient conditions for recurrence are provided, ergodicity will be examined, and stability will be studied.

|| Thursday, April 26 || || *Zhen-Qing Chen, * University of Washington || || *Discrete Approximations to Reflected Brownian Motion* ||

In this talk, I will present three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains $D$ in $R^n$ that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on $D\cap 2^{-k} Z^n$ with stationary initial distribution converge weakly in the space of $D([0, \infty), \overline D)$, equipped with the Skorokhod topology, to the law of the stationary reflected Brownian motion on $D$. We further show that the following myopic conditioning algorithm generates, in the limit, a reflected Brownian motion on any bounded domain $D$. For every integer $k\geq 1$, let $\{X^k_{j2^{-k}}, j=0, 1, 2, \dots \}$ be a discrete time Markov chain with one-step transition probabilities being the same as those for the Brownian motion in $D$ conditioned not to exit $D$ before time $2^{-k}$. We prove that the laws of $X^k$ converge to that of the reflected Brownian motion on $D$.

Joint work with Krzysztof Burdzy.

|| Thursday, May 3 || || ** || || ** ||

|| Thursday, May 10 || || *C�cile An�, * University of Wisconsin-Madison || || *Convergent but inconsistent intercept estimator from tree-structured data* ||

Linear regression is widely used on data collected from species. However, these data are not iid. Their correlation structure is related to the species genealogical tree. The most widely used model is that of Brownian motion along the tree. I will show that the asymptotic behavior of the intercept and group effect estimators is non-typical in these models.