# Past Probability Seminars Fall 2005

## UW Math Probability Seminar Fall 2005

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

Organized by Jason Swanson

### Schedule and Abstracts

|| Thursday, September 8 || || * Mike Reed,* Duke University || || * Cell Metabolism, Mathematics, and Public Health* ||

Folate and methionine metabolism, a small part of cell biochemistry, is crucial for cell replication and DNA methylation. There is mounting evidence that the mechanisms by which some gene polymorphisms or dietary deficiencies are statistically linked to heart disease and certain cancers involve disruptions of folate and methionine metabolism. Folate is also the target of several chemotheraputic agents and some antibiotics target folate metabolism in bacteria. A collaborative mathematical modeling project (with Cornelia Ulrich of the Fred Hutchinson Cancer Research Institute and Fred Nijhout of the Duke Department of Biology) has the goal of understanding the quantitative and qualitative emergent properties of the whole biochemical network. Analysis of the propagation of stochastic fluctuations through bio-chemical networks has proven to be a useful tool for understanding the biology and a source of new, interesting mathematical problems. Published and current work will be described as well as the difficulties involved. Several public health issues will also be discussed.

|| Thursday, September 15 || || * Krzysztof Burdzy,* University of Washington || || * On the Robin problem in fractal domains* ||

The Robin boundary conditions are the mathematical model of a semi-porous membrane. The model applies to man-made objects such as electrodes in car batteries and natural objects such as lungs. I will state a mathematical problem that is inspired by a practical and scientific question of which objects have "efficient" membranes. The problem is solved for a class of fractal domains. The solution of this analytic problem involves stochastic processes. The talk will be accessible to non-specialists and graduate students. Joint work with Rich Bass and Zhenqing Chen.

|| Thursday, September 22 || || * Panki Kim,* University of Illinois at Urbana-Champaign || || * Two-sided estimates on the density of (killed) Brownian Motion with Singular Drift.* ||

In this talk, we discuss two-sided estimates on the density of (killed) Brownian motion with singular drift (signed measure drift). The existence and uniqueness of this process was established by Bass and Chen recently. The potential theory for this process, for example, parabolic Harnack inequality, boundary Harnack inequality, Green function estimate will be discussed too. This is a joint work with R. Song.

|| Thursday, September 29 || || * Antal Járai,* Carleton University || || * A self-organized critical forest fire model (joint work with J. van den Berg)* ||

I will discuss a simple model that was proposed in the physics literature as an attempt to explain the apparent power law distribution of the sizes of large forest fires. Although various simulation studies have been performed, hardly anything is known about the model rigorously. We studied the one-dimensional case, which already shows interesting behaviour. In the introduction I will review related dynamical percolation-type models, such as Aldous' frozen percolation, and present open questions.

|| Thursday, October 6 || || * Christian Beneš, * Tufts University || || * Some Properties of the Complement of Planar Random Walk* ||

Consider simple random walk S in the plane and the continuous curve obtained from it by linear interpolation between integer times. If one defines the set of "holes" to be the set of connected components of C \ S[0,2n] and the set of "lattice holes" of S to be the set of connected components of Z^2 \ {S_j}_{0<= j<= 2n}, one can assign to each hole an "area", its Lebesgue measure, and to each lattice hole a "lattice area", its cardinality. In this talk, we will show that the number of holes (resp. lattice holes) of area (resp. lattice area) greater than A(n)*n is, up to a logarithmic correction term, asymptotic to A(n)^{-1}, if n^{d_0} < A(n) < f(n), where f(n) goes to 0 more slowly than any power function and d_0 > 0. This confirms an observation by Mandelbrot. A consequence is that the largest hole has an area which is logarithmically asymptotic to n. We will also mention the different and mysterious exponent of 5/3 observed by Mandelbrot for "small" lattice holes.

|| Note unusual date. || || Tuesday, October 11 in 901 Van Vleck Hall at 2:25 PM || || * Ruth Williams,* University of California, San Diego || || * Fluid limit of a network with fair bandwidth sharing and general document size distributions* ||

We consider a model of Internet congestion control (introduced by Roberts and Massoulie) that represents the randomly varying number of flows present in a network where bandwidth is shared fairly between document transfers.� In contrast to a prior work by Kelly and Williams, the present work allows interarrival times and document sizes to be generally distributed, rather than exponentially distributed.� To describe the evolution of the system, measure valued processes are used to keep track of the residual document sizes for all flows in the network. We propose a fluid model as a first order approximation for the system.� Under mild conditions we show that with law of large numbers scaling, the measure valued processes corresponding to a sequence of network models (with fixed network structure) are tight, and that any weak limit point of the sequence is almost surely a fluid model solution.� We also identify the invariant states for the fluid model.

|| Thursday, October 13 || || * David White,* Cornell University || || * Processes with inert drift* ||

A process with inert drift is a reflected diffusion process whose drift term changes only on the boundary of the process's domain. In the simplest form, it will be a pair (X,L) satisfying dX_t = dB_t + L_t + K L_t dt, where L(t) is the local time of X_t at zero, and X_t >= 0 for all t. I will discuss the original physical motivation for the process, construction of solutions, and some interesting behavior that arises from some simple versions of the process.

|| Thursday, October 20 || || * No seminar because of* || || * MIDWEST PROBABILITY COLLOQUIUM* ||

|| Thursday, October 27 || || * Sunder Sethuraman,* Iowa State University || || * Diffusivity of a tagged particle in 2D asymmetric simple exclusion* ||

The simple exclusion particle system follows the motion of a collection of dependent random walks on Z^d which interact by suppressing jumps to already occupied vertices. In this talk, we give recent results on diffusive variances for a tagged particle under equilibrium in simple exclusion with asymmetric jump probabilities on Z^2. We will also review the larger problem and point out some open questions.

|| Thursday, November 3 || || * Benedek Valkó,* University of Toronto || || * Perturbation of equilibria for interacting particle systems* ||

We consider a large family of interacting particle systems with conserved quantities. On the proper time and space scale the evolution of these quantities may be described by a system of partial differential equations, called hydrodynamic equations. We investigate the propagation of small order perturbations of equilibrium states in the one and two-component case, we show that under appropriate scaling we get a universal hydrodynamic limit for the evolution of perturbations. For one-component systems the result is the well-known Burgers equation. In the two-component case the result depends on the choice of the equilibrium state. Perturbations around a hyperbolic equilibrium point will result two decoupled Burgers-type equations. If the equilibrium point is singular then the limit is a member of a one-parameter family of two-by-two pde systems. (These are essentially perturbations of the constant solutions of the respective hydrodynamic equations.) Most of the presented results are joint work with Bálint Tóth.

|| Thursday, November 10 || || * Arnaud Doucet,* University of British Columbia || || * Sequential Monte Carlo Samplers* ||

We present a general methodology to sample from a sequence of probability distributions defined on a common space; i.e. in cases where one traditionally uses Markov Chain Monte Carlo. We propose to approximate these distributions by a large set of random samples which evolve over time using simple sampling and resampling mechanisms. This methodology yields a whole set of principled algorithms to make parallel Markov chain Monte Carlo runs interact. It also allows us to derive new algorithms to estimate ratio of normalizing constants, to perform global optimization or to solve sequential Bayesian estimation problems. Several convergence results and an extension to continuous time will be discussed.

|| Thursday, November 17 || || * Guillaume Bonnet,* University of California, Santa Barbara || || * The long time behavior of a stochastic Lotka-Voltera system with jumps* ||

The effect of environmental fluctuations on populations dynamics has been intensively investigated in theoretical ecology and applied probability. Recently, there has been great interests in the biological community to understand the effect of exceptionally large fluctuations in the environment (e.g drought) in species survival. I will briefly illustrate those effects with some ecological data. The main purpose of this talk is to present a stochastic Lotka-Voltera system with jumps as a model for that ecological context. I will first present some old and new results on the stationary distribution for the continuous diffusion case, some going back to Kesten and Ogura (1981). I will then turn to the model with jumps and show similar results. These results can be interpreted as conditions for species coexistence in the population dynamics context. If time permit, I will briefly mention some ongoing work in the spatial setting.

|| Thursday, November 24 || || * No seminar because of* || || * THANKSGIVING* ||

|| Thursday, December 1 || || * James Hegeman,* University of Wisconsin-Madison || || * Young Tableaux and last-passage growth models* ||

This will be an overview of Young Tableaux and the connection to an exclusion process via a growth model. I will give basic definitions and discuss the Robinson-Schensted-Knuth (RSK) correspondence and a formula for last-passage growth models.

|| Thursday, December 8 || || * Richard Sowers,* University of Illinois at Urbana-Champaign || || * Random Ducks* ||

We discuss the effect of noise on a classical dynamical system exhibiting canard-like behavior.� We identify the correct scale for noise and its influence on canards.� Along the way, we understand the importance of Fenichel-type manifolds for stochastic systems. We note that the nonanticipative demands of stochastic calculus require a slightly different perspective than the classical canard-like calculations.

|| Thursday, December 15 || || ** || || ** ||