# Past Probability Seminars Fall 2013

## Thursday, September 12, Tom Kurtz, UW-Madison

Title: Particle representations for SPDEs with boundary conditions

Abstract: Stochastic partial differential equations frequently arise as limits of finite systems of weighted interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations for the particle locations and weights. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness and convergence results. Following some discussion of general approaches to SPDEs, the talk will focus on situations where the particle locations are given by an iid family of diffusion processes, and the weights are chosen to obtain a nonlinear driving term and to match given boundary conditions for the SPDE. (Recent results are joint work with Dan Crisan.)

## Thursday, September 26, David F. Anderson, UW-Madison

Title: Stochastic analysis of biochemical reaction networks with absolute concentration robustness

Abstract: It has recently been shown that structural conditions on the reaction network, rather than a 'fine-tuning' of system parameters, often suffice to impart "absolute concentration robustness" on a wide class of biologically relevant, deterministically modeled mass-action systems [Shinar and Feinberg, Science, 2010]. Many biochemical networks, however, operate on a scale insufficient to justify the assumptions of the deterministic mass-action model, which raises the question of whether the long-term dynamics of the systems are being accurately captured when the deterministic model predicts stability. I will discuss recent results that show that fundamentally different conclusions about the long-term behavior of such systems are reached if the systems are instead modeled with stochastic dynamics and a discrete state space. Specifically we characterize a large class of models which exhibit convergence to a positive robust equilibrium in the deterministic setting, whereas trajectories of the corresponding stochastic models are necessarily absorbed by a set of states that reside on the boundary of the state space. The results are proved with a combination of methods from stochastic processes and chemical reaction network theory.

## Thursday, October 3, Lam Si Tung Ho, UW-Madison

Title: Asymptotic theory of Ornstein-Uhlenbeck tree models

Abstract: Tree models arise in evolutionary biology when sampling biological species, which are related to each other according to a phylogenetic tree. When observations are modeled using an Ornstein-Uhlenbeck (OU) process along the tree, the autocorrelation between two tips decreases exponentially with their tree distance. Under these models, tips represent biological species and the OU process parameters represent the strength and direction of natural selection. For the mean, we show that if the heights of the trees are bounded, then it is not microergodic: no estimator can ever be consistent for this parameter. On the other hand, if the heights of the trees converge to infinity, then the MLE of the mean is consistent and we establish a phase transition on the behavior of its variance. For covariance parameters, we give a general sufficient condition ensuring microergodicity. We also provide a $\sqrt{n}$-consistent estimators for them under some mild conditions.

## Tuesday, October 15, 4pm, Van Vleck B239, Distinguished Lecture Series in Mathematics: Alexei Borodin, MIT

Please note the non-standard time and day.

Title: Integrable Probability I

Abstract: The goal of the talks is to describe the emerging field of integrable probability, whose goal is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes.

## Wednesday October 16, 2:30pm, Van Vleck B139, Distinguished Lecture Series in Mathematics: Alexei Borodin, MIT

Please note the non-standard time and day.

Title: Integrable Probability II

Abstract: The goal of the talks is to describe the emerging field of integrable probability, whose goal is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes.

## Tuesday, October 22, 4pm, Van Vleck 901, Anton Wakolbinger, Goethe Universität Frankfurt

Please note the non-standard time and day, and the recently revised time and room.

Title: The time to fixation of a strongly beneficial mutant in a structured population

Abstract: We discuss a system that describes the evolution of the vector of relative frequencies of a beneficial allele in d colonies, starting in (0,...,0) and ending in (1,...,1). Its diffusion part consists of Wright-Fisher noises in all the components that model the random reproduction, its drift part consists of a linear interaction term coming from the gene flow between the colonies, together with a logistic growth term due to the selective advantage of the allele, and a term which makes the entrance from (0,...,0) possible. It turns out that there are d extremal ones among the solutions of the system, each of them corresponding to one colony in which the beneficial mutant originally appears. We then focus on the fixation time in the limit of a large selection coefficient, and explain how its asymptotic distribution can be analysed in terms of the so called ancestral selection graph. This is joint work with Andreas Greven, Peter Pfaffelhuber and Cornelia Pokalyuk.

## Thursday, October 24, Ke Wang, IMA

Title: Random weighted projections, random quadratic forms and random eigenvectors

Abstract: We start with a simple, yet useful, concentration inequality concerning random weighted projections in high dimensional spaces. The inequality is used to prove a general concentration inequality for random quadratic forms. In another application, we show the infinity norm of most unit eigenvectors of Hermitian random matrices with bounded entries is $O(\sqrt{\log n/n})$. This is joint work with Van Vu.

## Thursday, November 14, Miklos Z. Racz, UC Berkeley

Title: Multidimensional sticky Brownian motions as limits of exclusion processes

Abstract: I will talk about exclusion processes in one dimension where particles interact in a sticky fashion. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time and the entire particle system is slowed down until the "collision" is resolved. We show that under diffusive scaling of space and time such processes converge to what one might refer to as a sticky reflected Brownian motion in the wedge. The latter behaves as a Brownian motion with constant drift vector and diffusion matrix in the interior of the wedge, and reflects at the boundary of the wedge after spending an instant of time there. In particular, this leads to a natural multidimensional generalization of sticky Brownian motion on the half-line, which is of interest in both queueing theory and stochastic portfolio theory. For instance, this can model a market, which experiences a slowdown due to a major event (such as a court trial between some of the largest firms in the market) deciding about the new market leader. This is joint work with Mykhaylo Shkolnikov.

## Thursday, November 21, Amarjit Budhiraja, UNC-Chapel Hill

Title: Infinity Laplacian and Stochastic Differential Games

Abstract: A two-player zero-sum stochastic differential game(SDG), motivated by a discrete time random turn game of Peres, Schramm,Sheffield and Wilson(2006) known as the Tug of War, is introduced. The SDG is defined in terms of an m-dimensional state process that is driven by a one-dimensional Brownian motion, played until the state exits the domain. The players controls enter in a diffusion coefficient and in an unbounded drift coefficient of the state process. We show that the game has value, and characterize the value function as the unique viscosity solution of the inhomogeneous infinity Laplace equation introduced in Peres et al. A similar SDG is conjectured for the motion by curvature equation in the plane. Joint work with R. Atar.

## Thursday, December 5, Scott Hottovy, UW-Madison

Title: Small mass approximation for Brownian particles with applications to physical systems

Abstract:

In this talk a class of stochastic differential equations, from Newton's second law, is considered in the limit as mass tends to zero, called the Smoluchowski-Kramers limit. The Smoluchowski-Kramers approximation is useful in simplifying the dynamics of a system. For example, the problems of calculating of rates of chemical reactions, describing dynamics of complex systems with noise, and measuring ultra small forces, are simplified using the Smoluchowski-Kramers approximation. In this talk, I will give a limit theorem for the small mass limit for a multi-dimensional system with arbitrary state-dependent friction and noise coefficients, and show how it is applied to three different physical systems of interest: measuring micro forces of a Brownian particle in a diffusion gradient, movement of particles in a temperature gradient, and a noisey circuit. The main result is proved using a theory of convergence of stochastic integrals developed by Kurtz and Protter.

## Thursday, December 12, Nikos Zygouras, University of Warwick

Title: Polynomial Chaos and scaling limits of disordered systems

Abstract: We will formulate general conditions ensuring that a sequence of multi-linear polynomials of independent random variables converges to a limiting random variable, given by an explicit Wiener chaos expansion over the d-dimensional white noise. A key ingredient in our approach is a Lindeberg principle for polynomial chaos expansions, which extends earlier work of Mossel, O'Donnell and Oleszkiewicz. These results provide a unified framework to study the continuum and weak disorder scaling limits of statistical mechanics systems that are disorder relevant, including the disordered pinning model, the long-range directed polymer model in dimension 1+1, (generalizing the work of Alberts, Khanin, Quastel), and the two-dimensional random field Ising model.

Joint work with F. Caravenna and R.F. Sun

## Tuesday, December 17, 4pm Van Vleck B239 Perla Sousi, University of Cambridge

Department Colloquium. Note the room and time are not the same as the weekly probability seminar.

Title: The effect of drift on the volume of the Wiener sausage

Abstract: The Wiener sausage at time t is the algebraic sum of a Brownian path on [0,t] and a ball. Does the expected volume of the Wiener sausage increase when we add drift? How do you compare the expected volume of the usual Wiener sausage to one defined as the algebraic sum of the Brownian path and a square (in 2D) or a cube (in higher dimensions)? We will answer these questions using their relation to the detection problem for Poisson Brownian motions, and rearrangement inequalities on the sphere (with Y. Peres). We will also discuss generalizations of this to Levy processes (with A. Drewitz and R. Sun) as well as an adversarial detection problem and its connections to Kakeya sets (with Babichenko, Peres, Peretz and Winkler).