# Probability Seminar

## Fall 2013

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

**
Visit this page to sign up for the email list to receive seminar announcements.**

## 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 [math]\sqrt{n}[/math]-consistent estimators for them under some mild conditions.

## Thursday, October 10, NO SEMINAR

Midwest Probability Colloquium

## Tuesday, October 15, 4pm, VV B239, Distinguished Lecture Series in Mathematics: A. Borodin

## Wednesday October 16, 2:30pm, VVB139, Distinguished Lecture Series in Mathematics: A. Borodin

Please note the non-standard time and day.

Title: Integrable Probability I-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 , Anton Wakolbinger, Goethe Universität Frankfurt

Please note the non-standard time and day.

Title: TBA

Abstract:

## Thursday, October 24, Ke Wang, IMA

Title: TBA

Abstract:

## Thursday, October 31, TBA

Title: TBA

Abstract:

## Thursday, November 7, TBA

Title: TBA

Abstract:

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

Title: TBA

Abstract:

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

Title: TBA

Abstract:

## Thursday, November 28, NO SEMINAR