Probability SEMINAR


Probability Seminar

Fall 2013-2014

Time: Thursday 3-4pm

Room: MATH 350

October 17           
Sonin (UNC Charlotte) : 
Two Results in Probability Theory


October 24           
S. Molchanov (UNC Charlotte) : 
Intermittency in the population dynamics
ABSTR: Mathematical models in the population dynamics are dealing with the special class of the infinite dimensional Markov processes. Their phase space is a set of all locally finite configurations of the points (particles) in. Dynamics of such configurations includes the birth and death of particles and their random motion (diffusion, jumps etc.). The goal of the theory is to give the conditions of the ergodicity of such systems and explain several well-known empirical facts, for instance, the high level of non-uniformity in the spatial distribution of the particles (“patches”).

The talk will contain a review of the recent results in this area and the open problems.

November 7        
R. Schinazi (CU Colorado Springs) :
Is there an error threshold in virus replication?

ABSTR: Theoretical biology predicts that there
is a critical mutation probability above which a viral
population will go extinct. Above this threshold the virus
loses the ability to replicate the best adapted genotype,
leading to a population composed of low replicating mutants
that is eventually doomed. We propose a new branching model
that shows that this is not necessarily so.  A population
composed of ever changing mutants may survive.

November 14      2-3 pm (an hr earlier!)
T. Salisbury (York University, Canada) :  
Conditioned super Brownian motion

Super Brownian motion is an example of a diffusion in the space of measures, and also a model of a population that branches while migrating spatially. I will describe recent work with Deniz Sezer, about conditioning this process, using Dynkin's notions of exit measures and X-harmonic functions. Our results resemble certain problem in evolutionary biology, in that they involve reconstructing potential genealogies based on partial information about a population. 
-----------------  2014 --------------------

February 6           
L. Gilch (Graz, Austria) : 

Asymptotic Drift and Entropy of Random Walks on Groups and Regular Languages

In this talk I will present my recent results about asymptotic drift and entropy
and its properties of random walks on regular languages over a finite
alphabet. In particular, this setting applies to the case of random
walks on virtually free groups. Existence of the asymptotic entropy is
shown and formulas for it are presented. Moreover, I will show that the
entropy is the drift with respect to the Greenian metric and
varies analytically in terms of probability measures of constant support.


April 10
M. Freidlin (University of Maryland)

Long term influence of small term perturbations
I will consider small deterministic and stochastic perturbations of dynamical systems and stochastic processes. This is the natural generality of the problem since pure deterministic perturbations of deterministic systems can lead to a stochastic long time behavior. If the non-perturbed system has a unique normalized invariant measure the perturbed system (or process) can have many invariant measures, but these measures, under some small assumptions, will be close in the weak topology to the invariant measure of the original system. The long time behavior of the perturbed system in this case will be, in a sense, close to the behavior of the non-perturbed system. But if the original system has many stationary measures, the perturbations, in an appropriate time scale, will induce a motion on the cone of invariant measures. This motion determines the long time behavior of the perturbed system. Each cone is a convex envelope of its extreme points. A parameterization of the set of extreme points allows to describe the motion on the cone using the large deviation theory or a modified averaging principle. I will demonstrate how such an approach works for systems with a finite number of extreme invariant measures. I will consider the case of Hamiltonian systems, the Landau-Lifshitz equation, PDE problems  related to small perturbations of diffusion processes, and perturbations of incompressible 3D-flow with a conservation law .  In these cases the extreme points can be parameterized by points of a graph or of an “open book” space.

April 17
Valko (U. Wisconsin, Madison):

Large deviations for the Sine_beta process

The Sine_beta process is the bulk point process limit of the Gaussian beta-ensemble. For beta=2 it is a determinantal point process conjectured to behave similarly to the rescaled critical Riemann zeta zeroes. The process is translation invariant with an asymptotic constant density. We prove a large deviation principle for the average density of the process: i.e. we identify the asymptotic  probability of seeing an unusual average density in a large interval. Our approach is based on the representation of the counting function of the process using stochastic differential equations. Our techniques work for the full range of parameter values. The results are novel even in the classical beta = 1,2 and 4 cases. They are consistent with the existing rigorous results on large gap probabilities and confirm the (non-rigorous) predictions  made using log-gas arguments. 

(Joint with Diane Holcomb.)

April 24
K. Burdzy (U. Washington, Seattle): 

On the meteor process

The meteor process is a model of mass
redistribution on a graph. I will present results
on existence of the process and existence, uniqueness
and properties of the stationary distribution. I will also 
discuss special questions arising in the
case when the graph is a cycle or the set of integers.

Joint work with Sara Billey, Soumik Pal and Bruce Sagan.

C. Mueller (U. Rochester): 

Hitting questions and multiple points for stochastic PDE (SPDE) in the critical case