Dissipation and high disorder

Speaker: 

Michael Cranston

Institution: 

UCI

Time: 

Tuesday, November 18, 2014 - 11:00am to 12:00pm

Location: 

306 RH

We show that the total mass, i.e. the sum over all points in the d-dimensional integer lattice of the solution to the parabolic Anderson model with initial function the point mass at the origin goes to zero in the high disorder regime. This talk is basedon joint work with L. Chen, D. Khoshnevisan, and K. Kim.

Heavy tails and one-dimensional localization.

Speaker: 

Michael Cranston

Institution: 

UCI

Time: 

Tuesday, October 21, 2014 - 11:00am to 12:00pm

Location: 

RH 306

In this talk we address a question posed several years ago by G. Zaslovski: what is the effect of heavy tails of one-dimensional random potentials on the standard objects of localization theory: Lyapunov exponents, density of states, statistics of eigenvalues, etc. Professor G. Zaslovski always expressed a special interest in the models of chaos containing strong fluctuations, e.g. L ́evy flights? We’ll consider several models of potentials constructed by the use of iid random variables which belong to the domain of attraction of the stable distribution with parameter α < 1. This is a report on joint work with S. Molchanov.

Conditional Speed of Branching Brownian Motion, Skeleton Decomposition and Application to Random Obstacles

Speaker: 

Janos Englender

Institution: 

University of Colorado

Time: 

Tuesday, October 7, 2014 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

We study a d-dimensional branching Brownian motion, among obstacles scattered

according to a Poisson random measure with a radially decaying intensity. Obstacles

are balls with constant radius and each one works as a trap for the whole motion when

hit by a particle. Considering a general offspring distribution, we derive the decay

rate of the annealed probability that none of the particles hits a trap,

asymptotically, in time. 

 

This proves to be a rich problem, motivating the proof of a general result about the speed 

of branching Brownian motion conditioned on

non-extinction. We provide an appropriate `skeleton-decomposition' for the underlying

Galton-Watson process when supercritical, and show that the `doomed particles' do 

not contribute to the asymptotic decay rate.

 

This is joint work with Mine Caglar and Mehmet Oz.

Asymmetric Cauchy distribution and the destruction of large random recursive trees

Speaker: 

Jean Bertoin

Institution: 

Univ. Zurich

Time: 

Thursday, June 19, 2014 - 2:00pm to 3:00pm

Host: 

Location: 

RH 306

 The probability mass function $1/j(j+1)$ for $j\geq 1$ belongs to the domain of attraction of a completely asymmetric Cauchy distribution.

The purpose of the talk is to review some of applications of this simple observation to limit theorems related to the destruction of random recursive trees.

 

Specifically, a random recursive tree of size $n+1$ is a tree chosen uniformly at random amongst the $n!$ trees on the set of vertices $\{0,1, 2, ...,  n\}$ such that the sequence of vertices along any segment starting from the root $0$ increases. One destroys this tree by removing its edges one after the other in a uniform random order. It was first observed by Iksanov and M\"ohle that the central limit theorems for the random walk with step distribution given above explains the fluctuations of the number of cuts needed to isolate the root. We shall discuss further results in the same vein.

Viscosity Solutions for Forward SPDEs and PPDEs

Speaker: 

Jin Ma

Institution: 

USC

Time: 

Tuesday, May 20, 2014 - 11:00am to 12:00pm

Host: 

Location: 

RH 306

In this talk we introduce a notion of stochastic viscosity solution

for a class of fully nonlinear SPDEs and the corresponding Path-dependent

PDEs (PPDEs). The definition is based on our new accompanying work

on the pathwise stochastic Taylor expansion, using a variation of the path-

derivatives initiated by Dupire. As a consequence this new definition of the

viscosity solution is directly in the pathwise sense, without having to invoke

the stochastic characteristics for the localization. The issues of consistency,

stability, comparison principles, and ultimately the well-posedness of the

stochastic viscosity solutions will be discussed under this new framework.

This is a joint work with Rainer Buckdahn and Jianfeng Zhang.

A spatial evolutionary model of field cancerization

Speaker: 

Jasmine Foo

Institution: 

University of Minnesota

Time: 

Monday, May 5, 2014 - 10:00am to 11:00am

Location: 

340M

Cancer often arises through a sequence of genetic alterations.  Each of these alterations may confer a fitness advantage to the cell, resulting in a clonal expansion.  To model this process we consider a generalization of the biased voter process on a lattice which incorporates successive mutations modulating individual fitness.  We will study the rate of mutant spread and accumulation of oncogenic mutations in this process.  We then investigate the geometry and extent of premalignant fields surrounding primary tumors, and evaluate how the risk of secondary tumors arising from these fields may depend on the cancer progression pathway and tissue type.  (joint work w/K. Leder, R. Durrett, and M. Ryser).

Some Large-Scale Fractals

Speaker: 

: Davar Khoshnevisan

Institution: 

University of Utah

Time: 

Tuesday, May 6, 2014 - 11:00am to 12:00pm

Location: 

RH 306

We present a case study of the large-scale “fractal” behavior of concrete families of random processes that arise in complex systems. Among other things we will exhibit two random functions both of which are “multifractal” on large scales, but only one of which shows “intermittency.” This contradicts  the commonly-held view that “multifractality” and “intermittency” can be used interchangeably. This is based on joint work with K. Kim and Y. Xiao.

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