I will discuss aspects of a polymer model based on three-dimensional Brownian motion conditioned to hit (and keep returning to) the origin 

introduced by Mike Cranson and co-authors.   The construction and certain properties of this conditioned Brownian motion will be approached from two points of view (i) Dirichlet forms, and (ii) excursion theory. The latter gives a nice interpretation of the Johnson-Helms example from martingale theory. It turns out that this diffusion process is not a semimartingale, even though its radial part is just a one-dimensional Brownian motion reflected at the origin. 

(This talk is based on joint work with Liping Li of Fudan University.)

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.

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.

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.

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.