Abstract

A classical result of H.J Brascamp and E.H. Lieb says that the ground state eigenfunction for the Laplacian in convex regions (and of Schr ̈odinger operators with convex potentials on Rn) is log-concave. A proof can be given (interpreted) in terms of the finite dimensional distributions of Brownian motion. Some years ago the speaker raised similar questions (and made some con- jectures) when the Brownian motion is replaced by other stochastic processes and in particular those with transition probabilities given by the heat kernel of the fractional Laplacian–the rota- tionally symmetric stable processes. These problems (for the most part) remain open even for the unit interval in one dimension. In this talk we elaborate on this topic and outline a proof of a result of M. Kaßmann and L. Silvestre concerning superharmonicity of eigenfunctions for certain fractional powers of the Laplacian. Our proof is joint work with D. DeBlassie. 

Abstract: Limit shapes are an increasingly popular way to understand
the large—scale characteristics of a random ensemble.  The limit shape
of unrestricted integer partitions has been studied by many authors
primarily under the uniform measure and Plancherel measure.  In
addition, asymptotic properties of integer partitions subject to
restrictions has also been studied, but mostly with respect to
\emph{independent} conditions of the form ``parts of size $i$ can
occur at most $a_i$ times.”  While there has been some progress on
asymptotic properties of integer partitions under other types of
restrictions, the techniques are mostly ad hoc.  In this talk, we will
present an approach to finding limit shapes of restricted integer
partitions which extends the types of restrictions currently
available, using a class of asymptotically stable bijections.  This is
joint work with Igor Pak.

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.