We consider the stochastic heat equation with a multiplicative space-time white noise forcing term under standard "intermitency conditions.” The main byproduct of this talk is that, under mild regularity hypotheses, the a.s.-boundedness of the solution$x\mapsto u(t\,,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\Lambda:=\lim_{|x|\to\infty} \vert\log u_0(x)\vert/(\log|x|)^{2/3}$.

In this talk, we consider kinetic equations containing random

terms. The kinetic models contain a small parameter and it is well

known that, after scaling, when this parameter goes to zero the limit

problem is a diffusion equation in the PDE sense, ie a parabolic equation

of second order. A smooth noise is added, accounting for external perturbation.

It scales also with the small parameter. It is expected that the limit

equation is then a stochastic parabolic equation where the noise is in

Stratonovitch form.

Our aim is to justify in this way several SPDEs commonly used.

We first treat linear equations with multiplicative noise. Then show how

to extend the methods to nonlinear equations or to the more physical

case of a random forcing term.

The results have been obtained jointly with S. De Moor and J. Vovelle.

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.