School of Mathematics colloquia

An occasional series of talks on topics of wide interest.

Michael Roeckner (Universität Bielefeld), A new view of Fokker-Planck equations in finite and Infinite dimensional spaces

Friday 11th June 2010 at 15:00 in W 003

Fokker-Planck and Kolmogorov (backward) equations can be interpreted as linearizations of the underlying stochastic differential equations (SDE). It turns out that, in particular, on infinite dimensional spaces (i.e. for example if the SDE is a stochastic partial differential equation (SPDE) of evolutionary type), the Fokker-Planck equation is much better to analyze than the Kolmogorov (backward) equation. The reason is that the Fokker-Planck equation is a PDE for measures. Hence e.g. existence of solutions via compactness arguments is easier to show than for PDE on functions. On the other hand uniqueness appears to be much harder to prove.

In this talk we first give a quite elaborate introduction into the relations between S(P)DE, Fokker-Planck and Kolmogorov equations. Subsequently, we shall sketch a new method to prove uniqueness of solutions for Fokker-Planck equations.

David Elworthy (Warwick), Functions of finite energy in finite and infinite dimensions

Friday 5th March 2010 at 15:00 in RT 033 (Civil Engineering)

The simply looking question: if a function has derivative which is square integrable is the function (up to a constant) square integrable? introduces
deep analytical and geometric questions. A slight extension is to ask whether a vector field which is square integrable is the gradient of a square integrable function. These questions will be discussed for functions on $\mathbb{R}^n$, on finite dimensional manifolds, and then for functions on path spaces of Riemannian manifolds. For the latter techniques of stochastic analysis are used, and there are new results by X.Chen, Xue-Mei Li, & B.Wu, by S. Aida, and by myself with Y.Yang.

Christian Bär (Potsdam), Random walks in curved spaces

Friday 12th February 2010 at 15:00 in RT 033 (Civil Engineering)

The properties of random paths, more technically of Brownian motion, are closely related to the geometry of the space in which they move.
Typical questions are:
* Does a random path return to a given domain with positive probability (recurrence versus transience)?
* Does a random path leave the space in finite time with positive probability (stochastic completeness)?
We will discuss several results in this direction. We will also explore the relation to diffusion equations which can be solved by so-called path integrals.

Jens Marklof (Bristol), Chaos in Crystals -- postponed to a later date

No longer on: Friday 19th February 2010 at 15:00 in RT 033 (Civil Engineering)

Abstract: The periodic Lorentz gas describes a particle moving in a regular array of spherical scatterers, and is one of the fundamental mathematical models for chaotic diffusion in a crystal. In this lecture (aimed at a general mathematical audience) I describe the recent solution of a problem, posed by Y. Sinai in the early 1980s, on the nature of the diffusion when the scatterers are very small. The problem is closely related to some basic questions in number theory, in particular the distribution of lattice points visible from a given position, as discussed e.g. in Polya's 1918 paper on the visibility in a forest. The key technology in our approach is measure rigidity, a branch of ergodic theory that has proved valuable in recent solutions of other problems in number theory and mathematical physics, such as the value distribution of quadratic forms at integers, quantum unique ergodicity and questions of diophantine approximation. (This lecture is based on joint work with A. Strombergsson, Uppsala.)