Oberwolfach Seminar: The Ergodic Theory of Markov Processes

Date

October 24th - October 30th, 2010 (ID: 1043a)

Organizers

Martin Hairer, New York

Jonathan Mattingly, Durham

Programme

In recent years, much attention has been devoted to the study of stochastic partial differential equations (SPDEs) that arise in a number of areas like turbulence, quantum field theory, material science, spatial models in epidemiology. The solutions to such SPDEs form Markov processes, but on infinite-dimensional functional spaces. Stochastic delay equations from another class of stochastic dynamics which also produces a Markov process on an infinite dimensional space. While there are situations where the traditional tools of the theory of Markov processes can be applied to such systems, they do fail in many important cases. This spurred a number of recent advances in the theory. These have centered around the realization that the total variation metric, while very effective for countable state space Markov chains and for finite-dimensional diffusion processes, is much less useful in the infinite dimensional setting. The introduction of tools like asymptotic coupling and the asymptotic strong Feller property have lead to progress on a number of problems which had been previously out of reach.

The aim of the proposed seminar is to provide young researchers with a grounding in these recent advances as well as the foundations on which they are built. In addition, there will be a few advanced lectures on related topics.

Prerequisites and Literature

Participants are expected to be comfortable with measure theory, elementary stochastic analysis, functional analysis and basic semigroup theory. For those wishing to refresh their memory on these subjects, we recommend the introductory chapters of the following monographs:

• "Measure Theory" by V.I. Bogachev,
• "Continuous Martingales and Brownian Motion" by D. Revuz and M. Yor,
• and "One- Parameter Semigroups" by B. Davies.

Having a background in stochastic PDE theory will help but is not necessary to be able to follow the course. A good introduction to the subject can be found in the monographs by Da Prato and Zabczyk, "Stochastic equations in infinite dimensions" and "Ergodicity for infinite-dimensional systems". A much shorter but reasonably self-contained introduction can be found online here: An Introduction to Stochastic PDEs.

Deadline for applications

September 1st, 2010

The seminars take place at the Mathematisches Forschungsinstitut Oberwolfach. The number of participants is restricted to 25. The Institute covers accommodation and food. We are pleased that the Carl Friedrich von Siemens Foundation has decided to support the Oberwolfach Seminars from summer 2008 to summer 2013. By this support, travel expenses can be reimbursed up to 200 Euro in average per person. Participants can ask for travel support during their stay in Oberwolfach at the guest office against copy of travel receipts. Applications including

• full name and address, including e-mail address
• present position, university
• name of supervisor of Ph.D. thesis
• a short summary of previous work and interest

should be sent preferably by e-mail (.ps or .pdf file) to:

    Prof. Dr. Dr. h.c. Gert-Martin Greuel
    Universität Kaiserslautern
    Fachbereich Mathematik
    Erwin Schrödingerstr.
    67663 Kaiserslautern, Germany