Ergodic theorems for infinite dimensional Markov processes
June 26 - July 22, 2009
The aim of these lectures is to present a number of recent results on the long-time behaviour of Markov processes in infinite-dimensional spaces, with a focus on stochastic PDEs. We will start by giving a short introduction to the theory of stochastic evolution equations, followed by an introduction to ergodic theory for Markov processes. This will allow us to build some intuition on the problems at hand, as well as an understanding of the challenges that need to be tackled. The introduction will be relatively self-contained, aimed at students with knowledge of basic graduate analysis and some introduction to probability theory. We will then present a general theory that gives a framework in which many questions of interest can be solved.
The second half of the course will show how to apply the general theory to a large class of stochastic PDEs and stochastic delay equations. The central question will be "what is different in infinite dimensions?" and how can we overcome the added difficulties. One of the main achievements presented there will be an infinite-dimensional generalisation of Hörmander's celebrated "sums of squares" theorem. Along the way will touch on topics from Malliavin calculus and stochastic analysis.
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