Math 389: Ergodic theory of Dissipative Stochastic PDEs and Stochastic Analysis in infinite dimensions, Fall 2007

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This course will discuss the ergodic theory of dissipative stochastic partial differential equations through the study of the stochastically forced 2D Navier-Stokes equations (and maybe some reaction diffusion equations). The emphasis will be on understanding the inherent difficulties of a Markov process with an infinite dimensional phase space and the development of tools to overcome such difficulties.

The course will begin by introducing a few stochastic PDEs. Then we will return to the simples of Markov process (those with a finite number of states) and develop their ergodic theory. We will then add complications one at a time until we arrive at a theory capable of handling the non-locally compact phase space of a SPDE. This will require the introduction of a generalization of the classical strong Feller property of a Markov process called the asymptotic strong Feller property. The analysis of the systems will require a brief excursion into the stochastic calculus of variations (Malliavin calculus).

A basic outline of the course follows:

• Stochastically Navier Stokes and Reaction Diffusion equations
• Convergence of finite state Markov chains (Doeblin's condition)
• Coupling and Convergence of Harris Chains
• Optimal coupling and Distances on probability measures.
• Monge-Kantorovich duality
• Singularity and absolute continuity of measures in infinite dimensions
• Strong Feller and Asymptotic strong Feller Markov process
• Coupling and 1-Wasserstein metrics.
• Spectral Gaps in 1-Wasserstein metrics and exponential convergence to equilibrium.

This course will essentially be a reprise of my lectures in the 2007 Saint Flour probability summer school. Notes will be distributed.

If you have wondered what I spend much of my time working on, this is the course of you.