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Jonathan Mattingly : Ergodicity of Stochastically Forced PDEs

Stochastic PDEs have become important models for many phenomenon. Nonetheless, many fundamental questions about their behavior remain poorly understood. Often such SPDE contain different processes active at different scales. Not only does such structure give rise to beautiful mathematics and phenomenon, but I submit that it also contains the key to answering many seemingly unrelated questions. Questions such as ergodicity and Mixing. Given a stochastically forced dissipative PDE such as the 2D Navier Stokes equations, the Ginzburg-Landau equations, or a reaction diffusion equation; is the system Ergodic ? If so, at what rate does the system equilibrate ? Is the convergence qualitatively different at different physical scales ? Answers to these an similar questions are basic assumptions of many physical theories such as theories of turbulence. I will try both to convince you why these questions are interesting and explain how to address them. The analysis will suggest strategies to explore other properties of these SPDEs as well as numerical methods. In particular, I will show that the stochastically forced 2D Navier Stokes equations converges exponentially to a unique invariant measure. I will discuss under what minimal conditions one should expect ergodic behavior. The central ideas will be illustrated with a simple model systems. Along the way I will explain how to exploit the different scales in the problem and how to overcome the fact that the problem is an extremely degenerate diffusion on an infinite dimensional function space. The analysis points to a class of operators in between STRICTLY ELLIPTIC and HYPOELLIPTIC operators which I call EFFECTIVELY ELLIPTIC. The techniques use a representation of the process on a finite dimensional space with memory. I will also touch on a novel coupling construction used to prove exponential convergence to equilibrium.

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