An introduction to asymptotic coupling to prove unique ergodicity

An introduction to asymptotic coupling to prove unique ergodicity

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Jonathan Mattingly, Duke University
Fine Hall 322

I will give an overview of the idea of asymptotic coupling and how it can be used to prove unique ergodicity in a number of settings. In particular, I will consider some of Stochastically forced PDEs. Examples will include: the Navier stokes equations, a fractionally dissipative Euler equation, Stochastically Forced Euler-Voigt with damping, and a Damped Nonlinear Wave Equation. I will also discuss a Stochastic delay equations and the scaling limits of MCMC. These ideas date back to work with Ya Sinai and Weinan E, yet seem to have been under appreciated. Though in many cases, it is not much harder to prove exponential convergence to equilibrium, I will emphasis the simplest framework with an eye to proving only uniqueness of the invariant measure. I will make some comments about why these ideas are more central when working in infinite dimensions. Many of the examples will come from a joint work with Nathan Glatt-Holtz and Geordie Richards. The formulation asymptotic coupling mildly extends the framework developed in works with Martin Hairer and Michael Scheutzow.