Convergence of Markov processes to the invariant measure in the Wasserstein metric with applications to SPDEs

While ergodic behavior of finite-dimensional Markov processes (e.g., SDEs) is quite well understood by now, less is known about the ergodic behavior of infinite-dimensional Markov processes (e.g., SPDEs). In the first part of the talk we explain how the classical methods (which are based on the strong Feller property) can be extended to study convergence of infinite-dimensional Markov processes in the Wasserstein metric. This generalizes recent results of M. Hairer, J. Mattingly, M. Scheutzow (2011). In the second part of the talk we provide several specific applications to SPDEs (including 2D stochastic Navier-Stokes) and discuss the arising challenges. (Joint work with Alexey Kulik and Michael Scheutzow)

[1] O. Butkovsky (2014). Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Annals of Applied Probability, 24, 526-552.
[2] O. Butkovsky, M. Scheutzow (2017). Invariant measures for stochastic functional differential equations. arXiv:1703.05120.