We will present new coupling techniques for analyzing ergodicity of nonlinear stochastic PDEs with additive forcing. These methods complement the Hairer-Mattingly approach (2006, 2011). In the first part of the talk, we demonstrate how a generalized coupling approach can be used to study ergodicity for a broad class of nonlinear SPDEs, including 2D stochastic Navier-Stokes equations. This extends the results of [N. Glatt-Holtz, J. Mattingly, G. Richards, 2017]. The second part of the talk is devoted to SPDEs that satisfy comparison principle (e.g., stochastic reaction-diffusion equations). Using a new version of the coupling method, we establish exponential ergodicity of such SPDEs in the hypoelliptic setting and show how the corresponding Hairer-Mattingly results can be refined. (Joint work with Alexey Kulik and Michael Scheutzow)
 O. Butkovsky, A. Kulik, M. Scheutzow (2018). Generalized couplings and ergodic rates for SPDEs and other Markov models. arXiv:1806.00395; to appear in "The Annals of Applied Probability".