We establish subgeometric bounds on convergence rate of general Markov processes in the Wasserstein metric. In the discrete time setting we prove that the Lyapunov drift condition and the existence of a "good" d-small set imply subgeometric convergence to the invariant measure. In the continuous time setting we obtain the same convergence rate provided that there exists a "good" d-small set and the Douc-Fort-Guillin supermartingale condition holds. As an application of our results, we prove that the Veretennikov-Khasminskii condition is sufficient for subexponential convergence of strong solutions of stochastic delay differential equations. E P (t, dx)µ(dt).Recall (see, e.g., [2]) that if d is a semimetric on E, then the Wasserstein semidistance W d between probability measures µ, ν ∈ P(E) is given by W d (µ, ν) := inf λ∈C(µ,ν) E×E d(x, y)λ(dx, dy),

We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including 2D stochastic Navier-Stokes equations. Our main tool is a new version of the generalized coupling method.

We consider the stochastic differential equationwhere the drift b is a generalized function and L is a symmetric one dimensional α-stable Lévy processes, α ∈ (1, 2). We define the notion of solution to this equation and establish strong existence and uniqueness whenever b belongs to the Besov-Hölder space C β for β > 1/2−α/2.

We study ergodic properties of nonlinear Markov chains and stochastic McKean-Vlasov equations. For nonlinear Markov chains we obtain sufficient conditions for existence and uniqueness of an invariant measure and uniform ergodicity. We also prove optimality of these conditions. For stochastic McKean-Vlasov equations we establish exponential convergence of their solutions to stationarity in the total variation metric under Veretennikov-Khasminskii-type conditions.

We study stochastic reaction-diffusion equationwhere b is a generalized function in the Besov space B β q,∞ (R), D ⊂ R and Ẇ is a space-time white noise on R + × D. We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever β −. This class includes equations with b being measures, in particular, b = δ 0 which corresponds to the skewed stochastic heat equation. For β − 1/q > −3/2, we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalizes the results of to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020).