On Polynomial Mixing and Convergence Rate for Stochastic Difference and Differential Equations

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SUBGEOMETRIC ERGODICITY OF STRONG MARKOV PROCESSES (2005, background): “Veretennikov [36] and Malyshkin [19] deal with diffusion processes and can be read as a special application of the present paper.”
Subgeometric rates of convergence of Markov processes in the Wasserstein metric (2012, background): “(i) Assume additionally that for some constants α ∈ (0,1], M > 0, κ > 0, the generalized Veretennikov–Khasminskii condition holds, that is,〈 f (x), x(0) 〉 ≤ −κ∣∣x(0)∣∣α, x ∈ C, ∣∣x(0)∣∣ ≥ M.(3.6) Then SDDE (3.1) has a unique invariant measure π , and Law(Xt) converges to π in the Wasserstein metric ”
Examples of moderate deviation principle for diffusion processes (2005, methodology): “In a nonlinear case, we use Veretennikov - Khasminskii’s condition (see, [14] and [ 28 ]): for some positive numbers r, C and , (here hh·ii denotes the inner product) hhz,b(z)ii rkzk1+ , kzk > C”
Steady States of Fokker–Planck Equations: I. Existence (2015, unknown): “Then for any Ω′ ⊂⊂ Ω, sup x∈Ω′ u(x) ≤ C inf x∈Ω′ u(x), where C > 0 is a constant depending only on n, p, q, |λ−1|Lp(Ω), |Λ|Lq(Ω), |g|Lq(Ω) and dist(Ω′, ∂Ω).”