Exponential Convergence in $L^p$-Wasserstein Distance for Diffusion Processes without Uniformly Dissipative Drift

Abstract

By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups $(P_t{)}_{t\ge 0}$ with respect to the standard $L^p$-Wasserstein distance for all $p\in [1,\infty )$. In particular, we show that for the Itô stochastic differential equation $$\d X_t=\d B_t+b(X_t)\,\d t,$$ if the drift term $b$ satisfies that for any $x,y\in {\mathbb{R}}^d$, $$⟨b(x)-b(y),x-y⟩\le \{\begin{array}{llc}{K}_1|x-y{|}^2,& |x-y|\le L;-K_2|x-y{|}^2,& |x-y|>L\end{array}$$ holds with some positive constants $K_1$, $K_2$ and $L>0$, then there is a constant $\lambda :=\lambda (K_1,K_2,L)>0$ such that for all $p\in [1,\infty )$, $t>0$ and $x,y\in {\mathbb{R}}^d$, $$W_p(δ_x P_t,δ_y P_t)\leq Ce^{-λt/p} \begin{cases} |x-y|^{1/p}, & \mbox{if } |x-y|\le 1; |x-y|, & \mbox{if } |x-y|> 1. \end{cases}$$ where $C:=C(K_1,K_2,L,p)$ is a positive constant. This improves the main result in \cite{Eberle} where the exponential convergence is only proved for the $L^1$-Wasserstein distance.

22 pages

References in corpus (3)

• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations
• Subgeometric ergodicity of strong Markov processes
• Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Dini Continuous Drift

Cited by in corpus (3)

• Long-time behaviors of mean-field interacting particle systems related to McKean-Vlasov equations
• $L^p$-Wasserstein distance for stochastic differential equations driven by Lévy processes
• Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises