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 $(P_t)_{t\ge0}$ with respect to the standard $L^p$-Wasserstein distance for all $p\in[1,\infty)$. In particular, we show that for the It\^o stochastic differential equation $$\d X_t=\d B_t+b(X_t)\,\d t,$$ if the drift term $b$ satisfies that for any $x,y\in\R^d$, $$\langle b(x)-b(y),x-y\rangle\le \begin{cases} K_1|x-y|^2,& |x-y|\le L; -K_2|x-y|^2,& |x-y|> L \end{cases}$$ 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 $x,y\in\R^d$, $$W_p(\delta_x P_t,\delta_y P_t)\leq Ce^{-\lambda t/p} \begin{cases} |x-y|^{1/p}, & \mbox{if } |x-y|\le 1; |x-y|, & \mbox{if } |x-y|> 1. \end{cases}$$ 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.