Exponential Convergence in $L^p$-Wasserstein Distance for Diffusion Processes without Uniformly Dissipative Drift (1407.1986v3)

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.