Exponential convergence in ‐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 with respect to the standard ‐Wasserstein distance for all . In particular, we show that for the Itô stochastic differential equation urn:x-wiley:0025584X:media:mana201500351:mana201500351-math-0006 if the drift term b is such that for any , urn:x-wiley:0025584X:media:mana201500351:mana201500351-math-0008 holds with some positive constants K 1 , K 2 and , then there is a constant such that for all , and , urn:x-wiley:0025584X:media:mana201500351:mana201500351-math-0014 where is a positive constant. This improves the main result in where the exponential convergence is only proved for the L 1 ‐Wasserstein distance.