Exponential convergence in the Wasserstein metric $W_1$ for one dimensional diffusions

TL;DR

This paper establishes conditions under which one-dimensional diffusions exhibit exponential convergence in the Wasserstein $W_1$ metric, providing sharp estimates for convergence rates and illustrating with examples.

Contribution

It introduces general, efficient criteria for exponential convergence in $W_1$ for 1D diffusions and offers precise estimates for the convergence constants.

Findings

Exponential convergence in $W_1$ under specified conditions

Sharp bounds for convergence rate $\delta$ and constant $K$

Illustrative examples demonstrating the results

Abstract

In this paper, we find some general and efficient sufficient conditions for the exponential convergence $W_{1,d}(P_t(x,\cdot), P_t(y,\cdot) )\le Ke^{-\delta t}d(x,y)$ for the semigroup $(P_t)$ of one-dimensional diffusion. Moreover some sharp estimates of the involved constants $K\ge 1, \delta>0$ are provided. Those general results are illustrated by a series of examples.