Sub-exponential convergence to equilibrium for Gaussian driven Stochastic Differential Equations with semi-contractive drift

TL;DR

This paper investigates the rate at which certain Gaussian-driven stochastic differential equations converge to equilibrium, establishing sub-exponential bounds in Wasserstein and total variation distances under semi-contractive drift conditions.

Contribution

It introduces a novel coupling approach to prove sub-exponential convergence rates for SDEs with semi-contractive drifts in Wasserstein and total variation metrics.

Findings

Sub-exponential convergence bounds in Wasserstein distance.

Sub-exponential bounds in total variation distance.

Effective coupling strategies for semi-contractive SDEs.

Abstract

The convergence to the stationary regime is studied for Stochastic Differential Equations driven by an additive Gaussian noise and evolving in a semi-contractive environment, i.e. when the drift is only contractive out of a compact set but does not have repulsive regions. In this setting, we develop a synchronous coupling strategy to obtain sub-exponential bounds on the rate of convergence to equilibrium in Wasserstein distance. Then by a coalescent coupling close to terminal time, we derive a similar bound in total variation distance.