Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels

TL;DR

This paper studies how the invariant measures of diffusion processes change with variations in their coefficients, providing stability estimates under log-concavity assumptions and extending inequalities relating transport distances and Stein discrepancies to non-Gaussian contexts.

Contribution

It introduces a new stability analysis method for invariant measures of diffusions and extends existing inequalities to broader, non-Gaussian settings using Stein kernels.

Findings

Established stability estimates for invariant measures under $L^p$ perturbations.

Extended inequalities relating transport distances and Stein discrepancies beyond Gaussian cases.

Applied the method to non-Gaussian distributions via Stein kernel constructions.

Abstract

We investigate stability of invariant measures of diffusion processes with respect to $L^p$ distances on the coefficients, under an assumption of log-concavity. The method is a variant of a technique introduced by Crippa and De Lellis to study transport equations. As an application, we prove a partial extension of an inequality of Ledoux, Nourdin and Peccati relating transport distances and Stein discrepancies to a non-Gaussian setting via the moment map construction of Stein kernels.