Stein method for invariant measures of diffusions via Malliavin calculus

TL;DR

This paper develops a Stein method leveraging Malliavin calculus to quantify the distance between a Malliavin-regular random variable's law and various continuous probability laws, using bounds based on the Malliavin derivative.

Contribution

It introduces a novel approach combining Stein's method with Malliavin calculus to analyze invariant measures of diffusions, providing explicit bounds in terms of derivatives.

Findings

Derived bounds for distributional distances using Malliavin derivatives

Applicable to a wide class of continuous probability laws

Illustrated with multiple examples of diffusion processes

Abstract

Given a random variable $F$ regular enough in the sense of the Malliavin calculus, we are able to measure the distance between its law and almost any continuous probability law on the real line. The bounds are given in terms of the Malliavin derivative of $F$. Our approach is based on the theory of It\^o diffusions and the stochastic calculus of variations. Several examples are considered in order to illustrate our general results.