Note on local mixing techniques for stochastic differential equations

TL;DR

This paper explores various local mixing techniques for stochastic differential equations, emphasizing coupling methods and Markov chain embeddings to analyze convergence and mixing rates in different dimensions.

Contribution

It introduces and compares multiple local mixing techniques for SDEs, including trajectory intersections and Markov chain embeddings, expanding the toolkit for analyzing stochastic processes.

Findings

Techniques work in any dimension $d\,\geq 1$

In $d=1$, trajectory intersections suffice for coupling

For $d>1$, embedded Markov chains and MD conditions are effective

Abstract

This paper discusses several techniques which may be used for applying the coupling method to solutions of stochastic differential equations (SDEs). They all work in dimension $d\ge 1$, although, in $d=1$ the most natural way is to use intersections of trajectories, which requires nothing but strong Markov property and non-degeneracy of the diffusion coefficient. In dimensions $d>1$ it is possible to use embedded Markov chains either by considering discrete times $n=0,1,\ldots$, or by arranging special stopping time sequences and to use local Markov -- Dobrushin's (MD) condition. Further applications may be based on one or another version of the MD condition. For studies of convergence and mixing rates the (Markov) process must be strong Markov and recurrent; however, recurrence is a separate issue which is not discussed in this paper.