Markov chain approximations for nonsymmetric processes

TL;DR

This paper establishes conditions under which diffusion and nonsymmetric jump processes in multidimensional space can be approximated by Markov chains on scaled lattices, using PDE and Dirichlet form methods.

Contribution

It provides new theoretical results on approximating nonsymmetric diffusions and jump processes with Markov chains, including convergence criteria based on conductance conditions.

Findings

Markov chain approximations for nonsymmetric diffusions

Convergence conditions based on conductances

Application to nonsymmetric jump processes

Abstract

The aim of this article is to prove that diffusion processes in $\mathbb{R}^d$ with a drift can be approximated by suitable Markov chains on $n^{-1}\mathbb{Z}^d$. Moreover, we investigate sufficient conditions on the conductances which guarantee convergence of the associated Markov chains to such Markov processes. Analogous questions are answered for a large class of nonsymmetric jump processes. The proofs of our results rely on regularity estimates for weak solutions to the corresponding nonsymmetric parabolic equations and Dirichlet form techniques.