Discrete Approximation to Brownian Motion with Varying Dimension in Unbounded Domains

TL;DR

This paper develops a new method for approximating Brownian motion with varying dimension in unbounded domains using random walks, overcoming previous restrictions related to boundedness and initial distributions.

Contribution

It introduces a novel approach that removes the need for bounded state spaces and invariant initial distributions in approximating BMVD with random walks.

Findings

Established heat kernel upper bounds uniform in mesh size

Proved tightness of the approximating random walks

Provided a new method for discrete approximation in unbounded domains

Abstract

We establish the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) by random walks. The setting is very similar to that in [11], but here we use a different method allowing us to get rid the restrictions in [11] (or [3]) that the underlying state space has to be bounded, and that the initial distribution of the limiting continuous process has to be its invariant distribution. The approach in this paper is that we first obtain heat kernel upper bounds for the approximating random walks that are uniform in their mesh size, by establishing a Nash-type inequality based on their Dirichlet form characterization. Using the heat kernel upper bound, we then show the tightness of the approximating random walks by delicate analysis.