Discrete approximation to Brownian motion with varying dimension 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 sizes, by establishing an isoperimetric inequality and 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 via delicate analysis.