Reflecting random walk in fractal domains

TL;DR

This paper demonstrates that reflecting Brownian motion in bounded fractal domains can be approximated by simple random walks on discretized subsets, providing a new approach to understanding stochastic processes in complex geometries.

Contribution

It introduces a novel approximation method for reflecting Brownian motion in fractal domains using simple random walks on maximal connected lattice subsets.

Findings

Reflecting Brownian motion can be approximated by simple random walks as the lattice mesh size tends to zero.

The approximation uses maximal connected subsets of discretized domains within the original domain.

The method applies to any bounded domain D, including fractal and irregular geometries.

Abstract

In this paper, we show that reflecting Brownian motion in any bounded domain D can be approximated, as $k\to\infty$, by simple random walks on "maximal connected" subsets of $(2^{-k}\mathbb{Z}^d)\cap D$ whose filled-in interiors are inside of D.