Convergence of discrete Green functions with Neumann boundary conditions

TL;DR

This paper proves that discrete Green functions with Neumann boundary conditions for random walks converge to their continuous versions, and provides hitting estimates relevant for studying two-dimensional aggregation systems.

Contribution

It establishes convergence of discrete to continuous Green functions with Neumann boundary conditions and extends classical results to bounded geometry settings.

Findings

Convergence of discrete Green functions to continuous counterparts.

Derived Beurling type hitting estimates for random walks.

Applications to the analysis of the two-dimensional Competitive Erosion system.

Abstract

In this note we prove convergence of Green functions with Neumann boundary conditions for the random walk to their continuous counterparts. Also a few Beurling type hitting estimates are obtained for the random walk on discretizations of smooth domains. These have been used recently in the study of a two dimensional competing aggregation system known as $Competitive\, Erosion$. Some of the statements appearing in this note are classical for ${\mathbb{Z}}^2$. However additional arguments are needed for the proofs in the bounded geometry setting.