# Discrete harmonic functions in Lipschitz domains

## TL;DR

This paper establishes the existence and uniqueness of discrete harmonic functions in Lipschitz domains for certain random walks, using potential theory and comparison methods.

## Contribution

It introduces a new approach to prove existence and uniqueness of discrete harmonic functions in Lipschitz domains for finite-range, centered, elliptic random walks.

## Key findings

- Proves existence of discrete harmonic functions in Lipschitz domains.
- Establishes uniqueness of these functions under specified conditions.
- Develops a method based on comparison and potential theory.

## Abstract

We prove the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in $\mathbb{Z}^d$. Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.