# Conditioned two-dimensional simple random walk: Green's function and harmonic measure

## TL;DR

This paper investigates a conditioned 2D simple random walk, deriving explicit Green's function formulas and analyzing the convergence rate of the entrance measure to the harmonic measure, enhancing understanding of conditioned stochastic processes.

## Contribution

It provides explicit formulas for the Green's function and quantitative convergence results for the harmonic measure of the conditioned walk, advancing theoretical understanding.

## Key findings

- Explicit Green's function formula derived
- Quantitative convergence rate of entrance to harmonic measure established
- Enhanced understanding of conditioned 2D random walks

## Abstract

We study the Doob's $h$-transform of the two-dimensional simple random walk with respect to its potential kernel, which can be thought of as the two-dimensional simple random walk conditioned on never hitting the origin. We derive an explicit formula for the Green's function of this random walk, and also prove a quantitative result on the speed of convergence of the (conditional) entrance measure to the harmonic measure (for the conditioned walk) on a finite set.