Harmonic polynomials and other exactly computable characteristics for $2$-dimensional random walks in cones

TL;DR

This paper characterizes when harmonic polynomials exist for 2D lattice random walks in cones, providing explicit formulas and moments of exit times, with a focus on wedges with specific opening angles.

Contribution

It establishes a precise condition for the existence of harmonic polynomials in wedge-shaped domains and offers explicit formulas for these polynomials and exit time moments.

Findings

Harmonic polynomials exist if and only if the wedge angle is a multiple of π/m.

Explicit formulas for harmonic polynomials are provided for all such angles.

Exact expressions for all finite moments of the exit time are derived.

Abstract

In this note we consider $2$-dimensional lattice random walks killed at leaving a wedge with opening $\alpha\in(0,\pi]$. Assuming that the walk cannot jump over the boundary of the wedge we prove that there exists a harmonic polynomial if and only if $\alpha=\pi/m$ with some integer $m$. Our proof is constructive and allows one to give exact expressions for harmonic polynomials for every integer $m$. Furthermore, we give exact expressions for all finite moments of the exit time, this result is valid for all angles $\alpha$.