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

TL;DR

This work characterizes exactly computable harmonic characteristics for 2D lattice random walks killed upon exiting a wedge $K_α$ and provides explicit expressions for harmonic polynomials when the wedge angle satisfies $α=π/m$. The authors develop a constructive framework that reduces harmonicity to solving a finite linear system derived from the discrete generator, using a Brownian-prototype function $u_α$ to seed solutions. They also prove the existence and uniqueness (up to a scalar) of these polynomials under moment and no-overshoot conditions, and obtain closed-form expressions for all finite exit-time moments, including a recursive scheme for higher moments and the first two moments of the exit position. Additionally, an alternative approach and a detailed appendix offer numerical methods and proofs, linking to Brownian and Coxeter/Weyl-geometry perspectives, with implications for combinatorics in the quadrant.

Abstract

In this note we consider $2$-dimensional lattice random walks killed at leaving a wedge with opening $α\in(0,π]$. Assuming that the walk cannot jump over the boundary of the wedge we prove that there exists a harmonic polynomial if and only if $α=π/m$ 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 $α$.

Theorems & Definitions (14)

• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof
• proof : Proof of Theorem \ref{['thm:uniqueness']}