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

Abstract

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

25 pages