Random walks in Weyl chambers

TL;DR

This paper investigates the behavior of multi-dimensional random walks within Weyl chambers of types C and D, focusing on harmonic functions and exit time asymptotics under optimal conditions.

Contribution

It constructs positive harmonic functions for these random walks and derives tail asymptotics for their exit times, advancing understanding of their boundary behaviors.

Findings

Constructed positive harmonic functions for random walks in Weyl chambers.

Derived tail asymptotics for the exit time from Weyl chambers.

Established results under optimal moment assumptions.

Abstract

We study a $d$-dimensional random walk with zero mean and finite variance in the Weyl chambers of type C and D. Under optimal moment assumptions we construct positive harmonic functions for random walks killed on exiting Weyl chambers. We also find the tail asymptotics for the exit time of the random walk from Weyl chambers.