Green function for an asymptotically stable random walk in a half space

TL;DR

This paper derives the asymptotic behavior of the Green function for an asymptotically stable multidimensional random walk in a half-space, providing detailed local asymptotics as the points tend to infinity.

Contribution

It introduces new asymptotic formulas for the Green function of a multidimensional stable random walk in a half-space, extending understanding of boundary behaviors.

Findings

Asymptotics of transition probabilities $p_n(x,y)$ as $n$ tends to infinity.

Local asymptotics for the Green function $G(x,y)$ at large distances.

Characterization of the boundary behavior of the random walk.

Abstract

We consider an asymptotically stable multidimensional random walk $S(n)=(S_1(n),\ldots, S_d(n) )$. Let $\tau_x:=\min\{n>0: x_{1}+S_1(n)\le 0\}$ be the first time the random walk $S(n)$ leaves the upper half-space. We obtain the asymptotics of $p_n(x,y):= P(x+S(n) \in y+\Delta, \tau_x>n)$ as $n$ tends to infinity, where $\Delta$ is a fixed cube. From that we obtain the local asymptotics for the Green function $G(x,y):=\sum_n p_n(x,y)$, as $|y|$ and/or $|x|$ tend to infinity.