Green Function for an Asymptotically Stable Random Walk in a Half Space

Abstract

We consider an asymptotically stable multidimensional random walk $$S(n)=(S_1(n),\ldots , S_d(n) )$$ S ( n ) = ( S 1 ( n ) , … , S d ( n ) ) . For every vector $$x=(x_1\ldots ,x_d)$$ x = ( x 1 … , x d ) with $$x_1\ge 0$$ x 1 ≥ 0 , let $$\tau _x:=\min \{n>0: x_{1}+S_1(n)\le 0\}$$ τ x : = min { n > 0 : x 1 + S 1 ( n ) ≤ 0 } be the first time the random walk $$x+S(n)$$ x + S ( n ) leaves the upper half space. We obtain the asymptotics of $$p_n(x,y):= {\textbf{P}}(x+S(n) \in y+\Delta , \tau _x>n)$$ p n ( x , y ) : = P ( x + S ( n ) ∈ y + Δ , τ x > n ) as n tends to infinity, where $$\Delta $$ Δ is a fixed cube. From that, we obtain the local asymptotics for the Green function $$G(x,y):=\sum _n p_n(x,y)$$ G ( x , y ) : = ∑ n p n ( x , y ) , as $$|y |$$ | y | and/or $$|x |$$ | x | tend to infinity.