Can one condition a killed random walk to survive?

Abstract

We consider the simple random walk on ${\mathbb{Z}}^d$ killed with probability $p(|x|)$ at site $x$ for a function $p$ decaying at infinity. Due to recurrence in dimension $d=2$, the killed random walk (KRW) dies almost surely if $p$ is positive, while in dimension $d\ge 3$ it is known that the KRW dies almost surely if and only if ${\int }_0^{\infty }rp(r)dr=\infty$, under mild technical assumptions on $p$. In this paper we consider, for any $d\ge 2$, functions $p$ for which the KRW dies almost surely and we ask ourselves if the KRW conditioned to survive is well-defined. More precisely, given an exhaustion $(\hat{I}{›}_R{)}_{R\in \mathbb{N}}$ of ${\mathbb{Z}}^d$, does the KRW conditioned to leave $\hat{I}{›}_R$ before dying converges in distribution towards a limit which does not depend on the exhaustion? We first prove that this conditioning is well-defined for $p(r)=o(r^{-2})$, and that it is not for $p(r)=\min (1,r^{-\hat{I}\pm })$ for $\hat{I}\pm \in (14/9,2)$. This question is connected to branching random walks and the infinite snake. More precisely, in dimension $d=4$, the infinite snake is related to the KRW with $p(r)\asymp (r^2\log (r){)}^{-1}$, therefore our results imply that the infinite snake conditioned to avoid the origin in four dimensions is well-defined.

37 pages, 2 figures