Hitting times with taboo for a~random walk

E. Vl. Bulinskaya

Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 3-26

Résumé

For a symmetric homogeneous and irreducible random walk on the $d$-dimensional integer lattice, which have a finite variance of jumps, we study passage times (taking values in $[0,\infty]$) determined by a starting point $x$, a hitting state $y$, and a taboo state $z$. We find the probability that these passage times are finite, and study the distribution tail. In particular, it turns out that, for the above-mentioned random walks on $\mathbb Z^d$ except for a simple random walk on $\mathbb Z$, the order of the distribution tail decrease is specified by dimension $d$ only. In contrast, for a simple random walk on $\mathbb Z$, the asymptotic properties of hitting times with taboo essentially depend on mutual location of the points $x,y$, and $z$. These problems originated in recent study of a branching random walk on $\mathbb Z^d$ with a single source of branching.