One dimensional lattice random walks with absorption at a point / on a half line

TL;DR

This paper analyzes one-dimensional lattice random walks with absorption at a point or half line, providing asymptotic estimates for transition probabilities and entrance distributions, advancing understanding of their long-term behavior.

Contribution

It introduces asymptotic estimates for transition probabilities of absorbed lattice random walks, extending to half line absorption and first entrance distributions.

Findings

Asymptotic estimate of transition probability for walk absorbed at the origin

Extension of estimates to walks absorbed on a half line

Evaluation of the space-time distribution for first entrance into the half line

Abstract

This paper concerns a random walk that moves on the integer lattice and has zero mean and a finite variance. We obtain first an asymptotic estimate of the transition probability of the walk absorbed at the origin, and then, using the obtained estimate, that of the walk absorbed on a half line. The latter is used to evaluate the space-time distribution for the first entrance of the walk into the half line.