Random walks in $(\mathbb{Z}_+)^2$ with non-zero drift absorbed at the axes

TL;DR

This paper analyzes two-dimensional random walks with non-zero drift in the positive quadrant, deriving explicit formulas for absorption probabilities and Green functions, especially near the axes, to understand their asymptotic behavior.

Contribution

It provides explicit generating functions for absorption probabilities and detailed asymptotics of Green functions for random walks with drift in the quadrant.

Findings

Explicit absorption probability generating functions derived.

Asymptotic behavior of absorption probabilities along axes characterized.

Green functions asymptotics computed along various paths approaching axes.

Abstract

Spatially homogeneous random walks in $(\mathbb{Z}_{+})^{2}$ with non-zero jump probabilities at distance at most 1, with non-zero drift in the interior of the quadrant and absorbed when reaching the axes are studied. Absorption probabilities generating functions are obtained and the asymptotic of absorption probabilities along the axes is made explicit. The asymptotic of the Green functions is computed along all different infinite paths of states, in particular along those approaching the axes.