Random walks in \((\mathbb Z_+)^2\) with non-zero drift absorbed at the axes (Q2880695)

This paper is a detailed study on the 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 axis. The main tool of analysis is given by the book [\textit{G. Fayolle, R. Iasnogorodski} and \textit{V. Malyshev}, Random walks in the quarter-plane. Algebraic methods, boundary value problems and applications. Applications of Mathematics. 40. Berlin: Springer. (1999; Zbl 0932.60002)] which serves as a starting point for the present investigation. In the five sections of the paper the reader can found a generous introduction in the topic and an analytic approach of the subject. In a special case of a finite groups of the random walk the probability of absorption takes a particularly nice form. Explicit form of the absorption probabilities generating functions is given and the asymptotic of the absorption probabilities is studied. Also a study on the asymptotic of the Green function along all different infinite path is made.