About a possible analytic approach for walks in the quarter plane with arbitrary big jumps

TL;DR

This paper extends the analytic approach for studying random walks in the quarter plane to include walks with arbitrary large jumps, addressing new technical challenges via generalized boundary value problems on Riemann surfaces.

Contribution

It introduces a novel extension of the analytic method to handle big jumps in quarter plane walks, expanding the applicability of prior models.

Findings

Extended the analytic approach to big jumps

Addressed technical challenges with boundary value problems

Provided a framework for future analysis of complex walks

Abstract

In this note, we consider random walks in the quarter plane with arbitrary big jumps. We announce the extension to that class of models of the analytic approach of [G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter plane, Springer-Verlag, Berlin (1999)], initially valid for walks with small steps in the quarter plane. New technical challenges arise, most of them being tackled in the framework of generalized boundary value problems on compact Riemann surfaces.