Random walks in the quarter plane, harmonic functions and conformal mappings

Kilian Raschel (1, 2)

1 FDP - Fédération de recherche Denis Poisson
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

We propose here a new approach for finding harmonic functions of killed random walks with small steps in the quarter plane. It is based on solving a certain functional equation that satisfies the generating function of the values taken by the harmonic functions. As a first application of our results, we obtain a simple expression for the harmonic function that governs the asymptotic tail distribution of the first exit time from the quarter plane. As another corollary, we prove, in the zero drift case, the uniqueness of the harmonic function.

Mots clés: Random walk in the quarter plane; harmonic function; Martin boundary; exit time; conformal mapping

Domaines: Probabilités [math.PR]

Kilian Raschel. Random walks in the quarter plane, harmonic functions and conformal mappings. 2013. ⟨hal-00780452v1⟩