Martin boundary of random walks in convex cones

Jetlir Duraj (1) , Kilian Raschel (2, 3) , Pierre Tarrago (4) , Vitali Wachtel (5)

1 Department of Economics, Harvard University
2 IDP - Institut Denis Poisson
3 CNRS - Centre National de la Recherche Scientifique
4 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
5 Institut für Mathematik [Augsburg]

We determine the asymptotic behavior of the Green function for zero-drift random walks confined to multidimensional convex cones. As a consequence, we prove that there is a unique positive discrete harmonic function for these processes (up to a multiplicative constant); in other words, the Martin boundary reduces to a singleton.

Mots clés: Random walk; cone; exit time; Green function; harmonic function; Martin boundary; Brownian motion; coupling

Domaines: Probabilités [math.PR]

Jetlir Duraj, Kilian Raschel, Pierre Tarrago, Vitali Wachtel. Martin boundary of random walks in convex cones. Annales Henri Lebesgue, 2022, 5, pp.559-609. ⟨10.5802/ahl.130⟩. ⟨hal-02499786⟩