Harmonic functions on hyperbolic graphs

Camille Petit (1)

1 IF - Institut Fourier

We consider admissible random walks on hyperbolic graphs. For a given harmonic function on such a graph, we prove that asymptotic properties of non-tangential boundedness and non-tangential convergence are almost everywhere equivalent. The proof is inspired by the works of F. Mouton in the cases of Riemannian manifolds of pinched negative curvature and infinite trees. It involves geometric and probabilitistic methods.

Mots clés: Harmonic functions; hyperbolic graphs; random walks; boundary at infinity; Fatou's theorem; non-tangential convergence

Domaines: Géométrie métrique [math.MG]; Probabilités [math.PR]

Camille Petit. Harmonic functions on hyperbolic graphs. Proc. Amer. Math. Soc., 2012, 140, pp.235-248. ⟨10.1090/S0002-9939-2011-10931-6⟩. ⟨hal-00387558v2⟩