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Potential kernel, hitting probabilities and distributional asymptotics

Published online by Cambridge University Press:  26 January 2019

FRANÇOISE PÈNE
Affiliation:
Université de Brest and Institut Universitaire de France, Laboratoire de Mathématiques de Bretagne Atlantique, UMR CNRS 6205, 29238 Brest Cedex, France email francoise.pene@univ-brest.fr
DAMIEN THOMINE
Affiliation:
Département de Mathématiques d’Orsay, Université Paris-Sud, UMR CNRS 8628, F-91405 Orsay Cedex, France email damien.thomine@math.u-psud.fr

Abstract

$\mathbb{Z}^{d}$ -extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green–Kubo’s formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve, in turn, the assumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on Abelian covers of compact manifolds of negative curvature are discussed.

Type
Original Article
Copyright
© Cambridge University Press, 2019

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