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Potential kernel, hitting probabilities and distributional asymptotics

Part of: Ergodic theory Limit theorems Stochastic processes Markov processes Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  26 January 2019

FRANÇOISE PÈNE  and
DAMIEN THOMINE
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
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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.

Keywords

central limit theoremprobabilistic potential theoryclassical ergodic theoryinfinite measurenull recurrent process

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60J45: Probabilistic potential theory 60G50: Sums of independent random variables; random walks 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 37A05: Measure-preserving transformations

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 7 , July 2020 , pp. 1894 - 1967
Copyright
© Cambridge University Press, 2019

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