# Potential kernel, hitting probabilities and distributional asymptotics

## TL;DR

This paper develops a generalized central limit theorem for certain dynamical systems, relating hitting probabilities to potential kernels, with applications to Lorentz gases and geodesic flows, expanding theoretical understanding of these systems.

## Contribution

It introduces a generalized CLT for Z^d-extensions of dynamical systems, connecting hitting probabilities with potential kernels and improving assumptions for the theorem.

## Key findings

- Proves a generalized CLT under spectral assumptions.
- Shows invariance of Green-Kubo's formula under induction.
- Relates hitting probabilities to symmetrized potential kernels.

## Abstract

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 asumptions 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.