Potential theory and $\mathbb{Z}^d$-extensions

Abstract

We study hitting probabilities for $\mathbb{Z}^d$-extensions of Gibbs-Markov maps. The goal is to estimate, given a finite $\Sigma \subset \mathbb{Z}^d$ and $p$, $q \in \Sigma$, the probability $P_{pq}$ that the process starting from $p$ returns to $\Sigma$ at site $q$. Our study generalizes the methods available for random walks. We are able to give in many settings (square integrable jumps, jumps in the basin of a L\'evy or Cauchy random variable) asymptotics for the transition matrix $(P_{pq})_{p, q \in \Sigma}$ when the elements of $\Sigma$ are far apart. We use three main tools: a variant of the balayage identity using a transfer operator as a Markov transition kernel, a study inspired from fast-slow systems and the hitting time of small sets in hyperbolic systems to relate transfer operators and the transition matrices we seek to compute, and finally Fourier transform and perturbations of transfer operators \textit{\`a la} Nagaev-Guivarc'h to effectively compute these transition matrices in an asymptotic regime.