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

TL;DR

This paper investigates hitting probabilities in $bZ^d$-extensions of Gibbs-Markov maps, providing asymptotic estimates for transition matrices in various settings using advanced analytical tools.

Contribution

It generalizes methods for random walks to Gibbs-Markov extensions, offering new asymptotic estimates for transition probabilities in complex dynamical systems.

Findings

Asymptotic formulas for transition matrices when points are far apart.

Extension of random walk techniques to Gibbs-Markov map extensions.

Application of Fourier and perturbation methods to compute hitting probabilities.

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.