  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.
