Simple random walk on Z^2 perturbed on the axis (renewal case)

Pierre Andreoletti (1) , Pierre Debs (1)

1 IDP - Institut Denis Poisson

We study a simple random walk on Z^2 with constraints on the axis. Motivation comes from physics when particles (a gas for example, see [Dal88]) are submitted to a local field. In our case we assume that the particle evolves freely in the cones but when touching the axis a force pushes it back progressively to the origin. The main result proves that this force can be parametrized in such a way that a renewal structure appears in the trajectory of the random walk. This implies the existence of an ergodic result for the parts of the trajectory restricted to the axis.

Mots clés: renewal theorem; ergodic theorem; perturbed random walks

Domaines: Probabilités [math.PR]

Pierre Andreoletti, Pierre Debs. Simple random walk on Z^2 perturbed on the axis (renewal case). 2023. ⟨hal-03816534v2⟩