Strong averaging along foliated L\'evy diffusions with heavy tails on compact leaves

Abstract

This article shows a strong averaging principle for diffusions driven by discontinuous heavy-tailed L\'evy noise, which are invariant on the compact horizontal leaves of a foliated manifold subject to small transversal random perturbations. We extend a result for such diffusions with exponential moments and bounded, deterministic perturbations to diffusions with polynomial moments of order $p\geq 2$, perturbed by deterministic and stochastic integrals with unbounded coefficients and polynomial moments. The main argument relies on a result of the dynamical system for each individual jump increments of the corresponding canonical Marcus equation. The example of L\'evy rotations on the unit circle subject to perturbations by a planar L\'evy-Ornstein-Uhlenbeck process is carried out in detail.