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

TL;DR

This paper establishes a strong averaging principle for foliated Le9vy diffusions with heavy tails on compact leaves, extending previous results to cases with unbounded perturbations and polynomial moments.

Contribution

It extends the averaging results for heavy-tailed Le9vy diffusions to include unbounded deterministic and stochastic perturbations with polynomial moments.

Findings

Averaging principle proven for heavy-tailed Le9vy diffusions on foliated manifolds.

Extension from exponential to polynomial moments with unbounded coefficients.

Detailed example of Le9vy rotations on the circle with perturbations.

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.