A strong averaging principle for L\'evy diffusions in foliated spaces with unbounded leaves

TL;DR

This paper proves a strong averaging principle for Le9vy diffusions on non-compact foliated spaces, showing convergence of the perturbed system to an averaged system under certain moment and ergodic conditions.

Contribution

It extends the averaging principle to non-compact leaves by establishing $L^p$ convergence using new estimates and handling slower convergence rates.

Findings

Established $L^p$ convergence of the perturbed Le9vy diffusion to the averaged system.

Derived slower convergence rates depending on $p$ and $e9psilon$ due to non-compactness.

Utilized estimates of the Marcus equation increments and nonlinear Gronwall-Bihari bounds.

Abstract

This article extends a strong averaging principle for L\'evy diffusions which live on the leaves of a foliated manifold subject to small transversal L\'evy type perturbation to the case of non-compact leaves. The main result states that the existence of $p$-th moments of the foliated L\'evy diffusion for $p\geq 2$ and an ergodic convergence of its coefficients in $L^p$ implies the strong $L^p$ convergence of the fast perturbed motion on the time scale $t/\epsilon$ to the system driven by the averaged coefficients. In order to compensate the non-compactness of the leaves we use an estimate of the dynamical system for each of the increments of the canonical Marcus equation derived in da Costa and Hoegele (2017), the boundedness of the coefficients in $L^p$ and a nonlinear Gronwall-Bihari type estimate. The price for the non-compactness are slower rates of convergence, given as $p$-dependent powers of $\epsilon$ strictly smaller than $1/4$.