Averaging along foliated L\'evy diffusions

TL;DR

This paper investigates the behavior of solutions to stochastic differential equations driven by discontinuous Lévy processes on foliated manifolds, showing that small transversal perturbations lead to convergence towards a deterministic ODE on average.

Contribution

It extends existing results by analyzing Lévy-driven SDEs of Marcus type on foliated manifolds, providing convergence rates and illustrating with circle rotations.

Findings

Transversal component converges uniformly to a deterministic ODE as perturbation vanishes.

Average of the perturbing vector field determines the limiting dynamics.

Provides bounds for convergence rates and applies to random circle rotations.

Abstract

This article studies the dynamics of the strong solution of a SDE driven by a discontinuous L\'evy process taking values in a smooth foliated manifold with compact leaves. It is assumed that it is \textit{foliated} in the sense that its trajectories stay on the leaf of their initial value for all times a.s.. % Such a system is called a \textit{foliated L\'evy diffusion}. Under a generic ergodicity assumption for each leaf, % and a continuous variation of the ergodic measures among each others, we determine the effective behaviour of the system subject to a small smooth perturbation of order $\e>0$, which acts transversal to the leaves. The main result states that, on average, the transversal component of the perturbed SDE converges uniformly to the solution of a deterministic ODE as $\e$ tends to zero. This transversal ODE is generated by the average of the perturbing vector field with respect to the invariant measures of the unperturbed system and varies with the transversal height of the leaves. We give upper bounds for the rates of convergence % The right-hand side of this ODE is given as the average of the perturbing vector field % with respect to the unique invariant measures of the unperturbed system on the leaves. % We give upper bounds for the rates of convergence. and illustrate these results for the random rotations on the circle. This article %which are proved for pure jump L\'evy processes complements the results by Gargate and Ruffino for SDEs of Stratonovich type to general L\'evy driven SDEs of Marcus type.