Smooth invariant densities for random switching on the torus

TL;DR

This paper proves that a random switching system on the 2D torus, driven by two smooth vector fields with certain conditions, has a smooth invariant density, using Malliavin calculus techniques.

Contribution

It establishes the existence of smooth invariant densities for switched systems on the torus under broad conditions, extending previous results to stochastic switching scenarios.

Findings

The switched system has a smooth invariant density for all switching rates.

The proof uses an integration by parts formula inspired by Malliavin calculus.

Conditions include transversality and the absence of periodic orbits for the vector fields.

Abstract

We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal to each other at all points of the torus and that each of them allows for a smooth invariant density and no periodic orbits, we prove that the switched system also has a smooth invariant density, for every switching rate. Our approach is based on an integration by parts formula inspired by techniques from Malliavin calculus.