Invariant densities for dynamical systems with random switching

TL;DR

This paper investigates the existence and uniqueness of invariant measures for a class of random dynamical systems generated by differential equations with randomly switching vector fields, establishing conditions for absolute continuity.

Contribution

It demonstrates that hypoellipticity conditions ensure uniqueness and absolute continuity of invariant measures in systems with random switching.

Findings

Hypoellipticity guarantees unique invariant measures.

Invariant measures are absolutely continuous under certain conditions.

Results apply to non-autonomous differential equations with random switching.

Abstract

We consider a non-autonomous ordinary differential equation on a smooth manifold, with right-hand side that randomly switches between the elements of a finite family of smooth vector fields. For the resulting random dynamical system, we show that H\"ormander type hypoellipticity conditions are sufficient for uniqueness and absolute continuity of an invariant measure.