Regularity of invariant densities for random switching between two linear odes in Rd

TL;DR

This paper generalizes previous 2D results on invariant densities for randomly switched linear systems to higher dimensions, providing conditions for absolute continuity and smoothness of the invariant distribution.

Contribution

It extends the analysis of invariant densities to Rd, offering the first non-elliptic example with proven smoothness conditions in dimensions greater than 3.

Findings

Invariant distribution is absolutely continuous under certain conditions.

Provides the first non-elliptic example with smooth invariant density in higher dimensions.

Establishes sufficient conditions for the regularity of invariant densities.

Abstract

In a paper entitled singularities of invariant densities for random switching between two linear odes in 2D, Bakhtin et al [5], consider a Markov process obtained by random switching between two stable linear vector fields in the plane and characterize the singularities of the invariant density in terms of the switching and contraction rates. This paper considers a generalization of this model obtained by random switching between two stable linear vector fields in Rd and provides sufficient conditions ensuring that the invariant distribution is absolutely continuous and has a Cr density. In dimension greater than 3 it provides, to the best of our knowledge, the first fully non-elliptic example of random switching for which quantitative conditions guaranteeing smoothness of the invariant density can be proved.