Regularity of invariant densities for 1D-systems with random switching

TL;DR

This paper analyzes invariant measures for 1D dynamical systems with random switching, proving smoothness away from critical points and detailing the behavior at critical points.

Contribution

It provides new results on the regularity and asymptotic behavior of invariant densities in systems with random switching.

Findings

Invariant densities are smooth away from critical points.

Asymptotic behavior of densities at critical points is characterized.

The analysis enhances understanding of stochastic stability in 1D systems.

Abstract

This is a detailed analysis of invariant measures for one-dimensional dynamical systems with random switching. In particular, we prove smoothness of the invariant densities away from critical points and describe the asymptotics of the invariant densities at critical points.