Invariant Measures for Hybrid Stochastic Systems

TL;DR

This paper investigates the invariant measures of hybrid stochastic systems with discrete parameter changes, establishing their existence, relationships, and explicit calculations in examples.

Contribution

It introduces methods to prove the existence of invariant measures for embedded systems within hybrid stochastic systems and relates these measures through the flow.

Findings

Invariant measures exist for each embedded system.

Explicit invariant measures are calculated in several examples.

Relationships between measures of different embedded systems are established.

Abstract

In this paper, we seek to understand the behavior of dynamical systems that are perturbed by a parameter that changes discretely in time. If we impose certain conditions, we can study certain embedded systems within a hybrid system as time-homogeneous Markov processes. In particular, we prove the existence of invariant measures for each embedded system and relate the invariant measures for the various systems through the flow. We calculate these invariant measures explicitly in several illustrative examples.