Stability of Randomly Switched Diffusions

This paper provides a sufficient criterion for ε-moment stability (boundedness) and ergodicity for a class of systems comprising a finite set of diffusions among which switching is governed by a continuous time Markov chain. Stability/instability properties for each separate subsystem are assumed to be quantified by a Lyapunov function candidate and an associated growth rate equation. For the set of Lyapunov functions a compatibility criterion is assumed to be fulfilled bounding the ratio between pairs of Lyapunov functions. The established criterion is shown to be equivalent to an exact criterion for the almost sure convergence of an associated process bounding moments of the process under study. Examples are provided to illustrate the use of the established criterion.