On Strong Feller Property, Exponential Ergodicity and Large Deviations Principle for Stochastic Damping Hamiltonian Systems with State-Dependent Switching

TL;DR

This paper investigates stochastic damping Hamiltonian systems with state-dependent switching, establishing key properties like the strong Feller property, exponential ergodicity, and large deviations, with relaxed assumptions on switching rates.

Contribution

It proves the strong Feller property and exponential ergodicity for regime-switching stochastic damping Hamiltonian systems under mild conditions, extending existing results to measurable switching rates.

Findings

Established existence and uniqueness of global weak solutions.

Proved the strong Feller property using killing technique.

Provided conditions for exponential ergodicity and large deviations.

Abstract

This work focuses on a class of stochastic damping Hamiltonian systems with state-dependent switching, where the switching process has a countably infinite state space. After establishing the existence and uniqueness of a global weak solution via the martingale approach under very mild conditions, the paper next proves the strong Feller property for regime-switching stochastic damping Hamiltonian systems by the killing technique together with some resolvent and transition probability identities. The commonly used continuity assumption for the switching rates $q_{kl}(\cdot)$ in the literature is relaxed to measurability in this paper. Finally the paper provides sufficient conditions for exponential ergodicity and large deviations principle for regime-switching stochastic damping Hamiltonian systems. Several examples on regime-switching van der Pol and (overdamped) Langevin systems are studied in detail for illustration.