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This work focuses on a class of stochastic damping Hamiltonian systems withstate-dependent switching, where the switching process has a countably infinitestate space. After establishing the existence and uniqueness of a global weaksolution via the martingale approach under very mild conditions, the paper nextproves the strong Feller property for regime-switching stochastic dampingHamiltonian systems by the killing technique together with some resolvent andtransition probability identities. The commonly used continuity assumption forthe switching rates $q_{kl}(\cdot)$ in the literature is relaxed tomeasurability in this paper. Finally the paper provides sufficient conditionsfor exponential ergodicity and large deviations principle for regime-switchingstochastic damping Hamiltonian systems. Several examples on regime-switchingvan der Pol and (overdamped) Langevin systems are studied in detail forillustration.

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