Well-posedness and Feller property for functional stochastic Hamiltonian systems with singular coefficients and state-dependent switching

This work focuses on a class of functional stochastic Hamiltonian systems with singular coefficients and state-dependent switching, in which the switching process has a countably infinite state space. First, by Girsanov's transformation, we establish the martingale solution of the system in each fixed switching case. Then, since the two components of the system are intertwined and correlated, we investigate the special case when the discrete component is independent of the continuous component. Based on these results, we obtain the well-posedness of the system with the aid of a martingale process. Finally, in order to establish the Feller property, we take advantage of appropriate Radon-Nikodym derivative and introduce a vector valued elliptic equation to handle the effect of the singular coefficients.