  Motivated by networked systems in random environment and controlled hybrid
stochastic dynamic systems, this work focuses on modeling and analysis of a
class of switching diffusions consisting of continuous and discrete components.
Novel features of the models include the discrete component taking values in a
countably infinite set, and the switching depending on the value of the
continuous component involving past history. In this work, the existence and
uniqueness of solutions of the associated stochastic differential equations are
obtained. In addition, Markov and Feller properties of a function-valued
stochastic process associated with the hybrid diffusion are also proved. In
particular, when the switching rates depend only on the current state, strong
Feller properties are obtained. These properties will pave a way for future
study of control design and optimization of such dynamic systems.
