Well-posedness and ergodicity of functional stochastic partial differential equations with Markovian switching

This work focuses on a class of semi-linear functional stochastic partial differential equations with Markovian switching, in which the switching component may have finite or countably infinite states. The well-posedness of the underlying process is obtained by Skorokhod's representation of the switching component. Then, the exponential mixing of such processes in a finite state space is derived by using the so-called remote start method proposed firstly by Da Prato and Zabczyk Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series Vol. 229, Cambridge University Press, Cambridge, 1996. Finally, the corresponding result in a countable infinite state space is further obtained via the finite partition method.