Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

We study the inheritance of properties of free backward propagators associated with transition probability functions by backward Feynman-Kac propagators corresponding to functions and time-dependent measures from non-autonomous Kato classes. The inheritance of the following properties is discussed: the strong continuity of backward propagators on the space , the -smoothing property of backward propagators, and various generalizations of the Feller property. We also prove that a propagator on a Banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded.