  Focusing on stochastic systems arising in mean-field models, the systems
under consideration belong to the class of switching diffusions, in which
continuous dynamics and discrete events coexist and interact. The discrete
events are modeled by a continuous-time Markov chain. Different from the usual
switching diffusions, the systems include mean-field interactions. Our effort
is devoted to obtaining laws of large numbers for the underlying systems. One
of the distinct features of the paper is the limit of the empirical measures is
not deterministic but a random measure depending on the history of the
Markovian switching process. A main difficulty is that the standard martingale
approach cannot be used to characterize the limit because of the coupling due
to the random switching process. In this paper, in contrast to the classical
approach, the limit is characterized as the conditional distribution (given the
history of the switching process) of the solution to a stochastic McKean-Vlasov
differential equation with Markovian switching.
