Multivalued Stochastic McKean-Vlasov Equation

Multivalued stochastic McKean-Vlasov equation is a new problem in stochastic analysis.It is the combination form of classical McKean-Vlasov system and multival-ued stochastic differential equation. In this paper, we concern multivalued Stochastic McKean-Vlasov equation with a maximal monotone operator. When the maximal mono-tone operator is the sub-differential of the indicator of a convex set, multivalued stochastic McKean-Vlasov equation describes the reflected diffusion process.As the combination form seems not to be studied in the literature, we begin with studying the existence and uniqueness of the solution. The full paper mainly is divided into three parts:The first part is introducing some preliminaries ralated to our theme, including the notions and some characterizations of maximal monotone operaters, ralated knowledge of Stochastic differential equation and Stochastic analysis.In the second part, we devote ourselves to the solution of Multivalued Stochastic McKean-Vlasov equation included the existence and uniqueness of stong solutions under globle Lipschitz conditions and the existence of weak solutions under the linear growth condition. When the coefficient function are global Lipschitz continuous, we shall use the standard Picard's iteration to prove the existence and uniqueness of solution to equation. However, if the global Lipschitz assumption is broken, it is hard to solve equation. The main difficulty is that the coefficients globally depends on the solution and the usual stopping time technique of localization can not be used in this case. However, we still can obtain the existence of weak solutions. The key is to use the equivalence of weak solutions and martingale problems, Skorohod's representation theorem and Prohorov theorem on a complete separable metric space.In the third part the focus was mainly on the communication between the solutions of Multivalued Stochastic McKean-Vlasov equation and interaction particle system, and we can conclude that the solutions of interaction particle system converge in distribution to the solution of the original equation under globle Lipschitz conditions.