Large Deviation For Multivalued Stochastic Mckean-Vlasov Equation

The multivalued stochastic McKean-Vlasov equation can be regarded as a combi-nation of the classic McKean-Vlasov equation and the multivalued stochastic differential equation.The related research of it is relatively new to us.In this paper,we mainly fo-cuses on using the weak convergence methods recently proposed by Dupuis and Ellis^[9],which is using the equivalence of the large deviation principle and the Laplace princi-ple in Polish space to prove the large deviation principle of the multivalued stochastic McKean-Vlasov equation under the existence and uniqueness of the solution of the multivalued stochastic McKean-Vlasov equation,thereby we can avoid the complex ex-ponential estimation,the multivalued stochastic McKean-Vlasov equation considered here is under the multivalued maximal monotone operator.The first part of this article mainly introduce the multivalued stochastic McKean-Vlasov equation and the forms of the strong solution.First we present the related concepts and properties of the multi-valued maximal monotone operator.It is notices that when the operator is the sub-differentiation of the characteristic function of the convex set,the original equation describes the reflection and diffusion process.then we show some conclusion and assumptions of the strong solution like uniqueness of the solution and the uniqueness of the path and so on.The second part of this article focus on introducing the basic definition of large deviation principle and the weak convergence method.First we show the basic elements of large deviation principle and content of the theorem,which is the equivalence of large deviation principle and Laplace Principle,then we explain the weak convergence method which will be applied in the next proof and list two conditions of the weak convergence criterion.The third part of this paper is the main part of this article which presents sufficient conditions and criteria to establish the large deviation principle of multivalued McKean-Vlasov stochastic differential equation.The proof is divided into two parts according to the two condition criteria given by the weak convergence method.Important methods such as Girsanov's theorem,BDG inequality and Ito formula are mainly used in this part.Since the drift coefficient and the diffusion coefficient are respectively distributed dependent,the Girsanov's theorem may be affected in the proof,so we use the method of distribution solidification^[32]which guarantee the existence and uniqueness of Girsanov transform,thus we solving the proof of two weak convergence conditions.Finally,we obtain the large deviation principle of the multivalued stochastic McKean-Vlasov equation.