MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS IN HILBERT SPACES UNDER CARATHEODORY CONDITIONS

ABSTRACT. We establish results concerning the global existence, uniqueness, and controllability of mild solutions for a class of rst-order abstract McKean-Vlasov stochastic evolution equations with variable delay in a real separable Hilbert space. We allow the nonlinearities at a given time t to depend on the probability distribution at time t corresponding to the solution at time t. The results are obtained by imposing a so-called Caratheodory condition on the nonlinearities, which is weaker than the classical Lipschitz condition. Examples illustrating the applicability of the general theory are also provided. AMS (MOS) Subject Classication. 34K30, 34F05, 60H10. 1