We establish results concerning the global existence, uniqueness, approximate and exact controllability of mild solutions for a class of abstract second-order stochastic evolution equations in a real separable Hilbert space in which we allow the nonlinearities at a given time t to depend not only on the state of the solution at time t, but also on the corresponding probability distribution at time t. First-order equations of McKean-Vlasov type were first analyzed in the finite dimensional setting when studying diffusion processes, and then subsequently extended to the Hilbert space setting. The current manuscript provides a formulation of such results for second-order problems. Examples illustrating the applicability of the general theory are also provided.
Stochastic Analysis and Applications
Mahmudov, N. I., & McKibben, M. A. (2006). Abstract second-order damped McKean-Vlasov stochastic evolution equations. Stochastic Analysis and Applications, 24(2), 303-328. Retrieved from http://digitalcommons.wcupa.edu/math_facpub/3