On Interacting Systems of Hilbert-Space-Valued Diffusions

Abstract

A nonlinear Hilbert-space-valued stochastic differential equation where L -1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L -1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L -1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ 0 of the martingale problem posed by the corresponding McKean—Vlasov equation.