Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

Abstract

In this paper we shall consider the existence, uniqueness, and asymptotic behavior of mild solutions to stochastic partial functional differential equations with finite delay r>0:dX(t)=[−AX(t)+f(t, Xt)]dt+g(t, Xt)dW(t), where we assume that −A is a closed, densely defined linear operator and the generator of a certain analytic semigroup. f:(−∞, +∞)×Cα→H, g:(−∞, +∞)×Cα→L02(K, H) are two locally Lipschitz continuous functions, where Cα=C([−r, 0], D(Aα)), L02(K, H) are two proper infinite dimensional spaces, 0<α<1. Here, W(t) is a given K-valued Wiener process and both H and K are separable Hilbert spaces.