Existence, Uniqueness And Asymptotic Behavior Of Solutions To Stochastic Partial Functional Differential Equations With Infinite Delay

In this paper, we shall consider the existence and uniqueness and asymptotic behavior of mild solutions to stochastic partial functional differential equations with infinite delay in L~P(Ω, C_h~p) space (p > 2) :dX(t) = [-AX(t)+f(t,X_t)]dt + g(t,X_t)dW(t),where we assume -A is a closed, densely defined linear operator and the generator of a certain analytic semigroup, and / : R~+ × C_α~h→ H, g : R~+ ×C_α~h → (K,H) are two locally Lipschitz continuous functions. Here C_α~h = C(R~-,D(A~_α)) and (K, H) are two proper infinite dimensional spaces, 0 < α < 1, W(t) is a given K-value Wiener process, and both H and K are separable Hilbert spaces.This paper is composed of two parts. In the first chapter, we introduce the historical background of the problems which will be investigated and the main results of this paper. Also , we will state some preminary results which will be used in our paper. In the second chapter, in Section 2.1, the Banach spaces C_h~p and L~p(Ω, C_h~p) are studied which is fundamental for the subsequent developments. In Section 2.2, we shall discuss the existence and the uniqueness of solutions to stochastic partial functional equations with infinite delay . In Section 2.3, we devote to the study of p-th moment and almost sure Lyapunov exponential stability properties of mild solutions by using an estimate for stochastic convolution (see Lemma 2.3.1 below). Finally, we shall present in Section 2.4 some applications about Volterra stochastic integro-differential equation with infinite delay,In addition, we shall present an example for Volterra stochastic integro-differential reaction-diffusion equation which illustrates our main theorems.