Almost sure exponential stability for stochastic partial functional differential equations

In this paper we consider the almost, sure asymptotic behavior of mild solutions of the semilinear stochastic evolution equation with finite delays: dX(t) = [AX(t) + f(t, X-t)]dt + g(t, X-t)dW(t), where f,g have the Lipschitz condition and the linear growth condition. That is, we present the existence theorem, and the estimates of moment and almost sure Lyapunov exponent of the above equation. For illustrating the theorem we discuss a semilinear stochastic heat equation with finite delays.