On time-dependent stochastic evolution equations driven by fractional Brownian motion in a Hilbert space with finite delay (Q2922248)

The aim of the paper is to prove the existence and uniqueness of mild solutions to the fractional stochastic evolution equation NEWLINE\[NEWLINEdx(t)=[A(t)x(t) + f(t, x(t- r(t)))]dt + g(t, x(t- u(t)))dW(t) + \sigma (t)dB_Q^H(t)NEWLINE\]NEWLINE with initial data \(x(t)=\phi(t)\) for \(t\in [-\tau, 0]\) on a real separable Hilbert space \(H\). Here, \(A(t): D\subseteq H\to H\) is a family of densely defined closed linear operators generating a strong evolution operator \(U(t,s)\) with \(\|U(t,s)\|\leq Me^{-\delta(t-s)}\) for all \(0\leq s\leq t \leq T\), where \(M\) and \(\delta\) are positive costants. Moreover, \(f:[0,T]\times H\to H\) and \(g:[0,T]\times H\to L_Q(K, H)\) are measurable functions satisfying Lipschitz and linear growth conditions, and \(r, u:[0,+\infty)\to [0,\tau)\) are time delay continuous functions. The authors apply Banach's fixed point theorem to prove the main result of the paper. At last, an example is given to illustrate the applications of the theory developed in this paper.