The Existence Of Random Attractor For Stochastic Partial Functional Differential Equations Driven By The Fractional Brownian Motion

In a Banach space H, we investigate the following stochastic partial functional differ-ential equations driven by the fractional Brownian motion dx{t)= Ax(t)dt+f(xt)dt+dBH(t). Where A is a non-densely defined analytic linear operator, f(xt) is the term of time delay, BH is the fractional Brownian motion with Hurst parameter H ∈(1/2|,1).In part 1, we first recall some basic definitions and properties about fractional Brownian motion. Then, we present the definition of an integral concept with respect to fractional Brownian motion. Last, we construct the notion of the metric dynamical system.In part 2, we introduce the basic concepts and properties of the stochastic dynamical system. Then we give a sufficient condition to guarantee the existence of the random attr actor.In part 3, first we present some definitions and theorems for C0-semigroup and analytic semigroup theory. Then, we use the Dumford integral to define a semigroup T(t), which is generated by the non-densely defined linear operator A. Next, we use the semigroup T(t) to construct the mild solutions of the above equations. Finally, we use the mild solution to define the stochastic dynamical systems φ.In part 4, we prove the existence of the random absorbing set and the asymptotic compactness for φ, then we use them to prove the existence of the random attractor.