Abstract. In this paper we investigate the existence, uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional Brownian motion B H Q (t):with Hurst parameter H ∈ (1/2, 1). We also consider the existence of weak solutions.

Abstract. We first prove the existence and uniqueness of pullback and random attractors for abstract multi-valued non-autonomous and random dynamical systems. The standard assumption of compactness of these systems can be replaced by the assumption of asymptotic compactness. Then, we apply the abstract theory to handle a random reaction-diffusion equation with memory or delay terms which can be considered on the complete past defined by R − . In particular, we do not assume the uniqueness of solutions of these equations.1. Introduction. The intention of this article is to study the asymptotic behaviour of multi-valued non-autonomous and random dynamical systems. The long-time behaviour of these systems can be expressed by terms like pullback attractor and random attractor. The theories of these attractors are now well established as have been extensively developed over the last one and a half decades (see, e.g. Caraballo et al.

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This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder exponent in (1/2, 1), and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion B H with Hurst parameter H > 1/2. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder continuous functions with values in a Hilbert space V . If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Hölder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing B H as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.1991 Mathematics Subject Classification. Primary: 37L55; Secondary: 60H15, 60G22, 37H05, 35R60.

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