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We study pullback attractors of non-autonomous non-compact dynamical systemsgenerated by differential equations with non-autonomous deterministic as wellas stochastic forcing terms. We first introduce the concepts of pullbackattractors and asymptotic compactness for such systems. We then prove asufficient and necessary condition for existence of pullback attractors. Wealso introduce the concept of complete orbits for this sort of systems and usethese special solutions to characterize the structures of pullback attractors.For random systems containing periodic deterministic forcing terms, we show thepullback attractors are also periodic. As an application of the abstracttheory, we prove the existence of a unique pullback attractor forReaction-Diffusion equations on $\R^n$ with both deterministic and randomexternal terms. Since Sobolev embeddings are not compact on unbounded domains,the uniform estimates on the tails of solutions are employed to establish theasymptotic compactness of solutions.

Sufficient and Necessary Criteria for Existence of Pullback Attractors for Non-compact Random Dynamical Systems