Compactness In Lebesgue-bochner Spaces Of Random Variables And The Existence Of Mean-square Random Attractors

Let(?,F,P)be a probability space and let X be a Banach space.It is shown a subset V of Lp((?,F,P);X),where 1?p<?,is relatively compact in Lp((?,F,P);X)if and only if it is uniformly Lp-integrable and uniformly tight.The additional condition of scalarly relatively compact required in the literature is shown to hold by a probabilistic argument.The result is then used to establish the existence of a mean-square random attractor for dissipative stochastic differ-ential equations,stochastic parabolic partial differential equations and stochastic functional differential equations.