Abstract. The concept of square-mean almost automorphy for stochastic processes is introduced. The existence and uniqueness of square-mean almost automorphic solutions to some linear and non-linear stochastic differential equations are established provided the coefficients satisfy some conditions. The asymptotic stability of the unique square-mean almost automorphic solution in the square-mean sense is discussed.

The well-known Conley's theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow ϕ on the compact metric space X, i.e. X − CR(ϕ) = [B(A) − A], where CR(ϕ) denotes the chain recurrent set of ϕ, A stands for an attractor and B(A) is the basin determined by A. In this paper we show that by appropriately selecting the definition of random attractor, in fact we define a random local attractor to be the ω-limit set of some random pre-attractor surrounding it, and by considering appropriate measurability, in fact we also consider the universal σ-algebra F umeasurability besides F -measurability, we are able to obtain the random case of Conley's theorem.

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