Favard separation method is an important means to study almost periodic solutions to linear differential equations; later, Amerio applied Favards idea to nonlinear differential equations. In this paper, by appropriate choosing separation and almost p
eriodicity in distribution sense, we obtain the Favard and Amerio type theorems for stochastic differential equations.
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 square-mean sense is discussed.
In the first part of this paper, we generalize the results of the author cite{Liu,Liu2} from the random flow case to the random semiflow case, i.e. we obtain Conley decomposition theorem for infinite dimensional random dynamical systems. In the secon
d part, by introducing the backward orbit for random semiflow, we are able to decompose invariant random compact set (e.g. global random attractor) into random Morse sets and connecting orbits between them, which generalizes the Morse decomposition of invariant sets originated from Conley cite{Con} to the random semiflow setting and gives the positive answer to an open problem put forward by Caraballo and Langa cite{CL}.
