The random case of Conley's theorem: III. Random semiflow case and Morse decomposition.

In the first part of this paper, we generalize the results of the author (Liu 2006 Nonlinearity 19 277-91, 2007 Nonlinearity 20 1017-30) from the random flow case to the random semiflow case, i.e. we obtain the Conley decomposition theorem for infinite dimensional random dynamical systems. In the second part, by introducing the backward orbit for random semiflow, we are able to decompose the 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 (Conley 1978 Isolated Invariant sets and the Morse Index (Conf. Board Math. Sci. vol 38) (Providence, RI: American Mathematical Society)) to the random semiflow setting and gives a positive answer to an open problem put forward by Caraballo and Langa (Caraballo and Langa 2002 Stochastic Partial Differential Equations and Applications (Trento, 2002) (Lecture Notes in Pure and Applied Mathematics) vol 227 (New York: Dekker) pp 89-104). [ABSTRACT FROM AUTHOR]