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

TL;DR

This paper extends Conley's theorem to infinite-dimensional random semiflows, introducing Morse decompositions for invariant random sets and solving an open problem in the field.

Contribution

It generalizes Conley decomposition to random semiflows and introduces backward orbits for Morse decomposition in this setting.

Findings

Established Conley decomposition for infinite-dimensional random semiflows.

Decomposed invariant random sets into Morse sets and connecting orbits.

Provided a positive answer to an open problem on Morse decomposition in random semiflows.

Abstract

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 second 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}.