Topological properties of solution sets for stochastic evolution inclusions

ABSTRACT

In this paper, we investigate the topological structure of solution sets for stochastic evolution inclusions in Hilbert spaces when the semigroup is compact as well as noncompact. It is shown that the solution set is nonempty, compact, and an R_δ-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As applications of the obtained results, an example is given.

KEYWORDS:

• Stochastic evolution inclusions
• Brownian motions
• multimaps
• weak topology
• compact R_δ-set

MATHEMATICS SUBJECT CLASSIFICATION:

• 34G25
• 34F05
• 35R60
• 37L55
• 60H10
• 60H15