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Representation theorem for stochastic differential equations in Hilbert spaces and its applications. (English) Zbl 1118.37036

Summary: In this survey we recall the results obtained in [the author, Electron. J. Qual. Theory Differ. Equ. 2004, Paper No. 4, 1–22, electronic only (2004; Zbl 1072.60047)] where we gave a representation theorem for the solutions of stochastic differential equations in Hilbert spaces. Using this representation theorem we obtain deterministic characterizations of exponential stability and uniform observability in [loc. cit.] and [the author, Operator theory: Advances and Applications 153, 287–306 (2005; Zbl 1062.60064)] and we will prove a result of Datko type concerning the exponential dichotomy of stochastic equations.

MSC:

37L55 Infinite-dimensional random dynamical systems; stochastic equations
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C12 Foliations (differential geometric aspects)
51H25 Geometries with differentiable structure
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35B40 Asymptotic behavior of solutions to PDEs
93B07 Observability
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