Uniqueness and stability of invariant measures for stochastic differential equations in Hilbert spaces. (English) Zbl 0683.60037

The author investigates the asymptotic behavior of solutions for the stochastic differential equation in Hilbert space \[ (*)\quad d\xi_ t=(A\xi_ t+f(\xi_ t))dt+\Phi (\xi_ t)dw_ t \] for which the related notations and some fundamental results can be found in A. Ichikawa, J. Math. Anal. Appl. 90, No.1, 12-44 (1982; Zbl 0497.93055) and Stochastics 12, No.1, 1-39 (1984; Zbl 0538.60068). The particular results of interest are those related to the sufficient and/or necessary conditions for attractivity and stability of the system \(\{\) \({\mathcal S}_ t\}\), which is generated by the mild solutions of (*), as well as to uniqueness of invariant measures and strong attractivity of \(\{\) \({\mathcal S}_ t\}\).
Reviewer: Chengxun Wu


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
34D20 Stability of solutions to ordinary differential equations
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