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

Maslowski, Bohdan

Stochastics Stochastics Rep. 28, No. 2, 85-114 (1989).

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

MSC:

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

Keywords:

evolution equation; uniqueness; stochastic differential equation in Hilbert space; attractivity; stability; invariant measures; strong attractivity

Citations:

Zbl 0497.93055; Zbl 0538.60068