A dynamical system in a Hilbert space with a weakly attractive nonstationary point. (English) Zbl 0794.34054

Vrkoč, Ivo

Math. Bohem. 118, No. 4, 401-423 (1993).

In contrast to the title of this nice paper and the author’s choice of the classification number it deals with and is motivated by a problem which is closely related to the question of existence of invariant measures for stochastic evolution equations. Recently, G. Da Prato, D. Gatarek and J. Zabczyk [Stochastic Anal. Appl. 10, 387-408 (1992; Zbl 0758.60057)] proved a theorem, which in the (particular) deterministic case states that the differential equation \((*)\) \(x'(t)= Ax+ f(x)\) in a Hilbert space \(\mathbb{H}\) with \(f\) being Lipschitz continuous in \(\mathbb{H}\) and \(A\) generating a compact semigroup \(e^{At}\) on \(\mathbb{H}\) possesses a (not necessarily unique) invariant measure if at least one bounded mild solution to \((*)\) exists. The author of the paper under consideration shows that the compactness of \(e^{At}\) cannot be omitted. Moreover, he constructs an example indicating that there is no immediate relation between the invariant measure for the Galerkin approximation of an evolution equation and the invariant measure for the equation itself.

Reviewer: R.Manthey (Jena)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34F05	Ordinary differential equations and systems with randomness
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

invariant measures; stochastic evolution equations; Hilbert space; compact semigroup; Galerkin approximation

Citations:

Zbl 0758.60057