Periodic and almost periodic flows of periodic Itô equations. (English) Zbl 0765.60059

Tudor, C.

Math. Bohem. 117, No. 3, 225-238 (1992).

From the introduction: We consider finite-dimensional Itô equations with periodic coefficients. We introduce three stability concepts in \(L^ r\) \((L^ r\)-uniform stability, \(L^ r\)-asymptotic uniform stability and \(L^ r\)-uniform asymptotic stability in the large) which are natural extensions of the corresponding ones from the deterministic situation. Assuming the existence of an \(L^ p\)-bounded solution we prove that it is asymptotically almost periodic in distribution under the uniform stability hypothesis. Moreover, the existence of a trajectory of an almost periodic (periodic) flow defined on the space of all probability measures is also proved under the uniform asymptotic stability (uniform asymptotic stability in the large) assumption.

Reviewer: A.Ya.Dorogovtsev (Kiev)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20	Stochastic integral equations

Keywords:

trajectory of an almost periodic flow; uniform asymptotic stability; Itô equations