Abstract
The main purpose of this paper is to study a more general concept of uniform stability in mean in which the uniform behavior in the classical sense is replaced by a weaker requirement with respect to some probability measure. This concept includes, as particular cases, the concepts of uniform exponential stability in mean and uniform polynomial stability in mean. Extending techniques employed in the deterministic case, we obtain variants of some results for the general cases of uniform stability in mean for stochastic skew-evolution semiflows in Banach spaces.
Highlights
Considerable attention has been devoted to the problem of asymptotic behaviors of nonautonomous differential equations in Banach spaces
We note that the stochastic cocycles studied in [3] are particular cases of the concept below
The Main Results A first characterization of uniform exponential stability in mean is given by Theorem 3
Summary
Considerable attention has been devoted to the problem of asymptotic behaviors of nonautonomous differential equations in Banach spaces. Many results can be carried out for differential equations and evolution operators and for skew-evolution semiflows. The notion of skew-evolution semiflow was introduced in [1] and includes some particular cases of many well-known concepts in dynamical system theory, such as C0-semigroups, evolution operators and skew-product semiflows. We consider the case of stochastic skew-evolution semiflows studied in [2]. We note that the stochastic cocycles studied in [3] are particular cases of the concept below. Several important examples of stochastic evolution semiflows give rise to stochastic evolution equations, and the reader can refer to the monographs by [4,5]
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