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The real world phenomena are not always modeled by systems of deterministic differential equations, hence their ap-proach has to combine the classic study with methods of stochastic analysis. A non-trivial aspect refers to analytical tools from the theory of evolution equations, such as the cocycles approach, in order to study the existence problem and the long-time evolution for stochastic equations. The purpose of this study is to present some trichotomic behaviours in mean square for stochastic evo-lution cocycles, underlined by examples, characterizations, as well as connections between them. The classic instruments used to characterize asymptotic behaviours as stability, instability or dichotomy are generalized for the trichotomy property. Our study is conducted as an extension of the techniques from the deterministic framework for stochastic evolution cocycles on Hilbert spaces. [ABSTRACT FROM AUTHOR]

The exponential stability plays a central role in the theory of asymptotic behaviors for evolution equations. The aim of this paper is to define, to illustrate by examples and to characterize several general concepts of stability for skew-evolution semiflows on Banach spaces [ABSTRACT FROM AUTHOR]

In this paper, we intend to generalize the notion of exponential dichotomy to evolution equations with solutions that we only presume to exist in the future. Hence, we construct al framework for the exponential splitting that includes, as a particular case, the exponential dichotomy. The exponential splitting appears to be stronger than the exponential dichotomy, but, under adequate conditions, the notions are equivalent. [ABSTRACT FROM AUTHOR]