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In this paper, we intend to revisit and expand the concepts of Muldowney genre properties for the case of evolution cocycles, set in a nonuniform frame. Associations with the classic notions of asymptotic behaviours are also provided. Specific examples are used to highlight the approach. [ABSTRACT FROM AUTHOR]

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]