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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]

The aim of this paper is to emphasize some splitting asymptotic behaviours in mean square for stochastic cocycles, as well as connections between them. We define a general framework for the exponential splitting that includes, as a particular case, the exponential dichotomy. We will consider general splitting properties that consist in assuming the existence of a decomposition of the state space into invariant subspaces where the norms of the evolution trajectories are bounded by some functions that depend on the initial and final times. The additional conditions target the families of projections. [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]

The aim of this paper is to develop a framework for a unitary approach in a nonuniform setting for various concepts of dichotomy for skew-evolution semiflows on infinite dimensional spaces. The techniques that describe the stability and instability in infinite dimensional spaces are improved to characterize the property of dichotomy. Beside the classic concepts of uniform exponential dichotomy and exponential dichotomy, we propose also the notions of Barreira-Valls exponential dichotomy and (h, k)–dichotomy. Connections between notions are pointed out, several examples and counterexamples underlining the statements. [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]