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Existence of $$S^2$$-almost periodic solutions to a class of nonautonomous stochastic evolution equations. (English) Zbl 1183.34080
Summary: The paper studies the notion of Stepanov almost periodicity (or $$S^2$$-almost periodicity) for stochastic processes, which is weaker than the notion of quadratic-mean almost periodicity. Next, we make extensive use of the so-called Acquistapace and Terreni conditions to prove the existence and uniqueness of a Stepanov (quadratic-mean) almost periodic solution to a class of nonautonomous stochastic evolution equations on a separable real Hilbert space. Our abstract results will then be applied to study Stepanov (quadratic-mean) almost periodic solutions to a class of $$n$$-dimensional stochastic parabolic partial differential equations.

MSC:
 34G20 Nonlinear differential equations in abstract spaces 34K14 Almost and pseudo-almost periodic solutions to functional-differential equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 35B15 Almost and pseudo-almost periodic solutions to PDEs 34F05 Ordinary differential equations and systems with randomness 35R60 PDEs with randomness, stochastic partial differential equations
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