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The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of squaremean almost periodic solutions to some semilinear stochastic equations.

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In this paper we introduce and analyze an important class of (asymptotically) Stepanov almost periodic functions in the Lebesgue spaces with variable exponents, which generalizes in a natural fashion all the (asymptotically) almost periodic functions. We then make extensive use of these new functions to study some abstract Volterra integro-differential equations in Banach spaces including multi-valued ones.