Existence of S-almost Periodic Solutions to a Class of Nonautonomous Stochastic Evolution Equations

The paper studies the notion of Stepanov almost periodicity (or S-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 (quadraticmean) almost periodic solutions to a class of n-dimensional stochastic parabolic partial differential equations.