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Existence of almost periodic solutions to some stochastic differential equations. (English) Zbl 1130.34033
The concept of $$p$$th-mean almost periodicity for Banach-space-valued stochastic processes is studied. Some preliminary results are applied to verify existence and uniqueness of mean-square almost periodic mild solutions for semilinear stochastic evolution equations $dX(t) = [ A X(t) + F(t,X(t)) ] dt + G(t,X(t)) dW(t)$ with mean-square periodic, Lipschitz-continuous nonlinearity $$F$$ and driven by Brownian motion $$W$$. For this purpose, they make use of well-known Banach’s fixed point principle.

##### MSC:
 34F05 Ordinary differential equations and systems with randomness 34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations 35B15 Almost and pseudo-almost periodic solutions to PDEs 37L55 Infinite-dimensional random dynamical systems; stochastic equations 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60H20 Stochastic integral equations 60H25 Random operators and equations (aspects of stochastic analysis)
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