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Existence and exponential stability of almost automorphic mild solutions for stochastic functional differential equations. (English) Zbl 1221.60078

Summary: The concept of \(p\)-mean almost automorphy for stochastic processes is firstly introduced. Some properties of the \(p\)-mean almost automorphic stochastic processes are further studied. Based on these properties, a class of stochastic functional differential equations given by
\[ dx(t) = (Ax(t) + F(t, x(t), x_t))dt + g(t, x(t), x_t) \circ dW(t) \]
is investigated. Under suitable assumptions, existence, uniqueness and exponential stability of quadratic-mean almost automorphic mild solutions to the equations are discussed by means of semigroups of operators and by the Banach contraction principle. Moreover, the quadratic-mean almost automorphic mild solutions to semilinear stochastic partial functional differential equations, which is illustrated by an example, are in good agreement with the theoretical analysis.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E15 Stochastic stability in control theory
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