Abstract
The paper studies the notion of Stepanov almost periodicity (or S2-almost periodicity) for stochastic processes, which is weaker than the no- tion 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.
Highlights
Let (H, ·, ·, · ) be a separable real Hilbert space and let (Ω, F, P) be a complete probability space equipped with a normal filtration {Ft : t ∈ R}, that is, a rightcontinuous, increasing family of sub σ-algebras of F
The second sources is a paper Bezandry and Diagana [3], in which the authors made extensive use of the almost periodicity to study the existence and uniqueness of a quadratic-mean almost periodic solution to the class of nonautonomous semilinear stochastic evolution equations (1.2) dX(t) = A(t)X(t) dt + F (t, X(t)) dt + G(t, X(t)) dW (t), t ∈ R, where A(t) for t ∈ R is a family of densely defined closed linear operators satisfying the so-called Acquistapace and Terreni conditions [1], F : R×L2(P; H) → L2(P; H), G : R × L2(P; H) → L2(P; L02) are jointly continuous satisfying some additional conditions, and W (t) is a Wiener process
Note that the above-mentioned Acquistapace and Terreni conditions do guarantee the existence of an evolution family associated with A(t)
Summary
Let (H, · , ·, · ) be a separable real Hilbert space and let (Ω, F , P) be a complete probability space equipped with a normal filtration {Ft : t ∈ R}, that is, a rightcontinuous, increasing family of sub σ-algebras of F. The second sources is a paper Bezandry and Diagana [3], in which the authors made extensive use of the almost periodicity to study the existence and uniqueness of a quadratic-mean almost periodic solution to the class of nonautonomous semilinear stochastic evolution equations (1.2) dX(t) = A(t)X(t) dt + F (t, X(t)) dt + G(t, X(t)) dW (t), t ∈ R, where A(t) for t ∈ R is a family of densely defined closed linear operators satisfying the so-called Acquistapace and Terreni conditions [1], F : R×L2(P; H) → L2(P; H), G : R × L2(P; H) → L2(P; L02) are jointly continuous satisfying some additional conditions, and W (t) is a Wiener process. An example is given to illustrate our main results
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