Almost periodic solution in distribution for stochastic differential equations with Stepanov almost periodic coefficients

Fazia Bedouhene (1) , Nouredine Challali (1) , Omar Mellah (1) , Paul Raynaud de Fitte (2) , Mannal Smaali (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées [Tizi-Ouzou]
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

This paper deals with the existence and uniqueness of (µ-pseudo) almost periodic mild solution to some evolution equations with Stepanov (µ-pseudo) almost periodic coefficients, in both determinist and stochastic cases. After revisiting some known concepts and properties of Stepanov (µ-pseudo) almost periodicity in complete metric space, we consider a semilinear stochastic evolution equation on a Hilbert separable space with Stepanov (µ-pseudo) almost periodic coefficients. We show existence and uniqueness of the mild solution which is (µ-pseudo) almost periodic in 2-distribution. We also generalize a result by Andres and Pennequin, according to which there is no purely Stepanov almost periodic solutions to differential equations with Stepanov almost periodic coefficients.

Mots clés: Weighted pseudo almost periodic; Stepanov almost periodic; Stochastic evolution equations; Pseudo almost periodic in 2-distribution; Pseudo almost periodic in 2-UI distribution

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]

Fazia Bedouhene, Nouredine Challali, Omar Mellah, Paul Raynaud de Fitte, Mannal Smaali. Almost periodic solution in distribution for stochastic differential equations with Stepanov almost periodic coefficients. 2017. ⟨hal-01478199v3⟩