Almost automorphy and various extensions for stochastic processes

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

We compare different modes of pseudo almost automorphy and variants for stochastic processes: in probability, in quadratic mean, or in distribution in various senses. We show by a counterexample that square-mean (pseudo) almost automorphy is a property which is too strong for stochastic differential equations (SDEs). Finally, we consider two semilinear SDEs, one with almost automorphic coefficients and the second one with pseudo almost automorphic coefficients, and we prove the existence and uniqueness of a mild solution which is almost automorphic in distribution in the first case, and pseudo almost automorphic in distribution in the second case.

Mots clés: pseudo almost automorphic in distribution; stochastic evolution equation; semilinear stochastic differential equation; square-mean almost automorphic; Weighted pseudo almost automorphic; Besicovitch; Weyl; pseudo almost automorphic in quadratic mean; Ornstein-Uhlenbeck; Stepanov

Domaines: Probabilités [math.PR]; Systèmes dynamiques [math.DS]

Fazia Bedouhene, Nouredine Challali, Omar Mellah, Paul Raynaud de Fitte, Mannal Smaali. Almost automorphy and various extensions for stochastic processes. Journal of Mathematical Analysis and Applications, 2015, 429 (2), pp.1113-1152. ⟨10.1016/j.jmaa.2015.04.014⟩. ⟨hal-01081794v3⟩