Bochner-almost periodicity for stochastic processes

Fazia Bedouhene (1) , Omar Mellah (1, 2) , Paul Raynaud de Fitte (2)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées
2 LMRS - Laboratoire de Mathématiques Raphaël Salem

We compare several notions of almost periodicity for continuous processes defined on the time interval $I=R$ or $I=[0,+\infty)$ with values in a separable Banach space $E$ (or more generally a separable completely regular topological space): almost periodicity in distribution, in probability, in quadratic mean, almost sure almost periodicity, almost sure equi-almost periodicity. In the deterministic case, all these notions reduce to Bochner-almost periodicity, which is equivalent to Bohr-almost periodicity when $I=R$, and to asymptotic Bohr-almost periodicity when $I=[0,+\infty)$.

Mots clés: Almost periodic; Stochastic process; Bochner; Bohr

Domaines: Probabilités [math.PR]

Fazia Bedouhene, Omar Mellah, Paul Raynaud de Fitte. Bochner-almost periodicity for stochastic processes. Stochastic Analysis and Applications, 2012, 30 (2), pp.322-342. ⟨10.1080/07362994.2012.649628⟩. ⟨hal-00566659v2⟩