Received September 3, 2023. Published online October 29, 2024.
Abstract: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, where $\mathcal{F}$ is countably generated, and $X$ be a Polish space. Let $\varphi$ be a random dynamical system with time $\mathbb{T}$ on $X$. The skew product flow $ \{\Theta_t , t\in\mathbb{T} \} $ induced by $\varphi$ is a family of continuous operators acting on $\Pr_{\Omega}(X)$, the set of all probability measures on $X\times\Omega$ with marginal $\mathbb{P}$, which is a Polish space equipped with the narrow topology. In this work, we introduce and study the notion of narrow recurrence of the flow $\{\Theta_t, t\in\mathbb{T} \} $ on ${\rm Pr}_{\Omega}(X)$ and we give some results, which can be considered as an initiation of applications of properties of topological dynamics on stochastic process theory and random dynamical systems.
Keywords: hypercyclicity; transitivity; recurrence; the narrow topology; random dynamical system
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Affiliations: Ahmed Zaou (corresponding author), Mohamed Amouch, Chouaib Doukkali University, Faculty of Science, Department of Mathematics, Khalil Jabran Avenue, B.P. 299-24000, El Jadida, 24000, Morocco, e-mail: zaou.a@ucd.ac.ma, amouch.m@ucd.ac.ma
