Almost periodicity and periodicity for nonautonomous random dynamical systems

Abstract

We present a notion of almost periodicity which can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept allows for improvements of known results of almost periodicity in distribution, for general random processes and for solutions to stochastic differential equations.

Pseudo almost periodicity for stochastic differential equations in infinite dimensions

“…Definition 2.6. [15] Let X : R × Ω → B be a stochastic process. Assume that X(t) ∈ L p (Ω, B) for every t ∈ R. We say that X is p-mean θ-almost periodic (or simply θ p -almost periodic) if conditions (i) and (ii) below are satisfied:…”

“…where A : D(A) ⊂ H → H is a linear operator, F : R × H → H, and G : R × H → L(H) are continuous. Motivated by a recent work [15], where Raynaud de Fitte proposed a new method to study almost periodicity of equation (1.1), we introduce the concept of p-mean θpseudo almost periodicity and obtain the existence and uniqueness of square-mean θ-pseudo almost periodic solution to equation (1.1). Moreover, using the maximal inequality of stochastic convolution for semigroups, we show that the solution is also pseudo almost periodic in path distribution.…”

Almost Periodic and Periodic Solutions of Differential Equations Driven by the Fractional Brownian Motion with Statistical Application

“…By Assumption 3.1. ( 4), [23,Corollary 3.4] on the almost periodicity in product spaces, and by [23, Proposition 3.17], ensuring that a continuous process is θ-almost periodic if and only if its translation is an almost periodic map,…”

“…However, almost periodicity in probability or in square mean appeared to be inapplicable to stochastic differential equations, see [4,18]. Recently, a new definition of almost periodicity for stochastic processes has been introduced in Zhang and Zheng [28] and Raynaud de Fitte [23], namely θ-almost periodicity, where θ is the Wiener shift. One motivation of [23] was to circumvent the limitations of "plain" almost periodicity in square mean by introducing the action of a group θ of measure preserving transformations on the underlying probality space.…”

Weighted Pseudo-θ-Almost Periodic Sequence and Finite-Time Guaranteed Cost Control for Discrete-Space and Discrete-Time Stochastic Genetic Regulatory Networks with Time Delays

“…On the other hand, the global exponential stability and almost periodic nature of GRNs are significant and necessary dynamical behaviours that have been extensively researched by many authors in the last two decades, see the literature [18,[38][39][40][41][42]. Particularly in stochastic models, the notion of θ-almost periodicity was first introduced in the paper [43] on the basis of semi-flow and metric dynamical system theories, and the existence of θalmost periodicity for several continuous-time stochastic models was investigated [44,45]. First, pseudo-almost periodicity was introduced in the early 1990s by Zhang [46] as a natural extension of classical probability periodicity.…”