[Paper Review] Almost periodicity and periodicity for nonautonomous random dynamical systems

Paul Raynaud de Fitte|arXiv (Cornell University)|Jan 21, 2020

Nonlinear Differential Equations Analysis29 references4 citations

TL;DR

This paper introduces a new notion of $θ$-almost periodicity for random processes in Polish spaces, unifying almost periodicity in probability and $p$-mean while enabling stronger results for stochastic partial differential equations (SPDEs). It proves that under mild conditions, the unique bounded mild solution to a class of nonautonomous SPDEs in Hilbert spaces is both $θ$-almost periodic and almost periodic in path distribution (APPD), with periodic solutions arising as a special case when coefficients are periodic.

ABSTRACT

We present a notion of almost periodicity wich 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.

Motivation & Objective

• To address inconsistencies in prior claims about square-mean almost periodic solutions to SPDEs, which were shown to stem from invalid Itô integral variable changes.
• To develop a robust, general framework for almost periodicity in random dynamical systems that subsumes existing notions like almost periodicity in distribution and in $p$-mean.
• To establish a connection between metric dynamical systems and the almost periodic behavior of solutions to nonautonomous SPDEs.
• To prove that under suitable conditions, the unique bounded mild solution to a semilinear SPDE is both $θ$-almost periodic and almost periodic in path distribution (APPD).
• To show that periodicity of coefficients implies $θ$-$\tau$-periodicity of the solution, leading to periodicity in finite-dimensional distributions.

Proposed method

• Introduces $θ$-almost periodicity for random processes with values in a Polish space $\mathbb{X}$, defined via the shift $\theta$ on the underlying probability space.
• Uses the metric dynamical system framework with a group of measure-preserving transformations $\theta_t$ to define almost periodicity in the sense of $\theta$.
• Applies the concept to nonautonomous random dynamical systems defined as crude cocycles over metric dynamical systems, allowing for non-perfect and non-autonomous dynamics.
• Establishes a link between $\theta$-almost periodicity and almost periodicity in path distribution (APPD), showing that APPD is not implied by $\theta$-almost periodicity in general, but is implied under continuity and uniform integrability.
• Employs a contraction mapping argument in the space $\mathrm{CUB}(\Omega;\mathbb{H})$ of bounded measurable processes, using the operator $\Gamma$ defined via stochastic convolution.
• Derives a sufficient condition (5.22) involving the spectral gap $\mathfrak{d}$ and Lipschitz constants $\mathfrak{h}, \mathfrak{g}$ for the contraction of $\Gamma$, ensuring existence and uniqueness of the solution.

Experimental results

Research questions

• RQ1 Can a consistent and general notion of almost periodicity be defined for random processes that unifies existing concepts like almost periodicity in distribution and in $p$-mean?
• RQ2 Does $\theta$-almost periodicity imply almost periodicity in path distribution (APPD), and if not, what additional conditions ensure this?
• RQ3 Can the flawed claims about square-mean almost periodic solutions to SPDEs be corrected using a rigorous metric dynamical system approach?
• RQ4 Under what conditions does the unique bounded mild solution to a nonautonomous SPDE in a Hilbert space inherit $\theta$-almost periodicity and APPD?
• RQ5 Can periodicity of the coefficients $F$ and $G$ in the SPDE lead to $\theta$-$\tau$-periodicity and periodicity in finite-dimensional distributions of the solution?

Key findings

• The paper identifies and corrects a fundamental error in prior work on square-mean almost periodic solutions to SPDEs, which relied on invalid changes of variable in Itô integrals.
• It proves that under the condition $2\mathfrak{h}^2(1 + \frac{1}{2\mathfrak{d}}) < 1$, the solution operator $\Gamma$ is a contraction on $\mathrm{CUB}(\Omega;\mathbb{H})$, ensuring existence and uniqueness of a bounded mild solution.
• The unique bounded mild solution $X$ is shown to be $\theta$-almost periodic and almost periodic in path distribution (APPD), under the same contraction condition.
• If the coefficients $F(\cdot,x)$ and $G(\cdot,x)$ are $\tau$-periodic for all $x \in \mathbb{H}$, then the solution $X$ is $\theta$-$\tau$-periodic, implying periodicity in finite-dimensional distributions (PFD).
• The solution is also $\theta$-stationary when $F$ and $G$ are time-independent, showing that stationarity is a special case of $\theta$-almost periodicity.
• A counterexample is provided showing that $\theta$-almost periodicity does not imply APPD in general, but a sufficient condition is given for continuous processes under uniform integrability.