[Paper Review] Anticipating Random Periodic Solutions--II. SPDEs with Multiplicative Linear Noise

Chunrong Feng, Yue Wu+1 more|arXiv (Cornell University)|Mar 1, 2018

Advanced Mathematical Modeling in Engineering19 references3 citations

TL;DR

This paper establishes the existence of random periodic solutions for semilinear stochastic partial differential equations (SPDEs) with multiplicative linear noise on bounded domains using coupled forward-backward infinite horizon stochastic integral equations in $L^2(\mathcal{O})$. By employing generalized Schauder's fixed point theorem, relative compactness in Wiener-Sobolev spaces, and a localization argument, the authors prove existence results, which are applied to the stochastic Allen-Cahn equation with periodic potential.

ABSTRACT

In this paper, we study the existence of random periodic solutions for semilinear stochastic partial differential equations with multiplicative linear noise on a bounded open domain ${\cal O}\subset {\mathbb R}^d$ with smooth boundary. We identify them with the solutions of coupled forward-backward infinite horizon stochastic integral equations in $L^2({\cal O})$. We then use generalized Schauder's fixed point theorem, the relative compactness of Wiener-Sobolev spaces in $C^0([0, T], L^2(Ω imes{\cal O}))$ and a localization argument to prove the existence of solutions of the infinite horizon integral equations, which immediately implies the existence of the random periodic solution to the corresponding SPDEs. As an example, we apply our result to the stochastic Allen-Cahn equation with a periodic potential and prove the existence of a random periodic solution using a localisation argument.

Motivation & Objective

• To establish the existence of random periodic solutions for semilinear SPDEs with multiplicative linear noise on bounded domains with smooth boundary.
• To address the challenge that random periodic solutions in multiplicative noise settings are anticipating, requiring Skorokhod-type stochastic integrals.
• To overcome difficulties in Malliavin derivative estimation by introducing stochastic linear evolution operators and reformulating the problem as coupled forward-backward infinite horizon random integral equations (IHRIEs).
• To extend the theory of random periodic solutions to infinite-dimensional settings, particularly for SPDEs.
• To demonstrate the applicability of the framework through an application to the stochastic Allen-Cahn equation with periodic potential.

Proposed method

• Formulate the SPDE as a coupled system of forward-backward infinite horizon stochastic integral equations (IHSIEs) in $L^2(\mathcal{O})$.
• Use the stochastic linear evolution operator to handle the multiplicative noise and avoid direct treatment of Skorokhod integrals.
• Apply generalized Schauder's fixed point theorem to prove existence of solutions to the infinite horizon integral equations.
• Leverage relative compactness of Wiener-Sobolev spaces in $C^0([0,T], L^2(\Omega \times \mathcal{O}))$ to ensure convergence of approximating sequences.
• Implement a localization argument to control nonlinearities and ensure uniform bounds in the fixed point argument.
• Utilize the spectral decomposition of the uniformly elliptic operator $\mathcal{L}$ via eigenfunctions $\{\phi_k\}$ and eigenvalues $\{\mu_k\}$ in the analysis.

Experimental results

Research questions

• RQ1 Can random periodic solutions exist for semilinear SPDEs with multiplicative linear noise, given the anticipating nature of such solutions?
• RQ2 How can one overcome the analytical challenges posed by Skorokhod-type stochastic integrals in the context of random periodic solutions?
• RQ3 What conditions ensure the existence of random periodic solutions in infinite-dimensional SPDEs with multiplicative noise?
• RQ4 Can the framework be applied to physically relevant models such as the stochastic Allen-Cahn equation with periodic potential?
• RQ5 How does the relative compactness of Wiener-Sobolev spaces contribute to the existence proof in the infinite-dimensional setting?

Key findings

• The existence of random periodic solutions for SPDEs with multiplicative linear noise is established via the solution of coupled forward-backward infinite horizon random integral equations.
• The method avoids direct handling of Skorokhod integrals by introducing stochastic linear evolution operators, enabling the use of fixed point arguments.
• Relative compactness of Wiener-Sobolev spaces in $C^0([0,T], L^2(\Omega \times \mathcal{O}))$ is used to extract convergent subsequences in the fixed point argument.
• A localization argument is successfully applied to control the nonlinear term in the SPDE, ensuring uniform bounds necessary for the fixed point theorem.
• For the stochastic Allen-Cahn equation with periodic potential $F(t,u) = u - u^3 + \sin t$, the weakly dissipative condition is satisfied, and the existence of a random periodic solution of period $2\pi$ is proven.
• The key estimate $\tilde{M} = M - \epsilon > \sigma^2/2$ is used to ensure dissipativity, which is essential for the existence result.