[Paper Review] Irreducibility of stochastic real Ginzburg-Landau equation driven by $α$-stable noises and applications

Ran Wang, Jie Xiong+1 more|arXiv (Cornell University)|Oct 7, 2015

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper establishes the irreducibility of the stochastic real Ginzburg-Landau equation driven by $α$-stable Lévy noise on the torus $\mathbb{T}$ using a maximal inequality and a control problem approach. The key contribution is proving exponential ergodicity under a stronger topology than total variation and deriving the moderate deviation principle via Lyapunov test functions, extending ergodic theory to non-Wiener Lévy noise settings.

ABSTRACT

We establish the irreducibility of stochastic real Ginzburg-Landau equation with $α$-stable noises by a maximal inequality and solving a control problem. As applications, we prove that the system converges to its equilibrium measure with exponential rate under a topology stronger than total variation and obeys the moderate deviation principle by constructing some Lyapunov test functions.

Motivation & Objective

• To establish irreducibility of the stochastic real Ginzburg-Landau equation driven by $α$-stable Lévy noise with $α \in (3/2, 2)$.
• To prove exponential convergence to the invariant measure under a topology stronger than total variation.
• To derive the moderate deviation principle for the system using Lyapunov test functions.
• To extend ergodic theory results from Wiener noise to non-Lipschitz, non-Gaussian $α$-stable noise settings.

Proposed method

• Prove irreducibility by solving a control problem using a maximal inequality to handle the non-Lipschitz nonlinearity and discontinuous paths of $α$-stable noise.
• Construct a Lyapunov test function to establish exponential ergodicity under a stronger-than-total-variation topology.
• Apply results from [21] to derive the moderate deviation principle by verifying the same Lyapunov condition used for exponential ergodicity.
• Use the mild solution framework and Fourier analysis in $H = L^2(\mathbb{T}; \mathbb{R})$ with zero mean, leveraging the orthonormal basis $\{e_k = e^{i2\pi k\xi}\}_{k \in \mathbb{Z}_*}$.
• Define the generator $\mathcal{L}$ and verify $\Psi \in \mathbf{D}_e(\mathcal{L})$ via Itô's formula and convergence arguments in $L^1$ and $L^2$ martingales.
• Apply Fatou’s lemma and the Lebesgue dominated convergence theorem to pass limits in approximating sequences $X^m$ to $X$, ensuring convergence of integrals involving $\|X_s\|_V^2$ and $\langle X_s, N(X_s)\rangle_H$.

Experimental results

Research questions

• RQ1 Can the irreducibility of the stochastic real Ginzburg-Landau equation driven by $α$-stable noise be established despite the lack of second moments and discontinuous paths?
• RQ2 Does the system converge to its invariant measure with an exponential rate under a topology stronger than total variation?
• RQ3 Can the moderate deviation principle be derived for this non-Wiener, non-Lipschitz SPDE system?
• RQ4 How can Lyapunov test functions be constructed to prove exponential ergodicity and moderate deviations in the absence of second-order moment structure?

Key findings

• The stochastic real Ginzburg-Landau equation driven by $α$-stable noise with $\alpha \in (3/2, 2)$ is irreducible on the torus $\mathbb{T}$.
• The system converges to its unique invariant measure $\pi$ with exponential rate under a topology stronger than total variation.
• The moderate deviation principle holds for the system, as established via the same Lyapunov condition used for exponential ergodicity.
• The irreducibility is proven by solving a control problem using a maximal inequality, overcoming the challenges posed by the non-Lipschitz nonlinearity and non-Gaussian noise.
• The existence and uniqueness of the invariant measure $\pi$ are inherited from prior results, and the strong Feller property is used to strengthen ergodicity results.
• The convergence of approximating sequences $X^m$ to $X$ in $L^1$ and $L^2$ martingale terms is rigorously justified using Fatou’s lemma and the dominated convergence theorem.