Moderate Deviations Principle and Central Limit Theorem for Stochastic Cahn-Hilliard Equation in Holder Norm

Résumé

We consider a stochastic Cahn-Hilliard partial differential equation driven by a space-time white noise. In this paper, we prove a Central Limit Theorem (CLT) and a Moderate Deviation Principle (MDP) for a perturbed stochastic Cahn-Hilliard equation in Holder norm. The techniques are based on Freidlin-Wentzell’s Large Deviations Principle. The exponential estimates in the space of Holder continuous functions and the Garsia-Rodemich-Rumsey’s lemma plays an important role, an another approach than the Li.R. ¨and Wang.X. Finally, we estabish the CLT and MDP for stochastic Cahn-Hilliard equation with uniformly Lipschitzian coefficients.

Keywords: Large Deviations Principle, Moderate Deviations Principle, Central Limit Theorem, Holder space, Stochastic Cahn-Hilliard equation, Green’s function, Freidlin-Wentzell’s method.
MSC: 60H15, 60F05, 35B40, 35Q62

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Communicated Editor: Chala Adel
Manuscript received Dec 07, 2023; revised Fb 09, 2024; accepted Feb 16, 2024; published May 13, 2024.