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June 2021 Conservative stochastic two-dimensional Cahn–Hilliard equation
Michael Röckner, Huanyu Yang, Rongchan Zhu
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Ann. Appl. Probab. 31(3): 1336-1375 (June 2021). DOI: 10.1214/20-AAP1620

Abstract

We consider the stochastic two-dimensional Cahn–Hilliard equation which is driven by the derivative in space of a space-time white noise. We use two different approaches to study this equation. First we prove that there exists a unique solution Y to the shifted equation (1.4). Then X:=Y+Z is the unique solution to the stochastic Cahn–Hilliard equation, where Z is the corresponding O-U process. Moreover, we use the Dirichlet form approach in (Probab. Theory Related Fields 89 (1991) 347–386) to construct a probabilistically weak solution to the original equation (1.1) below. By clarifying the precise relation between the two solutions, we also get the restricted Markov uniqueness of the generator and the uniqueness of the martingale solutions to the equation (1.1). Furthermore, we also obtain exponential ergodicity of the solutions.

Citation

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Michael Röckner. Huanyu Yang. Rongchan Zhu. "Conservative stochastic two-dimensional Cahn–Hilliard equation." Ann. Appl. Probab. 31 (3) 1336 - 1375, June 2021. https://doi.org/10.1214/20-AAP1620

Information

Received: 1 June 2018; Revised: 1 May 2020; Published: June 2021
First available in Project Euclid: 23 June 2021

Digital Object Identifier: 10.1214/20-AAP1620

Subjects:
Primary: 60H15, 82C28

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 3 • June 2021
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