Electronic Journal of Probability

Triviality of the 2D stochastic Allen-Cahn equation

Martin Hairer, Marc Ryser, and Hendrik Weber

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Abstract

We consider the stochastic Allen-Cahn equation driven by mollified space-time white noise. We show that, as the mollifier is removed, the solutions converge weakly to 0, independently of the initial condition. If the intensity of the noise simultaneously converges to 0 at a sufficiently fast rate, then the solutions converge to those of the deterministic equation. At the critical rate, the limiting solution is still deterministic, but it exhibits an additional damping term.

Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 39, 14 pp.

Dates
Accepted: 30 May 2012
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465062361

Digital Object Identifier
doi:10.1214/EJP.v17-1731

Mathematical Reviews number (MathSciNet)
MR2928722

Zentralblatt MATH identifier
1245.60063

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 81T08: Constructive quantum field theory

Keywords
SPDEs Allen-Cahn equation white noise stochastic quantisation

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Hairer, Martin; Ryser, Marc; Weber, Hendrik. Triviality of the 2D stochastic Allen-Cahn equation. Electron. J. Probab. 17 (2012), paper no. 39, 14 pp. doi:10.1214/EJP.v17-1731. https://projecteuclid.org/euclid.ejp/1465062361


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