Electronic Journal of Probability

Triviality of the 2D stochastic Allen-Cahn equation

Martin Hairer, Marc Ryser, and Hendrik Weber

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

SPDEs Allen-Cahn equation white noise stochastic quantisation

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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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