The Annals of Probability

Sharp interface limit for stochastically perturbed mass conserving Allen–Cahn equation

Tadahisa Funaki and Satoshi Yokoyama

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This paper studies the sharp interface limit for a mass conserving Allen–Cahn equation, added an external noise and derives a stochastically perturbed mass conserving mean curvature flow in the limit. The stochastic term destroys the precise conservation law, instead the total mass changes like a Brownian motion in time. For our equation, the comparison argument does not work, so that to study the limit we adopt the asymptotic expansion method, which extends that for deterministic equations used originally in de Mottoni and Schatzman [Interfaces Free Bound. 12 (2010) 527–549] for the nonconservative case and then in Chen et al. [Trans. Amer. Math. Soc. 347 (1995) 1533–1589] for the conservative case. Differently from the deterministic case, each term except the leading term appearing in the expansion of the solution in a small parameter $\varepsilon$ diverges as $\varepsilon$ tends to $0$, since our equation contains the noise which converges to a white noise and the products or the powers of the white noise diverge. To derive the error estimate for our asymptotic expansion, we need to establish the Schauder estimate for a diffusion operator with coefficients determined from higher order derivatives of the noise and their powers. We show that one can choose the noise sufficiently mild in such a manner that it converges to the white noise and at the same time its diverging speed is slow enough for establishing a necessary error estimate.

Article information

Ann. Probab., Volume 47, Number 1 (2019), 560-612.

Received: October 2016
Revised: December 2017
First available in Project Euclid: 13 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35K93: Quasilinear parabolic equations with mean curvature operator 74A50: Structured surfaces and interfaces, coexistent phases

Sharp interface limit Allen–Cahn equation mean curvature flow asymptotic expansion mass conservation law stochastic perturbation


Funaki, Tadahisa; Yokoyama, Satoshi. Sharp interface limit for stochastically perturbed mass conserving Allen–Cahn equation. Ann. Probab. 47 (2019), no. 1, 560--612. doi:10.1214/18-AOP1268.

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Supplemental materials

  • Supplement to “Sharp interface limit for stochastically perturbed mass conserving Allen–Cahn equation”. The supplementary file provides proofs of all the lemmas, with some exceptions, stated in Sections 3–5 of the present paper.