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January 2019 Sharp interface limit for stochastically perturbed mass conserving Allen–Cahn equation
Tadahisa Funaki, Satoshi Yokoyama
Ann. Probab. 47(1): 560-612 (January 2019). DOI: 10.1214/18-AOP1268


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.


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Tadahisa Funaki. Satoshi Yokoyama. "Sharp interface limit for stochastically perturbed mass conserving Allen–Cahn equation." Ann. Probab. 47 (1) 560 - 612, January 2019.


Received: 1 October 2016; Revised: 1 December 2017; Published: January 2019
First available in Project Euclid: 13 December 2018

zbMATH: 07036344
MathSciNet: MR3909976
Digital Object Identifier: 10.1214/18-AOP1268

Primary: 60H15
Secondary: 35K93, 74A50

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 1 • January 2019
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