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

TL;DR

This paper investigates the sharp interface limit of a stochastically perturbed, mass conserving Allen-Cahn equation, revealing how noise affects the limiting mean curvature flow and developing new analytical techniques for error estimation.

Contribution

It introduces an asymptotic expansion method for stochastic equations with white noise, extending deterministic approaches and establishing error estimates under mild noise conditions.

Findings

Derivation of a stochastic mean curvature flow as the sharp interface limit.

Development of Schauder estimates for diffusion operators with noise-dependent coefficients.

Identification of conditions under which noise converges to white noise while allowing error control.

Abstract

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 in Chen et al. [3]. Differently from the deterministic case, each term except the leading term appearing in the expansion of the solution in a small parameter $\epsilon$ diverges as $\epsilon$ 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.