  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.
