| Abstract: |
| In this talk, we introduce the sharp interface limit for a mass conserving Allen-Cahn equation with a small parameter $\epsilon>0$ added an external noise.
The stochastic term destroys the precise conservation law, instead the total mass changes like a Brownian motion in time.
The evolutional law of the limit hypersurface is described by the mass conserving mean curvature flow with a multiplicative white noise.
In order to study the limit, we use the asymptotic expansion method.
However, in our case,
when $\epsilon$ tends to $0$, each term except the leading term appearing in the expansion of the solution in a small parameter $\epsilon$ diverges.
This is because our equation contains the noise which converges to a white noise and the products or the powers of the white noise diverge when $\epsilon$ tends to $0$.
We introduce how to control those terms when $\epsilon$ tends to $0$ to obtain our goal.
Reference: T. Funaki and S. Yokoyama, Sharp interface limit for stochastically perturbed mass conserving Allen-Cahn equation, to appear in Ann. Probab. |
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