A Stochastically Perturbed Mean Curvature Flow by Colored Noise

Abstract

We study the motion of the hypersurface $(\gamma _t)_{t\ge 0}$ evolving according to the mean curvature perturbed by $\dot{w}^Q$, the formal time derivative of the Q-Wiener process ${w}^Q$, in a two-dimensional bounded domain. Namely, we consider the equation describing the evolution of $\gamma _t$ as a stochastic partial differential equation (SPDE) with a multiplicative noise in the Stratonovich sense, whose inward velocity V is determined by $V=\kappa \,+\,G \circ \dot{w}^Q$, where $\kappa$ is the mean curvature and G is a function determined from $\gamma _t$. Already known results in which the noise depends on only the time variable are not applicable to our equation. To construct a local solution of the equation describing $\gamma _t$, we derive a certain second-order quasilinear SPDE with respect to the signed distance function determined from $\gamma _0$. Then we construct the local solution making use of probabilistic tools and the classical Banach fixed point theorem on suitable Sobolev spaces.

Appendix

1.1 Regularity for the Fundamental Solution

Friedman [10] shows that the fundamental solution g(t, x, y), $x,y\in {\mathbb {R}}$ of the second-order parabolic linear differential operator $\frac{\partial }{\partial t}-A$ has the following estimate:

$$\begin{aligned} \Bigl |D_t^j D_x^\alpha D_y^\beta g(t,x,y)\Bigr |\le t^{-\frac{\alpha +\beta }{2}-j}{\overline{g}}(t,x,y), \end{aligned}$$

where

$$\begin{aligned} {\overline{g}}(t,x,y)=K_1t^{-\frac{1}{2}}\exp \Bigl ( {-K_2\frac{|x-y|^{2}}{t}\Bigr )}, \end{aligned}$$

(4.64)

and $K_1$, $K_2>0$ are constants depending on $\alpha$, $\beta$, $j \in {\mathbb {Z}}_+$. In our case, we make use of the estimate in [10] (see also Funaki [12] and Yokoyama [25]).

Lemma 4.5

$$\begin{aligned} \Bigl |D_y^{-k} D_x^\alpha g(t,x,y)\Bigr |=\le t^{-\frac{\alpha }{2}+\frac{k}{2}}{\overline{g}}(t,x,y),\quad t>0,\,x,y\in M \end{aligned}$$

(4.65)

holds for $k,\alpha \in \{0,1,2\}$ with $k\le \alpha$ and suitable $K_1$ and $K_2>0$.

Proof

This is obtained by the estimates for the Gaussian kernel, hence we only sketch the outline of the proof. Take a sufficiently small $z_0 < z$ satisfying $|z-x|\le |z_0-x|$ for every x, z. Then, we get

$$\begin{aligned} \Bigl |D_x^2 \int _{z_0}^z g(t,x,{\tilde{z}}) \hbox {d}{\tilde{z}}\Bigr | \le t^{-1} \int _{z_0}^z {\overline{g}}(t,x,{\tilde{z}}) \hbox {d}{\tilde{z}} \le K_1' t^{-\frac{1}{2}} t^{-\frac{1}{2}} e^{-{K_2'}\frac{|z-x|^2}{t}}, \end{aligned}$$

(4.66)

for some $K_1'>0$ and $K_2'\in (0,K_2)$ and by changing of variables. Proceeding the similar argument again, the estimates in the case of $k=2$ are also obtained. $\square$