A Stochastically Perturbed Mean Curvature Flow by Colored Noise

Abstract

We study the motion of the hypersurface $(\gamma_t)_{t\geq 0}$ 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 time variable is not applicable to our equation. To construct a local solution of the equation describing $\gamma_t$, we will 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.