A stochastically perturbed mean curvature flow by colored noise

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 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.