A Stochastically Perturbed Mean Curvature Flow by Colored Noise

We study the motion of the hypersurface $$(\gamma _t)_{t\ge 0}$$ ( γ t ) t ≥ 0 evolving according to the mean curvature perturbed by $$\dot{w}^Q$$ w ˙ Q , the formal time derivative of the Q-Wiener process $${w}^Q$$ w Q , in a two-dimensional bounded domain. Namely, we consider the equation describing the evolution of $$\gamma _t$$ γ t as a stochastic partial differential equation (SPDE) with a multiplicative noise in the Stratonovich sense, whose inward velocity V is determined by $$V=\kappa \,+\,G \circ \dot{w}^Q$$ V = κ + G ∘ w ˙ Q , where $$\kappa $$ κ is the mean curvature and G is a function determined from $$\gamma _t$$ γ 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 $$\gamma _t$$ γ t , we derive a certain second-order quasilinear SPDE with respect to the signed distance function determined from $$\gamma _0$$ γ 0 . Then we construct the local solution making use of probabilistic tools and the classical Banach fixed point theorem on suitable Sobolev spaces.