Stochastic Motion by Mean Curvature

Abstract.

We prove the existence of a continuously time‐varying subset K(t) of R ⁿ such that its boundary ∂K(t), which is a hypersurface, has normal velocity formally equal to the (weighted) mean curvature plus a random driving force. This is the first result in such generality combining curvature motion and stochastic perturbations. Our result holds for any C ² convex surface energy. The K(t) can have topological changes. The randomness is introduced by means of stochastic flows of diffeomorphisms generated by Brownian vector fields which are white in time but smooth in space. We work in the context of geometric measure theory, using sets of finite perimeter to represent K(t). The evolution is obtained as a limit of a time‐stepping scheme. Variational minimizations are employed to approximate the curvature motion. Stochastic calculus is used to prove global energy estimates, which in turn give a tightness statement of the approximating evolutions.