A stochastic representation for mean curvature type geometric flows

Abstract

A smooth solution $\{ \Gamma(t)\}_{t \in[0,T]}\subset \R^d $ of a parabolic geometric flow is characterized as the reachability set of a stochastic target problem. In this control problem the controller tries to steer the state process into a given deterministic set $\Tc$ with probability one. The reachability set, $V(t)$, for the target problem is the set of all initial data x from which the state process $\xx(t) \in \Tc$ for some control process $\nu$. This representation is proved by studying the squared distance function to $\Gamma(t)$. For the codimension k mean curvature flow, the state process is $dX(t)= \sqrt{2} P \,dW(t)$, where $W(t)$ is a d-dimensional Brownian motion, and the control P is any projection matrix onto a $(d-k)$-dimensional plane. Smooth solutions of the inverse mean curvature flow and a discussion of non smooth solutions are also given.