A stochastic representation for geometric flows

A Feynman-Kac type representation formula for the weak solutions of parabolic geometric flows will be discussed. These flows are initial value problems for sets in the Euclidean space and the evolution rule is a possibly nonlinear function of the curvature. Typical examples are the general codimension mean curvature flow, inverse mean curvature flow and the smooth crystalline flow of Gurtin. This representation characterizes the solution at time t as the set of points in the ambient space from which a carefully chosen stochastic process reaches the initial set with probability one. This is a stochastic controllability problem. In the case of mean curvature flow the stochastic process is a Brownian motion living on an N-d dimensional plane. Here the orientation of the plane is a control parameter we get to choose and d is the codimension and N is the dimension of the ambient space.

After a brief introduction to Ito calculus, the representation will be proved for smooth flows. This is joint work with Nizar Touzi from Paris.