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A smooth solution {Γ(t)}t∈[0,T]⊂\Rd 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)∈\Tc for some control process ν. This representation is proved by studying the squared distance function to Γ(t). For the codimension k mean curvature flow, the state process is dX(t)=2√PdW(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.

A stochastic representation for mean curvature type geometric flows