### Abstract

A smooth solution {γ(t)}_{t ∈[0, T]} ⊂ 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 T 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 X_{x}^{v}(t) ∈ T for some control process v. 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) = √2P 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.

Original language | English (US) |
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Pages (from-to) | 1145-1165 |

Number of pages | 21 |

Journal | Annals of Probability |

Volume | 31 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1 2003 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Codimension -k mean curvature flow
- Geometric flows
- Inverse mean curvature flow
- Stochastic target problem

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## Cite this

*Annals of Probability*,

*31*(3), 1145-1165. https://doi.org/10.1214/aop/1055425773