Abstract
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 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 Xxv(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 2003 |
Keywords
- Codimension -k mean curvature flow
- Geometric flows
- Inverse mean curvature flow
- Stochastic target problem
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty