A stochastic representation for mean curvature type geometric flows

H. Mete Soner, Nizar Touzi

Research output: Contribution to journalArticlepeer-review

27 Scopus citations

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 languageEnglish (US)
Pages (from-to)1145-1165
Number of pages21
JournalAnnals of Probability
Volume31
Issue number3
DOIs
StatePublished - Jul 1 2003
Externally publishedYes

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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