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July 2003 A stochastic representation for mean curvature type geometric flows
H. Mete Soner, Nizar Touzi
Ann. Probab. 31(3): 1145-1165 (July 2003). DOI: 10.1214/aop/1055425773


A smooth solution $\{ \Gamma(t)\}_{t \in[0,T]}\subset \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 $\Tc$ 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 $\xx(t) \in \Tc$ for some control process $\nu$. This representation is proved by studying the squared distance function to $\Gamma(t)$. For the codimension k mean curvature flow, the state process is $dX(t)= \sqrt{2} P \,dW(t)$, 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.


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H. Mete Soner. Nizar Touzi. "A stochastic representation for mean curvature type geometric flows." Ann. Probab. 31 (3) 1145 - 1165, July 2003.


Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1080.60076
MathSciNet: MR1988466
Digital Object Identifier: 10.1214/aop/1055425773

Primary: 49J2060J60 , 60J60
Secondary: 35K55 , 49L20

Keywords: codimension $-k$ mean curvature flow , Geometric flows , inverse mean curvature flow , stochastic target problem.

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 3 • July 2003
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