The Annals of Probability

A stochastic representation for mean curvature type geometric flows

H. Mete Soner and Nizar Touzi

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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.

Article information

Ann. Probab., Volume 31, Number 3 (2003), 1145-1165.

First available in Project Euclid: 12 June 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 49J2060J60 60J60: Diffusion processes [See also 58J65]
Secondary: 49L20: Dynamic programming method 35K55: Nonlinear parabolic equations

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


Soner, H. Mete; Touzi, Nizar. A stochastic representation for mean curvature type geometric flows. Ann. Probab. 31 (2003), no. 3, 1145--1165. doi:10.1214/aop/1055425773.

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