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Soner, H. Mete; Touzi, Nizar

A stochastic representation for the level set equations. (English) Zbl 1036.49010

Commun. Partial Differ. Equations 27, No. 9-10, 2031-2053 (2002).
Summary: A Feynman-Kac representation is proved for geometric partial differential equations. This representation is in terms of a stochastic target problem. In this problem the controller tries to steer a controlled process into a given target by judicious choices of controls. The sublevel sets of the unique level set solution of the geometric equation are shown to coincide with the reachability sets of the target problem whose target is the sublevel set of the final data.

Cited in 1 Review
Cited in 17 Documents

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
60J60 Diffusion processes
49L20 Dynamic programming in optimal control and differential games
35K55 Nonlinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)

Keywords:

Feynman-Kac representation; stochastic target problem; sublevel sets; level set solution; reachability sets; geometric flows; codimension \(k\) mean curvature flow; inverse mean curvature flow
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References:

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