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