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March  2010, 9(2): 307-326. doi: 10.3934/cpaa.2010.9.307

Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach

Nicolas Dirr 1, , Federica Dragoni 2, and  Max von Renesse 3,

1. 

Department of Mathematical Sciences, University of Bath, Bath, BA1 7AY, United Kingdom
2. 

Department of Pure Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
3. 

Institut für Mathematik, Technische Universität Berlin, Strasse des 17. Juni 136, 10632 Berlin, Germany

Received  January 2009 Revised  August 2009 Published  December 2009

We study evolution by horizontal mean curvature flow in sub- Riemannian geometries by using stochastic approach to prove the existence of a generalized evolution in these spaces. In particular we show that the value function of suitable family of stochastic control problems solves in the viscosity sense the level set equation for the evolution by horizontal mean curvature flow.
Keywords: stochastic processes and control., level set equation, sub-Riemannian geometries, Mean curvature flow.
Mathematics Subject Classification: Primary: 53C44, 53C17, 93E03; Secondary: 35K65, 49L2.
Citation: Nicolas Dirr, Federica Dragoni, Max von Renesse. Evolution by mean curvature flow in sub-Riemannian geometries: A stochastic approach. Communications on Pure and Applied Analysis, 2010, 9 (2) : 307-326. doi: 10.3934/cpaa.2010.9.307
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