Comptes Rendus
no. 23-24
Probability Theory
Viability property on Riemannian manifolds

Presented by: Marc Yor

Shige Peng1; Xuehong Zhu1, 2
1 Institute of Mathematics, Shandong University, Jinan, 250100, China
2 School of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China
Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1423-1428.

This Note studies a sufficient and necessary condition for the viability property of a state system in a closed subset K of a finite-dimensional compact Riemannian manifold without boundary. Our result is: the system enjoys the viability property in K if and only if the square of the distance function of K is a viscosity supersolution of a second-order partial differential equation in some neighborhood of K.

Dans cette Note on donne une condition nécessaire et suffisante pour que soit satisfaite la propriété de viabilité d'un système sur un sous-ensemble K d'une variété riemannienne, de dimension finie, sans bord. Le résultat s'énonce ainsi : le système sur K possède la propriété de viabilité si et seulement si le carré de la fonction distance à K est une sursolution de viscosité d'une équation aux dérivées partielles du second ordre définie sur un voisinage de K.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2009.10.007

Shige Peng 1; Xuehong Zhu 1, 2

1 Institute of Mathematics, Shandong University, Jinan, 250100, China
2 School of Science, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China 
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Shige Peng; Xuehong Zhu. Viability property on Riemannian manifolds. Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1423-1428. doi : 10.1016/j.crma.2009.10.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.10.007/

[1] D. Azagra; J. Ferrera; B. Sanz Viscosity solutions to second order partial differential equations on Riemannian manifolds, J. Differential Equations, Volume 245 (2008), pp. 307-336

[2] R. Buckdahn; S. Peng; M. Quincampoix; C. Rainer Existence of stochastic control under state constraints, C. R. Acad. Sci. Paris, Volume 327 (1998), pp. 17-22

[3] Elton P. Hsu Stochastic Analysis on Manifolds, American Mathematical Society, 2002

[4] X. Zhu, Viscosity solutions to second order parabolic PDEs on Riemannian manifolds, preprint

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