Repeated games for non-linear parabolic integro-differential equations and integral curvature flows
Serfaty, Sylvia; Imbert, Cyril (2011), Repeated games for non-linear parabolic integro-differential equations and integral curvature flows, Discrete and Continuous Dynamical Systems. Series A, 29, 4, p. 1517-1552. http://dx.doi.org/10.3934/dcds.2011.29.1517
TypeArticle accepté pour publication ou publié
External document linkhttp://arxiv.org/abs/0911.0240v1
Journal nameDiscrete and Continuous Dynamical Systems. Series A
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Abstract (EN)The main purpose of this paper is to approximate several non-local evolution equations by zero-sum repeated games in the spirit of the previous works of Kohn and the second author (2006 and 2009): general fully non-linear parabolic integro-differential equations on the one hand, and the integral curvature flow of an on the other hand. In order to do so, we start by constructing such a game for eikonal equations whose speed has a non-constant sign. This provides a (discrete) deterministic control interpretation of these evolution equations. In all our games, two players choose positions successively, and their final payoff is determined by their positions and additional parameters of choice. Because of the non-locality of the problems approximated, by contrast with local problems, their choices have to "collect" information far from their current position. For parabolic integro-differential equations, players choose smooth functions on the whole space. For integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces.
Subjects / Keywordsgeometric flows; viscosity solutions; parabolic integro-differential equations; integral curvature flows; Repeated games
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