Repeated games for  non-linear parabolic integro-differential equations and integral  curvature flows

Cyril Imbert; Sylvia Serfaty

doi:10.3934/dcds.2011.29.1517

Article Contents

2011, Volume 29, Issue 4: 1517-1552. Doi: 10.3934/dcds.2011.29.1517

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Repeated games for non-linear parabolic integro-differential equations and integral curvature flows

Cyril Imbert^1, and
Sylvia Serfaty^2,

1.
CEREMADE, UMR CNRS 7534, université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cedex 16
2.
UPMC Univ Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F-75005

Received: December 31, 2009

Revised: August 31, 2010

Published: September 2011

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Abstract

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.

Keywords:

Repeated games,
integral curvature flows,
parabolic integro-differential equations,
viscosity solutions,
geometric flows.

Mathematics Subject Classification: 35C99, 53C44, 90D10, 49K25.

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