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

Cyril Imbert, Sylvia Serfaty

Research output: Contribution to journalArticlepeer-review

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 nonlinear 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.

Original languageEnglish (US)
Pages (from-to)1517-1552
Number of pages36
JournalDiscrete and Continuous Dynamical Systems
Volume29
Issue number4
DOIs
StatePublished - Apr 2011

Keywords

  • Geometric flows
  • Integral curvature flows
  • Parabolic integro-differential equations
  • Repeated games
  • Viscosity solutions

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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