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

Cyril Imbert (1) , Sylvia Serfaty (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CIMS - Courant Institute of Mathematical Sciences [New York]

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 interface (Imbert, 2008) 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 integral curvature flows, players choose hypersurfaces in the whole space and positions on these hypersurfaces. For parabolic integro-differential equations, players choose smooth functions on the whole space.

Mots clés: Repeated games; integral curvature flows; parabolic integro-differential equations; viscosity solutions; geometric flows

Domaines: Equations aux dérivées partielles [math.AP]

Cyril Imbert, Sylvia Serfaty. Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations. Discrete and Continuous Dynamical Systems - Series S, 2011, 29 (4), pp.1517-1552. ⟨10.3934/dcds.2011.29.1517⟩. ⟨hal-00429218⟩