A game theoretical approximation for a parabolic/elliptic system with different operators

TL;DR

This paper introduces a game-theoretic approach to approximate solutions of a coupled parabolic and elliptic PDE system using a two-player zero-sum game with different rules on each board, converging to viscosity solutions.

Contribution

It develops a novel game-based method to approximate solutions of a coupled PDE system involving the infinity Laplacian and the Laplacian, with convergence proof.

Findings

Game value functions converge uniformly to viscosity solutions

Two different game rules model the PDE system effectively

Method provides a new link between game theory and PDE systems

Abstract

In this paper we find viscosity solutions to a coupled system composed by two equations, the first one is parabolic and driven by the infinity Laplacian while the second one is elliptic and involves the usual Laplacian. We prove that there is a two-player zero-sum game played in two different boards with different rules in each board (in the first one we play a Tug-of-War game taking the number of plays into consideration and in the second board we move at random) whose value functions converge uniformly to a viscosity solution to the PDE system.