Game Theoretical Methods in Nonlinear PDEs

TL;DR

This paper explores how solutions to certain nonlinear PDEs, specifically of p-Laplacian type, can be understood as limits of values from a specific stochastic game called Tug-of-War as the step size approaches zero.

Contribution

It demonstrates the connection between nonlinear PDE solutions and stochastic differential games, providing a game-theoretic interpretation of PDEs of p-Laplacian type.

Findings

Solutions to p-Laplacian PDEs can be obtained as limits of Tug-of-War game values.

The step size in the game influences the convergence to PDE solutions.

The approach links stochastic game theory with nonlinear PDE analysis.

Abstract

Nonlinear PDEs, mean value properties, and stochastic differential games are intrinsically connected. In this short expository note, we will describe how the solutions to certain PDEs (of $p$-Laplacian type) can be interpreted as limits of values of a specific Tug-of-War game, when the step-size $\epsilon$ determining the allowed length of move of a token, decreases to $0$.