The obstacle problem for the $p$-laplacian via optimal stopping of Tug-of-War games

TL;DR

This paper introduces a probabilistic method using tug-of-war games with noise to approximate solutions to the obstacle problem for the p-Laplacian, connecting game theory with nonlinear PDEs.

Contribution

It develops a novel approach that models the obstacle problem via tug-of-war games, extending stochastic game methods to nonlinear PDEs with obstacles.

Findings

Solutions approximated by tug-of-war processes converge as step size decreases

Maximization over stopping times yields the obstacle problem solution

Probabilistic approach links game theory with nonlinear PDEs

Abstract

We present a probabilistic approach to the obstacle problem for for the $p$-Laplace operator. The solutions are approximated by running processes determined by tug-of-war games plus noise, and letting the step size go to zero, not unlike the case when Brownian motion is approximated by random walks. Rather than stopping the process when the boundary is reached, the value function is obtained by maximizing over all possible stopping times that are smaller than the exit time of the domain.