A Projected Tug-of-War Game for the Regularized $p$-Laplacian

TL;DR

This paper interprets the regularized p-Laplacian through a tug-of-war game framework, establishing a connection with p-harmonic functions and providing convergence results for the associated discrete scheme.

Contribution

It introduces a novel tug-of-war interpretation for the regularized p-Laplacian and proves existence, uniqueness, and convergence of solutions via a discrete dynamic programming approach.

Findings

Established a tug-of-war interpretation for the regularized p-Laplacian.

Proved existence, uniqueness, and measurability of solutions with boundary data.

Demonstrated convergence of the discrete scheme to the viscosity solution as the discretization parameter tends to zero.

Abstract

We give a tug-of-war interpretation of the regularized $p$-Laplacian $\divgg\big((1+|Dv|^2)^{p/2-1}Dv\big)=0$ in a bounded domain $\Omega\subset\R^n$, $p\ge 2$. The key is the linear lift $w(x,x_{n+1})=v(x)+x_{n+1}$, which identifies this equation with $\Delta_p w=0$ in $\R^{n+1}$. Projecting the standard $(n+1)$-dimensional $p$-harmonious scheme onto $\R^n$ yields a discrete dynamic programming principle for which we prove existence, uniqueness, and Borel measurability of solutions with strip boundary data, identify the unique fixed point with the value of the projected game, and establish convergence to the viscosity solution as $\varepsilon\to 0$.