H\"older estimate for a tug-of-war game with $1<p<2$ from Krylov-Safonov regularity theory

TL;DR

This paper introduces a new tug-of-war game model linked to the p-Laplacian for 1<p<2, deriving H"older continuity of solutions from Krylov-Safonov regularity theory without boundary corrections.

Contribution

It presents a novel version of the tug-of-war game and a dynamic programming principle for the singular p-Laplacian case, establishing regularity and existence results.

Findings

H"older continuity of solutions derived from Krylov-Safonov theory

Existence of measurable solutions without boundary corrections

Comparison principle established

Abstract

We propose a new version of the tug-of-war game and a corresponding dynamic programming principle related to the $p$-Laplacian with $1<p<2$. For this version, the asymptotic H\"older continuity of solutions can be directly derived from recent Krylov-Safonov type regularity results in the singular case. Moreover, existence of a measurable solution can be obtained without using boundary corrections. We also establish a comparison principle.