Connections between coupling and Ishii-Lions methods for tug-of-war with noise stochastic games

TL;DR

The article unifies stochastic coupling methods from tug-of-war with noise with the Ishii-Lions regularity framework for viscosity solutions by deriving and analyzing $2n$-dimensional dynamic programming principles ($2n$-DPPs) and their associated $2n$-PDEs. It demonstrates how reflection-type couplings in ${f R}^{2n}$ reproduce the key analytic structures of the Laplacian, infinity-Laplacian, and normalized $p$-Laplacians, and shows how the Ishii-Lions barrier/maximum principle translates to a robust Hölder regularity theory across these operators. The work provides a coherent probabilistic-analytic pathway that links stochastic game dynamics to the classical PDE regularity toolbox, suggesting a unified approach to regularity via doubled-variable PDEs. These connections enhance intuition for both stochastic games and viscosity solutions and may inform future cross-disciplinary techniques in nonlinear potential theory and stochastic control.

Abstract

We present a streamlined account of two different regularity methods as well as their connections. We consider the coupling method in the context of tug-of-war with noise stochastic games, and consider viscosity solutions of the $p$-Laplace equation in the context of the Ishii-Lions method.

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Lemma 2.2
• Example 2.3
• Theorem 3.1: Ishii-Lions
• proof
• Theorem 5.1: Ishii-Lions
• proof
• Theorem 5.2
• proof