Tug-of-war games related to $p$-Laplace type equations with zeroth order terms

TL;DR

The paper analyzes tug-of-war games with a fixed per-move discount and their link to $p$-Laplacian type equations with a zeroth-order term. By formulating a dynamic programming principle and its associated stochastic game, it proves existence, uniqueness, and regularity of the game value function, and derives a mean-value interpretation supporting the connection to the normalized $p$-Laplacian with discount. It then proves interior and boundary regularity estimates for the value functions and shows that as the step size vanishes, subsequential limits are viscosity solutions of $\Delta_p^N u-(p+n)\gamma u=0$ in $\Omega$ with boundary data. These results establish a rigorous bridge between discounted tug-of-war games and $p$-Laplace type PDEs, including convergence and regularity theory for the associated value functions.

Abstract

In this paper, we investigate a class of tug-of-war games that incorporate a constant payoff discount rate at each turn. The associated model problems are $p$-Laplace type partial differential equations with zeroth-order terms. We establish existence, uniqueness, and regularity results for the corresponding game value functions. Furthermore, we explore properties of the solutions to the model PDEs, informed by the analysis of the underlying games.

Figures (1)

• Figure 1: The solutions of $u_1"=0$ (red) and $u_2"-u_2=0$ (blue).

Theorems & Definitions (15)

• Remark 2.1
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• proof
• Lemma 4.2
• Theorem 4.3
• proof