Time-dependent tug-of-war games and normalized parabolic $p$-Laplace equations

TL;DR

This paper studies the connection between time-dependent tug-of-war games and normalized parabolic p-Laplace equations, establishing existence, uniqueness, regularity, and convergence of game values to PDE solutions.

Contribution

It proves the existence and uniqueness of value functions for these games and shows their convergence to viscosity solutions of the normalized parabolic p-Laplace equation.

Findings

Existence and uniqueness of game value functions

Asymptotic behavior of game values as T approaches infinity

Convergence of game values to viscosity solutions of PDE

Abstract

This paper concerns value functions of time-dependent tug-of-war games. We first prove the existence and uniqueness of value functions and verify that these game values satisfy a dynamic programming principle. Using the arguments in the proof of existence of game values, we can also deduce asymptotic behavior of game values when $T \to \infty$. Furthermore, we investigate boundary regularity for game values. Thereafter, based on the regularity results for value functions, we deduce that game values converge to viscosity solutions of the normalized parabolic $p$-Laplace equation.