A game-theoretic approach to the parabolic normalized p-Laplacian obstacle problem

TL;DR

The paper develops a probabilistic representation for the parabolic obstacle problem associated with the normalized $p$-Laplacian by introducing a zero-sum tug-of-war game with noise and an optimal stopping rule in a space-time cylinder. It proves that the game has a value, satisfies a dynamic programming principle, and that the corresponding value functions $u^{\varepsilon}$ converge uniformly to the unique viscosity solution of the continuous parabolic obstacle problem as the step size $\varepsilon \to 0$, specifically to the solution of $\min\{ (n+p)u_t - [(p-2)\Delta_\infty u + \Delta u], u-\psi \} = 0$ in $\Omega_T$, with boundary data $u=F$ on $\partial_p \Omega_T$ and the constraint $u\ge\psi$. The analysis combines martingale methods, barrier constructions, and viscosity techniques to establish existence, uniqueness, and convergence, providing a robust link between stochastic game representations and fully nonlinear parabolic obstacle PDEs.

Abstract

This paper establishes a probabilistic representation for the solution of the parabolic obstacle problem associated with the normalized $p$-Laplacian. We introduce a zero-sum stochastic tug-of-war game with noise in a space-time cylinder, where one player has the option to stop the game at any time to collect a payoff given by an obstacle function. We prove that the value functions of this game exist, satisfy a dynamic programming principle, and converge uniformly to the unique viscosity solution of the continuous obstacle problem as the step size $\varepsilon$ tends to zero.

Theorems & Definitions (19)

• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Theorem 3.3
• proof
• Lemma 4.1
• Definition 4.2
• Lemma 4.3
• proof