[Paper Review] Random Tug of War games for the ${\mathbf p}$-Laplacian: ${\mathbf{1<p<{\boldsymbol \infty}}}$

Marta Lewicka|arXiv (Cornell University)|Oct 8, 2018

Advanced Mathematical Modeling in Engineering9 references4 citations

TL;DR

This paper introduces a novel noisy Tug of War game for the p-Laplacian ($1 < p < \infty$) that combines deterministic min-max averaging over balls with stochastic averaging over N-dimensional ellipses whose aspect ratio depends on $p$ and $N$. The resulting dynamic programming principle ensures continuous solutions that converge uniformly to the unique viscosity solution of the Dirichlet problem for $\Delta_p u = 0$, establishing game regularity for domains satisfying the exterior corkscrew condition.

ABSTRACT

We propose a new finite difference approximation to the Dirichlet problem for the homogeneous $\mathbf{p}$-Laplace equation posed on an $N$-dimensional domain, in connection with the Tug of War games with noise. Our game and the related mean-value expansion that we develop, superposes the ``deterministic averages'' ``$\frac{1}{2}(\inf +\sup)$'' taken over balls, with the ``stochastic averages'' ``$\fint$'', taken over $N$-dimensional ellipsoids whose aspect ratio depends on $N,\mathbf{p}$ and whose orientations span all directions while determining $\inf / \sup$. We show that the unique solutions $u_\epsilon$ of the related dynamic programming principle are automatically continuous for continuous boundary data, and coincide with the well-defined game values. Our game has thus the min-max property: the order of supremizing the outcomes over strategies of one player and infimizing over strategies of their opponent, is immaterial. We further show that domains satisfying the exterior corkscrew condition are game regular in this context, i.e. the family $\{u_\epsilon\}_{\epsilon o 0}$ converges uniformly to the unique viscosity solution of the Dirichlet problem.

Motivation & Objective

• To develop a finite difference approximation for the homogeneous p-Laplacian $\Delta_p u = 0$ using a novel stochastic game-theoretic framework.
• To resolve the lack of regularity and measurability in prior noisy Tug of War games by constructing a game with intrinsic continuity and well-defined game values.
• To establish that solutions to the dynamic programming principle are continuous and coincide with game values, ensuring the min-max property holds.
• To prove that the family of solutions $\{u_\epsilon\}_{\epsilon \to 0}$ converges uniformly to the unique viscosity solution of the Dirichlet problem on game-regular domains.

Proposed method

• Proposes a mean value expansion combining deterministic $\frac{1}{2}(\inf + \sup)$ over balls with stochastic averaging over $N$-dimensional ellipsoids $E(x, r; \alpha, \nu)$, where the aspect ratio $\alpha_p$ depends on $p$ and $N$.
• Derives a key expansion (1.3) that identifies $\Delta_p u$ via a weighted average involving $\gamma_p$ and $\alpha_p$, with the compatibility condition $\frac{N+2}{\gamma_p^2} + a_p^2 = p - 1$.
• Introduces a two-player zero-sum game where moves are updated via a coin toss and a noise vector uniformly distributed on a codimension-2 sphere orthogonal to the last move, with scaling factor $\gamma_p = \sqrt{\frac{N-1}{p-1}}$.
• Uses the corkscrew condition to prove game regularity: for domains satisfying this geometric condition, the game stops almost surely and solutions converge uniformly.
• Applies the Ascoli-Arzela theorem and viscosity solution theory to identify the limit of $u_\epsilon$ as the unique viscosity solution of $\Delta_p u = 0$ in $D$, $u = F$ on $\partial D$.
• Establishes that the game values are continuous and coincide with solutions to the dynamic programming principle, ensuring the min-max property holds regardless of strategy order.

Experimental results

Research questions

• RQ1 Can a Tug of War game be constructed such that its value function is continuous and coincides with the solution of a finite difference scheme for $\Delta_p u = 0$?
• RQ2 Does the proposed game with ellipsoidal averaging and noise ensure the min-max property, i.e., that the order of player strategies does not affect the outcome?
• RQ3 For which domains is the game regular, meaning that the solutions $u_\epsilon$ converge uniformly to the viscosity solution as $\epsilon \to 0$?
• RQ4 Can the mean value expansion (1.3) be derived without prior knowledge of $\nabla u(x)$, enabling the identification of $p$-harmonic functions that are only continuous?
• RQ5 Is the convergence of $u_\epsilon$ to the viscosity solution uniform and unique under the corkscrew condition?

Key findings

• The proposed game ensures that solutions $u_\epsilon$ to the dynamic programming principle are continuous for continuous boundary data, inheriting the regularity of the boundary function $F$.
• The game values are well-defined and coincide with the solutions $u_\epsilon$, resolving the measurability and regularity issues present in earlier formulations.
• The min-max property holds: the order of supremizing and infimizing over player strategies is immaterial, ensuring a consistent game value.
• For domains satisfying the exterior corkscrew condition, the family $\{u_\epsilon\}_{\epsilon \to 0}$ converges uniformly to the unique viscosity solution of the Dirichlet problem for $\Delta_p u = 0$.
• The convergence is uniform and the limit is uniquely identified as the viscosity solution, with the proof relying on viscosity sub- and supersolution verification via test functions.
• The compatibility condition $\frac{N+2}{\gamma_p^2} + a_p^2 = p - 1$ ensures the validity of the mean value expansion (1.3) for all $p \in (1, \infty)$, making the framework applicable across the full range of $p$.