## A finite difference approach to the infinity Laplace equation and tug-of-war games

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- by Scott N. Armstrong and Charles K. Smart PDF
- Trans. Amer. Math. Soc.
**364**(2012), 595-636 Request permission

## Abstract:

We present a modified version of the two-player “tug-of-war” game introduced by Peres, Schramm, Sheffield, and Wilson (2009). This new tug-of-war game is identical to the original except near the boundary of the domain $\partial \Omega$, but its associated value functions are more regular. The dynamic programming principle implies that the value functions satisfy a certain finite difference equation. By studying this difference equation directly and adapting techniques from viscosity solution theory, we prove a number of new results.

We show that the finite difference equation has unique maximal and minimal solutions, which are identified as the value functions for the two tug-of-war players. We demonstrate uniqueness, and hence the existence of a value for the game, in the case that the running payoff function is nonnegative. We also show that uniqueness holds in certain cases for sign-changing running payoff functions which are sufficiently small. In the limit $\varepsilon \to 0$, we obtain the convergence of the value functions to a viscosity solution of the normalized infinity Laplace equation.

We also obtain several new results for the normalized infinity Laplace equation $-\Delta _\infty u = f$. In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous $f$, and continuous boundary data, as well as the uniqueness of solutions to this problem in the generic case. We present a new elementary proof of uniqueness in the case that $f>0$, $f< 0$, or $f\equiv 0$. The stability of the solutions with respect to $f$ is also studied, and an explicit continuous dependence estimate from $f\equiv 0$ is obtained.

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## Additional Information

**Scott N. Armstrong**- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- Email: sarm@math.berkeley.edu
**Charles K. Smart**- Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
- MR Author ID: 893148
- Email: smart@math.berkeley.edu
- Received by editor(s): July 8, 2009
- Published electronically: September 14, 2011
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**364**(2012), 595-636 - MSC (2000): Primary 35J70, 91A15
- DOI: https://doi.org/10.1090/S0002-9947-2011-05289-X
- MathSciNet review: 2846345