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


Authors: Scott N. Armstrong and Charles K. Smart
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
Published electronically: September 14, 2011
MathSciNet review: 2846345
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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
Email: smart@math.berkeley.edu

DOI: https://doi.org/10.1090/S0002-9947-2011-05289-X
Keywords: Infinity Laplace equation, tug-of-war, finite difference approximations
Received by editor(s): July 8, 2009
Published electronically: September 14, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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