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

DOI10.1090/S0002-9947-2011-05289-XzbMATH Open1239.91011arXiv0906.2871MaRDI QIDQ3116566

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Publication date: 24 February 2012

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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. This new tug-of-war game is identical to the original except near the boundary of the domain $partialOmega$, 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 $epsilono0$, 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 $-Delt{a}_{i}nftyu=f$. 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 $fequiv0$. The stability of the solutions with respect to $f$ is also studied, and an explicit continuous dependence estimate from $fequiv0$ is obtained.

Full work available at URL: https://arxiv.org/abs/0906.2871

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