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 , 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 , 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 . In particular, we demonstrate the existence of solutions to the Dirichlet problem for any bounded continuous , 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 , , or . The stability of the solutions with respect to is also studied, and an explicit continuous dependence estimate from is obtained.
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