Nonlocal Tug‐of‐War and the Infinity Fractional Laplacian

Motivated by the “tug‐of‐war” game studied by Peres et al. in 2009, we consider a nonlocal version of the game that goes as follows: at every step two players pick, respectively, a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount ϵ > 0 (as is done in the classical case), it is an s‐stable Levy process that chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically derive a deterministic nonlocal integrodifferential equation that we call the “infinity fractional Laplacian.” We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double‐obstacle problem, both problems having a natural interpretation as tug‐of‐war games. © 2011 Wiley Periodicals, Inc.