Nonlocal Tug‐of‐War and the Infinity Fractional Laplacian
By Clayton Bjorland, Luis Caffarelli, Alessio Figalli. 2010
124 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/f44982a7ef0ba8d0989ca24806aa36ab603a0a6e
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In Applied Mathematics
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Citations in the corpus
- Higher Hölder regularity for the fractional p-Laplacian in the superquadratic case (2017, background): “In [5] and [6] an interesting connection to a nonlocal tug-of-war is found, while the operator introduced in [11] is connected to Lévy processes.”
- Nonlocal Equations with Measure Data (2014, background): “Notice that, upon taking the special case Φ(t) = |t|t, we recover the fractional p-Laplacean operator with measurable coefficients (see [5, 20, 21]).”
- A Hölder infinity Laplacian (2012, background): “Very recently, in [6], a closely related operator has been studied.”
- Nonlocal Diffusion and Applications (2015, background): “Many fundamental topics slipped completely out of these notes: just to name a few, the topological methods and the fine regularity theory in the fractional cases are not presented here, the fully nonlinear or singular/degenerate equations are not taken into account, only very few applications are di”
- Hölder regularity for nonlocal double phase equations (2019, background): “For Hölder estimates and some higher regularity result for related (s, p)-Laplace equations when s approach 1, we refer to [4].”