Non-Local Tug-of-War and the Infinity Fractional Laplacian

2010/11/09 by Bjorland, Clayton, Caffarelli, Luis, Figalli, Alessio · 3 citations
#35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1011.1966

Abstract

Motivated by the "tug-of-war" game studied in [12], we consider a "non-local" version of the game which 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 of a fixed amount ε>0 (as is done in the classical case), it is a s-stable Levy process which chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically we derive a deterministic non-local integro-differential equation that we call "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.

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• Exponential Absolute Minimizing extension and biased infinity Laplacian
• On a gradient term for a class of second-order PDEs and applications to the infinity Laplace equation

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