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

TL;DR

This paper introduces a non-local tug-of-war game driven by a stable Levy process, leading to the definition and analysis of the 'infinity fractional Laplacian' PDE, with results on existence, uniqueness, and regularity.

Contribution

It formulates a new non-local game model and derives the associated infinity fractional Laplacian PDE, analyzing its fundamental properties.

Findings

Established existence and uniqueness of solutions

Proved regularity results for the PDE

Connected the game dynamics to the non-local PDE framework

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 $\epsilon>0$ (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.