Evolution Driven by the Infinity Fractional Laplacian

Abstract

We consider the evolution problem associated to the infinity fractional Laplacian introduced by Bjorland, Caffarelli and Figalli (2012) as the infinitesimal generator of a non-Brownian tug-of-war game. We first construct a class of viscosity solutions of the initial-value problem for bounded and uniformly continuous data. An important result is the equivalence of the nonlinear operator in higher dimensions with the one-dimensional fractional Laplacian when it is applied to radially symmetric and monotone functions. Thanks to this and a comparison theorem between classical and viscosity solutions, we are able to establish a global Harnack inequality that, in particular, explains the long-time behavior of the solutions.

26 pages, 5 figures

References in corpus (1)

• Evolution Driven by the Infinity Fractional Laplacian

Cited by in corpus (2)

• Evolution Driven by the Infinity Fractional Laplacian
• The convergence rate of $p$-harmonic to infinity-harmonic functions