Evolution Driven by the Infinity Fractional Laplacian

TL;DR

This paper studies the evolution problem for the infinity fractional Laplacian, establishing existence, equivalence with one-dimensional cases, and long-term behavior of solutions through a global Harnack inequality.

Contribution

It constructs viscosity solutions for the infinity fractional Laplacian evolution problem and proves their equivalence to one-dimensional cases for radially symmetric functions.

Findings

Constructed viscosity solutions for the evolution problem.

Established equivalence with one-dimensional fractional Laplacian for symmetric functions.

Proved a global Harnack inequality explaining long-time behavior.

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.