Coupling Levy measures and comparison principles for viscosity solutions

TL;DR

This paper establishes new comparison principles for viscosity solutions of non-linear integro-differential equations, utilizing optimal transport to couple Levy measures and extend the applicability of comparison techniques.

Contribution

It introduces a novel method using optimal transport maps to couple Levy measures, enhancing the comparison principles for viscosity solutions of complex integro-differential equations.

Findings

Proves comparison principles for Levy-type operators

Extends applicability to a broader class of non-linear equations

Utilizes optimal transport in the analysis

Abstract

We prove new comparison principles for viscosity solutions of non-linear integro-differential equations. The operators to which the method applies include but are not limited to those of L\'evy-It\^o type. The main idea is to use an optimal transport map to couple two different L\'evy measures, and use the resulting coupling in a doubling of variables argument