Regularity theory for fully nonlinear integro-differential equations

TL;DR

This paper develops a regularity theory for fully nonlinear integro-differential equations, extending classical PDE results to nonlocal equations with jump processes, including ABP estimates, Harnack inequality, and interior regularity.

Contribution

It introduces a nonlocal regularity framework that remains uniform as the order approaches two, bridging the gap between PDE and nonlocal integro-differential equations.

Findings

Established a nonlocal ABP estimate

Proved a Harnack inequality for solutions

Achieved interior $C^{1,eta}$ regularity

Abstract

We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump L\`evy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior $C^{1,\alpha}$ regularity for general fully nonlinear integro-differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.