Harnack inequality for weakly coupled nonlocal systems

TL;DR

This paper establishes a scale-invariant Harnack inequality for weakly coupled systems of nonlocal operators with both diffusion and jump components, using probabilistic methods and Green function estimates.

Contribution

It introduces a novel probabilistic approach to prove Harnack inequalities for complex nonlocal systems with general jump kernels and coupling structures.

Findings

Proves scale-invariant Harnack inequality for nonlocal systems

Establishes full rank Harnack inequality under irreducibility

Utilizes Green function estimates for nonlocal operators

Abstract

In this paper, we consider a weakly coupled system of nonlocal operators which contain both diffusion part with uniformly elliptic diffusion matrices and bounded drift vectors and the jump part with relatively general jump kernels. We use the two-sided scale-invariant Green function estimation to prove the scale-invariant Harnack inequality for the weakly coupled nonlocal systems. In the case where the switching rate matrix is strictly irreducible, the scale-invariant full rank Harnack inequality is proved. Our approach is mainly probabilistic.