Towards a characterization of elliptic Harnack inequality for jump processes

TL;DR

This paper develops probabilistic methods to determine when isotropic unimodal Lévy jump processes satisfy the elliptic Harnack inequality, providing new insights and examples, including the first known subordinated Brownian motion that does not satisfy EHI.

Contribution

It introduces probabilistic criteria based on jump kernels and moments for EHI, and presents the first example of an SBM failing EHI, expanding understanding of jump process regularity.

Findings

Probabilistic methods can determine EHI satisfaction from jump kernels.

Constructed the first SBM example that does not satisfy EHI.

Bounded perturbations of known EHI-satisfying SBMs also satisfy EHI.

Abstract

Let $X$ be an isotropic unimodal L\'{e}vy jump process on $\mathbb{R}^d$. We develop probabilistic methods which in many cases allow us to determine whether $X$ satisfies the elliptic Harnack inequality (EHI), by looking only at the jump kernel of $X$, and its truncated second moments. Both our positive results and our negative results can be applied to subordinated Brownian motions (SBMs) in particular. We produce the first known example of an SBM that does \textit{not} satisfy EHI. We show that for many SBMs that were previously known to satisfy EHI (such as the geometric stable process, the iterated geometric stable process, and the relativistic geometric stable process), bounded perturbations of them also satisfy EHI (which was not previously clear). We show that certain SBMs with Laplace exponent $\phi(\lambda) = \tilde{\Omega}(\lambda)$ satisfy EHI, which previous methods were unable to determine.