Towards a characterization of elliptic Harnack inequality for jump processes

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.