Counterexamples to elliptic Harnack inequality for isotropic unimodal Lévy processes

Until now, it has been an open question whether every subordinated Brownian motion (SBM) satisfies the elliptic Harnack inequality (EHI). In this paper, we show that the answer is ``no." In our first theorem, we show that if $X=(X_t{)}_{t\ge 0}$ is an isotropic unimodal Lévy process, and $X$ satisfies certain criteria (involving the jump kernel of $X$ and the distribution of the location upon first exiting balls of various sizes) then $X$ does not satisfy EHI. (Note that the class of isotropic unimodal Lévy processes is larger than the class of SBMs.) We then check that many specific SBMs do indeed satisfy the criteria, and thus do not satify EHI.