Counterexamples to elliptic Harnack inequality for isotropic unimodal L\'{e}vy processes

TL;DR

This paper demonstrates that many isotropic unimodal Lévy processes, including certain subordinated Brownian motions, do not satisfy the elliptic Harnack inequality, challenging previous assumptions in the field.

Contribution

It provides the first counterexamples showing that not all isotropic unimodal Lévy processes satisfy the elliptic Harnack inequality, expanding understanding beyond subordinated Brownian motions.

Findings

Many specific subordinated Brownian motions do not satisfy EHI.

Established criteria under which isotropic unimodal Lévy processes violate EHI.

Counterexamples challenge previous beliefs about EHI applicability.

Abstract

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 $X=(X_t)_{t \geq 0}$ is an isotropic unimodal L\'{e}vy 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 L\'{e}vy 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.