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Towards a characterization of elliptic Harnack inequality for jump processes

Jens Malmquist

TL;DR

The paper advances understanding of the elliptic Harnack inequality (EHI) for isotropic unimodal Lévy jump processes by linking EHI to the jump kernel $j(r)$ and truncated second moments $m_2(r)$. It develops a probabilistic framework based on Meyer decompositions (small/large, small/flat, SBM) and decompositions of Green's and Poisson kernels, enabling positive results for broad classes of jump processes, including bounded perturbations of SBMs. A key contribution is the first known subordinate Brownian motion that does not satisfy EHI, illustrating limitations of prior analytic criteria and highlighting the delicate balance between small- and large-scale jump structures. The results provide a structured path toward a kernel-based characterization of EHI for non-local processes and supply technical tools (Green's function and Poisson kernel decompositions) for future exploration and refinement.

Abstract

Let $X$ be an isotropic unimodal Lévy 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 $φ(λ) = \tildeΩ(λ)$ satisfy EHI, which previous methods were unable to determine.

Towards a characterization of elliptic Harnack inequality for jump processes

TL;DR

The paper advances understanding of the elliptic Harnack inequality (EHI) for isotropic unimodal Lévy jump processes by linking EHI to the jump kernel and truncated second moments . It develops a probabilistic framework based on Meyer decompositions (small/large, small/flat, SBM) and decompositions of Green's and Poisson kernels, enabling positive results for broad classes of jump processes, including bounded perturbations of SBMs. A key contribution is the first known subordinate Brownian motion that does not satisfy EHI, illustrating limitations of prior analytic criteria and highlighting the delicate balance between small- and large-scale jump structures. The results provide a structured path toward a kernel-based characterization of EHI for non-local processes and supply technical tools (Green's function and Poisson kernel decompositions) for future exploration and refinement.

Abstract

Let be an isotropic unimodal Lévy jump process on . We develop probabilistic methods which in many cases allow us to determine whether satisfies the elliptic Harnack inequality (EHI), by looking only at the jump kernel of , 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 satisfy EHI, which previous methods were unable to determine.

Paper Structure

This paper contains 22 sections, 44 theorems, 341 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Subordinate Brownian motions
  4. Counterexample and negative result
  5. TEXT
  6. Meyer decompositions
  7. The zoo of Meyer decompositions
  8. Decompositions of the Green's function and Poisson kernel
  9. Properties of the small/flat decomposition
  10. Preliminary facts about isotropic unimodal Lévy jump processes with bounded jumps
  11. Basic observations
  12. TEXT
  13. Quadratic variation and Berry-Esseen
  14. Proof of main results and examples
  15. Proof of Theorem \ref{['EHI:t:main']}
  16. ...and 7 more sections

Key Result

Lemma 1.3

Let $X$ be an isotropic unimodal Lévy process on $\mathbb{R}^d$. There exists a universal constant $C_1=C_1(d)>0$ (depending only on $d$) such that for all $r>0$, If $X$ satisfies EHI:e:regularJumps, then there exists a $C_2=C_2(c_j, d)>0$ (depending only on $c_j$ and $d$) such that for all $r>0$,

Figures (2)

  • Figure 3.1: An illustration of the small/large and small/flat decompositions
  • Figure 4.1: A visual aide for the proof of Lemma \ref{['EHI:l:FactorOf2ForExits']}. The dotted ball is the reflection of $B(0, R)$ across $X_{\tau_{B(0, R)}}$. The conditional distribution of $X_T$ given $\tau_{B(0, R)} \leq T$ is isotropic unimodal, centered at $X_{\tau_{B(0, R)}}$. Therefore, it assigns equal mass to each of the disjoint balls in the figure. In particular, it assigns mass at most $1/2$ to $B(0, R)$.

Theorems & Definitions (109)

  • Definition 1.1: Harmonicity
  • Definition 1.2: Elliptic Harnack inequality
  • Lemma 1.3
  • proof
  • Theorem 1.4
  • Corollary 1.5
  • proof
  • Corollary 1.6
  • Example 1.7
  • Example 1.8
  • ...and 99 more