Mathematics > Probability
[Submitted on 12 Nov 2025]
Title:Towards a characterization of elliptic Harnack inequality for jump processes
View PDFAbstract: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 $\phi(\lambda) = \tilde{\Omega}(\lambda)$ satisfy EHI, which previous methods were unable to determine.
References & Citations
export BibTeX citation
Loading...
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.