Invariant measures and symmetry property of Lévy type operators
type: Item
instance of: scholarly article
title: Invariant measures and symmetry property of Lévy type operators
zbMATH Open document ID: 0978.60096
DOI: 10.1023/A:1008705820024
author: Barbara Rüdiger
author: Jiang-Lun Wu
author: S. Albeverio
published in: Potential Analysis
publication date: 2002-01-24T00:00:00Z
review text: Infinitesimal invariant probability measures of elliptic integro-differential operators on \(\mathbb{R}^d\) of type \[ Lf= a_1\Delta f+ a_2(-\Delta)^{\alpha/2}f+ \beta\cdot\nabla f,\qquad f\in{\mathcal S}(\mathbb{R}^d), \] with \(a_1,a_2\in [0,\infty)\), \(a_1+ a_2> 0\), \(\beta: \mathbb{R}^d\to \mathbb{R}^d\) are studied, i.e. measures \(\mu\) satisfying the equation \(L^*\mu= 0\). Under some conditions existence of regular infinitesimal invariant probability measures is proved. It is also shown that for such a measure \(\mu(L,{\mathcal S}(\mathbb{R}^d))\) can only be symmetric on \(L^2(\mathbb{R}^d,\mu)\) if \(a_2= 0\).
Mathematics Subject Classification ID: 60J75
Mathematics Subject Classification ID: 47G20
Mathematics Subject Classification ID: 28C10
zbMATH DE Number: 1546779
zbMATH Keywords: infinitesimal invariant probability measures
zbMATH Keywords: elliptic integro-differential operators
MaRDI profile type: Publication
full work available at URL: a:1008705820024
OpenAlex ID: W47395632
description: scientific article; zbMATH DE number 1546779