Feller processes generated by pseudo-differential operators: On the Hausdorff dimension of their sample paths
type: Item
instance of: scholarly article
title: Feller processes generated by pseudo-differential operators: On the Hausdorff dimension of their sample paths
zbMATH Open document ID: 0910.60028
DOI: 10.1023/A:1022678219821
published in: Journal of Theoretical Probability
publication date: 1999-04-06T00:00:00Z
review text: Let \((X_t)_{t\geq 0}\) be a Feller process with state space \(\mathbb{R}^n\). The corresponding generator is under minimal assumptions on its domain a pseudo-differential operator with symbol \(-q(x,\xi)\), \(q:\mathbb{R}^n \times \mathbb{R}^n \to\mathbb{C}\) is a locally bounded function with respect to \(x\) and a continuous negative definite function with respect to \(\xi\). Suppose that for a fixed continuous negative definite function \(\psi\) we have the estimate \(\sup_{x\in \mathbb{R}^n} | q(x,\xi) |\leq c(1+ \psi(\xi))\). The Hausdorff dimension of the set \(\{X_t;t\in E\}\), \(E\subset [0,1]\) is any analytic set, is almost surely bounded above by \(\beta_\psi \dim E\). Here \(\beta_\psi\) is the Blumenthal-Getoor upper index of the Lévy process associated with \(\psi(\xi)\), and \(\dim E\) is the Hausdorff dimension of \(E\).
reviewed by: Niels Jacob
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 35S99
Mathematics Subject Classification ID: 60J25
zbMATH DE Number: 1208786
zbMATH Keywords: Feller process
zbMATH Keywords: Lévy process
zbMATH Keywords: path properties
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: pseudo-differential operator
author: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: a:1022678219821
OpenAlex ID: W16026228
description: scientific article; zbMATH DE number 1208786