Conservativeness of semigroups generated by pseudo differential operators
type: Item
instance of: scholarly article
title: Conservativeness of semigroups generated by pseudo differential operators
zbMATH Open document ID: 0917.60066
DOI: 10.1023/A:1008664419747
published in: Potential Analysis
publication date: 1999-07-18T00:00:00Z
review text: Let \(q(x,D)\) be a pseudo-differential operator with a symbol \(q(x,\xi)\). In a series of papers by Jacob and Hoh sufficient conditions were found for \(q(x,D)\) to be a generator of a Feller semigroup; see in particular \textit{N. Jacob} [Math. Z. 215, No. 1, 151-166 (1994; Zbl 0795.35154)]. The author assumes that \(q(x,D)\) is a Feller generator and that an essential part of the above conditions holds (so that \(q(x,D)\) is a sum of a constant coefficient Lévy generator and a certain perturbation). It is shown that \(q(x,D)\) is conservative if and only if \(q(x,0)\equiv 0\). The sufficiency of this condition was already known [see \textit{W. Hoh}, Stochastics Stochastics Rep. 55, No. 3/4, 225-252 (1995; Zbl 0880.47029)], but the author gives a new proof. The necessity was established by \textit{N. Jacob} [Potential Anal. 8, No. 1, 61-68 (1998; Zbl 0908.60041)] only under strong additional assumptions.
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 35S05
zbMATH DE Number: 1241119
zbMATH Keywords: conservative semigroup
zbMATH Keywords: Feller semigroup
zbMATH Keywords: Markov semigroup
zbMATH Keywords: pseudo-differential operator
author: R. L. Schilling
reviewed by: Anatoly N. Kochubei
MaRDI profile type: Publication
full work available at URL: a:1008664419747
OpenAlex ID: W185846395
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description: scientific article; zbMATH DE number 1241119