Pseudo differential operators and Markov processes. In 3 vol. Vol. 1: Fourier analysis and semigroups
type: Item
instance of: scholarly article
zbMATH Open document ID: 0987.60003
publication date: 2002-02-19T00:00:00Z
full work available at URL: toc.shtml
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 47D07
Mathematics Subject Classification ID: 60J25
Mathematics Subject Classification ID: 31C25
zbMATH DE Number: 1707584
zbMATH Keywords: Fourier analysis
zbMATH Keywords: pseudo-differential operator
zbMATH Keywords: Dirichlet form
zbMATH Keywords: negative definite function
zbMATH Keywords: completely monotone function
zbMATH Keywords: Bernstein function
zbMATH Keywords: operator semigroup
zbMATH Keywords: Markov semigroup
zbMATH Keywords: Feller semigroup
author: Niels Jacob
MaRDI profile type: Publication
title: Pseudo differential operators and Markov processes. In 3 vol. Vol. 1: Fourier analysis and semigroups
review text: This is the first of three volumes in a treatise on ``Pseudo differential operators and Markov processes''. Under this common heading three parts are planned, ``Fourier analysis and semigroups'' (the volume under review), ``Generators and potential theory'' (vol. 2, to appear in winter 2002) and ``Markov processes and applications'' (vol. 3). The intention of these monographs is to give a complete and self-contained account of the theory of stochastic processes which are generated by pseudo-differential operators, in particular on Lévy-type and Feller processes. Many contributions to the field were developed over the last ten years by the author and the treatise is a state-of-the art summary of the analytic side of the theory; various new and unpublished results are included. NEWLINENEWLINENEWLINEThe present work elaborates a previous research note by the author which appeared some years earlier under the same title (1996; Zbl 0860.60002) and which was a non-technical survey of the topic, essentially without proofs. The new monograph remedies this shortcoming. It is written in a style that is accessible to non-specialists and novices and strives to be as self-contained as possible. There are some good reasons for this: the subject under investigation requires tools from quite different disciplines, extending from modern methods of PDEs, harmonic analysis, semigroup theory, to potential theory, Dirichlet forms and, later, stochastic processes. Results which are needed as prerequisites are collected with careful references to accessible sources and more central results are developed in the text. This digest of material is not only an invaluable service for the reader but it serves also a further purpose: different theories have different notations and normalizations for the same objects. NEWLINENEWLINENEWLINEThe present exposition gives a uniform and consistent approach aiming to close the (language-) gap between analysis and probability theory. Volume 1 contains most…
reviewed by: R. L. Schilling
description: scientific article; zbMATH DE number 1707584