Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators
This book addresses both probabilists working on diffusion processes and analysts interested in linear parabolic partial differential equations with singular coefficients. The central question discussed is whether a given diffusion operator, i.e., a secon
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1718
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1718
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Andreas Eberle
Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators
Springer
Author Andreas Eberle Faculty of Mathematics University of Bielefeld P.O. Box 100131 33501 Bielefeld, Germany E-mail: [email protected] Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Eberle, Andreas: Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators / Andreas Eberle. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo : Springer, 1999 (Lecture notes in mathematics; 1718) Zugl: Bielefeld, Univ., Diss., 1998 ISBN 3-540-66628-1
Mathematics Subject Classification (1991): 60J60, 35KIO, 31C25, 47F05, 60135, 47B25, 60H30, 60H15, 34B24 ISSN 0075-8434 ISBN 3-540-66628-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1999 Printed in Germany
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10700296 41/3143-543210
Preface This book is aimed at both probabilists working on diffusion processes and analysts interested in linear parabolic partial differential equations with singular coefficients. The central question discussed is whether a given diffusion operator, i.e., a second order linear elliptic differential operator without zeroth order term, which is a priori only defined on test functions over some ( finite or infinite dimensional) state space, uniquely determines a strongly continuous semigroup on a corresponding weighted LP space. This general uniqueness problem is related to several specific questions arising in different areas, e.g., uniqueness of martingale problems, existence of operator cores consisting of "nice" functions, essential selfadjointness and uniqueness problems in mathematical physics, as well as uniqueness problems for Dirichlet forms. We give particular emphasis to pointing out phenomena responsible for nonuniqueness, as well as to cla
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