Stochastic Spectral Theory for Selfadjoint Feller Operators A functi
A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Ham
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Series Editors Thomas Liggett Charles Newman Loren Pitt
Michael Demuth Jan A. van Casteren Stochastic Spectral Theory for Selfadjoint Feller Operators A functional integration approach
Springer Basel AG
Authors' addresses Michael Demuth Technische Universitat Clausthal Institut fUr Mathematik Erzstr. 1 38678 Clausthal-Zellerfeld Germany
Jan A. van Casteren University of Antwerp (UlA) Department of Mathematics and Computer Science 2610 Antwerp Belgium
2000 Mathematics Subject Classification: 47D06, 47D02, 47AlO, 47A40, 60125, 60135, 60145, 81Q1O (primary); 47B07, 47B38, 47B65, 47010, 46N50 (secondary)
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data Demuth, Michael: Stochastic spectral theory for selfadjoint feller operators : a functional integration approach / Michael Demuth ; Jan A. van Casteren. - Basel ; Boston; Berlin: Birkhliuser, 2000 (Probability and its applications) ISBN 978-3-0348-9577-4 ISBN 978-3-0348-8460-0 (eBook) DOI 10.1007/978-3-0348-8460-0
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation, reprinting, re-use of ilIustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2000 Springer Basel AG Originally published by Birkhiiuser Verlag in 2000 Softcover reprint of the hardcover 1st edition 2000 Printed on acid-free paper produced from chlorine-free pulp. TCF 00
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Contents Preface
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Vll
A readers guideline
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x
Acknowledgement
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Xl
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1
A Introduction
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1
B Assumptions and Free Feller Generators
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5
C Examples
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11
D Heat kernels
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25
E Summary of Schrodinger semigroup theory E.1 Gaussian processes E.2 Brownian motion and related processes E.3 Kato-Feller potentials for the Laplace operator E.4 Schrodinger semigroups E.5 Generalizations and modifications
. . . . . .
33 33 39
Chapter 1 Basic Assumptions of Stochastic Spectral Analysis: Free Feller Operators
Chapter 2 Perturbations of Free Feller Operators
47 49 51
53
The framework of stochastic spectral analysis
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56
A Regular perturbations
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57
B Integral kernels, martingales, pinned measures
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76 87
C Singular perturbations
Chapter 3 Proof of Continuity and Symmetry of Feynman-Kac Kernels..............
103
Chapter 4 Resolvent and Semigroup Differences for Feller Operators: Operator Norms.........................................................
129
A Regular perturbations. ..
. . . . ..
B Singular perturbations
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129 145
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CONTENTS
vi
Chapter 5 Hilbert-Schmidt Properties of Resolvent and Semigroup Differences A Regular perturbations B Singular perturbations . 0 0 0 0 0 0
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Chapter 6 Trace Class Properties of Semigroup Dif
