On the Geometry of Diffusion Operators and Stochastic Flows
Stochastic differential equations, and Hoermander form representations of diffusion operators, can determine a linear connection associated to the underlying (sub)-Riemannian structure. This is systematically described, together with its invariants, and t
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1720
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris
1720
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
K. D. Elworthy Y. Le Jan Xue-Mei Li
On the Geometry of Diffusion Operators and Stochastic Flows
Springer
Authors K. David Elworthy Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom E-mail: [email protected]
Xue-Mei Li Department of Mathematics University of Connecticut 196 Auditorium Road Storrs, CT 06269, USA E-mail: [email protected]
Yves Le Jan Departernent de Mathematique Universite Paris Sud 91405 Orsay, France E-mail: [email protected] Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Elworthy, David: On the geometry of diffusion operators and stochastic flows / D. Elworthy ; Y. Le Jan ; X.-M. Li. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo : Springer, 1999 (Lecture notes in mathematics ; 1720)
ISBN 3-540-66708-3
Mathematics Subject Classification (1991): 58G32, 53B05, 60HlO, 60H07, 58B20, 58030, 53C05, 53C21, 93EI5 ISSN 0075- 8434 ISBN 3-540-66708-3 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: 10700319 4113143du-543210
Contents Introduction 1
Construction of connections 1.1 Construction of connections . . . . . . . . . 1.2 Basic Classes of Examples . . . . . . . . . . 1.3 Adjoint connections, torsion skew symmetry, 1.4 Example: Homogeneous spaces continued .
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. . . . . . . . . . . . . . . . . . basic formulae . . . . . . .
infinitesimal generators and associated operators The irrelevance of drift in dimension greater than 1 . Torsion Skew Symmetry . . . . . . . . . . . . . . The 'divergence operator' J . . . . . . . . . . . . Horrnander form generators on differential forms On the infinitesimal generator . . . . . . . 2.5.1 Example............... 2.5.2 Symmetricity of the generator Aq .
7 7 14 18 26
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The 2.1 2.2 2.3 2.4 2.5
30 30 35 37 42 47 47 49
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Decomposition of n
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