A Concise Course on Stochastic Partial Differential Equations
These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. All kinds of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. To keep the tec
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Claudia Prévôt Michael Röckner
A Concise Course on Stochastic Partial Differential Equations
1905
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1905
Claudia Pr´evˆot · Michael R¨ockner
A Concise Course on Stochastic Partial Differential Equations
ABC
Authors Michael R¨ockner Fakult¨at f¨ur Mathematik Universit¨at Bielefeld Universit¨atsstr. 25 33615 Bielefeld Germany e-mail: [email protected]
Claudia Pr´evˆot Fakult¨at f¨ur Mathematik Universit¨at Bielefeld Universit¨atsstr. 25 33615 Bielefeld Germany e-mail: [email protected]
Departments of Mathematics and Statistics Purdue University 150 N. University St. West Lafayette, IN 47907-2067 USA e-mail: [email protected]
Library of Congress Control Number: 2007925694 Mathematics Subject Classification (2000): 35-XX, 60-XX ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-70780-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-70780-6 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-70781-3 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, reuse of illustrations, recitation, broadcasting, reproduction on microfilm 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. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 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 by the authors and SPi using a Springer LATEX macro package Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper
SPIN: 11982159
VA41/3100/SPi
543210
Contents 1. Motivation, Aims and Examples 2. Stochastic Integral in Hilbert Spaces 2.1. Infinite-dimensional Wiener processes . . . . . . . . . . . . 2.2. Martingales in general Banach spaces . . . . . . . . . . . . . 2.3. The definition of the stochastic integral . . . . . . . . . . . 2.3.1. Scheme of the construction of the stochastic integral 2.3.2. The construction of the stochastic integral in detail . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Properties of the stochastic integral . . . . . . . . . . . . . . 2.5. The stochastic integral for cylindrical Wiener processes . . 2.5.1. Cylindrical Wiener processes . . . . . . . . . . . . . 2.5.2. The definition of the stochastic integral . . . . . . .
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3. Stochastic Differential Equations in Finite Dimensions 4
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