Stochastic Partial Differential Equations A Modeling, White Noise Fu
This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera tion between the Norwe
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The front cover shows the time development of a vibrating string under the influence of a local 2-parameter white noise force, i.e., a sample path of a solution to the wave equation
where Wx .t denotes a 2-parameter white noise. Note in particular the wave propagation along the characteristics x + t = 0.25 and x - t = 0.75.
HeJge Holden Bernt 0ksendal Jan Ub~e Tusheng Zhang
St()chastic Partial Differential Equations A ~1odeling, White Noise Functional Approach
1996 Birkhauser Boston • Basel • Berlin
Helge Holden Dept. of Mathematical Sciences Norwegian University of Science and Technology N-7034 Trondheim Norway
Jan Ub¢e Tusheng Zhang StordlHaugesund College Skaregaten 103 N-5500 Haugesund Norway
Bernt 0ksendal Department of Mathematics University of Oslo N-0316 Oslo Norway
Library of Congress Cataloging-in-Publication Data Stochastic partial differential equations: a modeling, white noise functional analysis approach / Helge Holden ... [et a1.]. p. cm. -- (Probability and its applications) Includes bibliographical references (p. 217-222) and index. ISBN 978-1-4684-9217-0 1. Stochastic partial differential equations. (Helge), 1956II. Series. QA274.25.S744 1996 519.2--dc20
Printed on acid-free paper. Birkhiiuser © 1996 Birkhauser Boston Sof'tcover reprint of the hardcover 1st edition 1996
I. Holden, H.
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All rights reserved. This work may not be translated or copied in whole or in part vithoul the written permission of the publisher (Birkhauser Boston, clo Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief exc~rpts in connection with reviews or scholarly analysis. Use in connection with any torm of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this pub1;cation, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly l)e used freely by anyone. ISBN 978-1-4684-9217-0 ISBN 978-1-4684-9215-6 (eBook) DOI 10.1007/978-1-4684-9215-6
Typeset by the Authors in 'lEX.
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To Ingvill, Eva and Qinghua
ALVl.iUvJw~ Wt.CW\I Lk.J (t) }. In this ca~e the solution u of the corresponding SDE will depend on cP also, so we may cnnsider the solution as a function U(cp,t):
S(IIl) x III - (S)-1.
Similarly, for the multiparameter equation the solution u of the corresponding SPDE will be a function
Such prot esses are called functional processes. We stress that from a modeling point of "iew these processes are of interest in their own right, not just as technically convenient "approximations". In fact, there may be cases where it is not even )lhysically relevant to ask what happens if cP - 8 0 (the Dirac measure
8 Chapter 1: Introduction
at 0). Nevertheless, such questions may be mathematically intert~st1ng, both from the point of view of approx
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