Stochastic Partial Differential Equations

Universitext Editorial Board (North America): S. Axler K.A. Ribet For other titles published in this series, go to http://www.springer.com/series/223 Helge Holden Bernt Øksendal Jan Ubøe Tusheng Zhang Stochastic Partial Differential Equations A Modeling, White Noise Functional Approach Second Edition 123 Helge Holden Bernt Øksendal Department of Mathematical Sciences Department of Mathematics Norwegian University of Science University of Oslo and Technology 0316 Oslo 7491 Trondheim Blindern Norway Norway [email protected] and Center of Mathematics and Applications University of Oslo 0316 Oslo Norway [email protected] Jan Ubøe Tusheng Zhang Department of Economics University of Manchester Norwegian School of Economics School of Mathematics and Business Administration Manchester M13 9PL 5045 Bergen United Kingdom Norway [email protected] [email protected] Editorial Board: Sheldon Axler, San Francisco State University Vincenzo Capasso, Universitá degli Studi di Milano Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Claude Sabbah, CNRS, Ecole´ Polytechnique Endre Süli, University of Oxford Wojbor Woyczynski, ´ Case Western Reserve University ISBN 978-0-387-89487-4 e-ISBN 978-0-387-89488-1 DOI 10.1007/978-0-387-89488-1 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009938826 Mathematics Subject Classification (2000): 60H15, 35R60, 60H40, 60J60, 60J75 c First Edition published by Birkhäuser Boston, 1996 c Second Edition published by Springer Science+Business Media, LLC, 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To Eva, Qinghua, and Tatiana At vide hvad man ikke véd, er dog en slags alvidenhed. Knowing what thou knowest not is in a sense omniscience. Piet Hein Piet Hein gruk and grooks reprinted with kind permission of Piet Hein a.s., DK-Middelfart c . Preface to the Second Edition Since the first edition of this book appeared in 1996, white noise theory and its applications have expanded to several areas. Important examples are (i) White noise theory for fractional Brownian motion. See, e.g., Biagini et al. (2008) and the references therein. (ii) White noise theory as a tool for Hida–Malliavin calculus and anticipative stochastic calculus, with applications to finance. See, e.g., Di Nunno et al. (2009) and the references therein. (iii) White noise theory for Lévy processes and Lévy random fields, with applications to SPDEs. The last area (iii) fits well into the scope of this book, and it is natural to include an account of this interesting development in this second edition. See the new Chapter 5. Moreover, we have added a remarkable, new result of Lanconelli and Proske (2004), who use white noise theory to obtain a striking general solution formula for stochastic differential equations. See the new Section 3.7. In the new Chapter 5 we provide an introduction to the more general theory of white noise based on Lévy processes and Lévy random fields, and we apply this theory to the study SPDEs driven by this type of noise. This is an active area of current research. We show that the white noise machinery developed in the previous chapters is robust enough to be adapted, after some basic modifications, to the new type of noise. In particular, we obtain the corresponding Wick product, generalized Skorohod integration and Hermite transform in the Lévy case, and we get the same general solution procedure for SPDEs. The method is illustrated by a study of the (stochastic) Poisson equation, the wave equation and the heat equation involving space or space-time Lévy white noise. In this second edition we have also improved the presentation at some points and corrected misprints. We are grateful to the readers for their posi- tive responses and constructive remarks. In particular we would like to thank (in alphabetical order) Atle Gyllensten, Jørgen Haug, Frank Proske, Mikael Signahl, and Gjermund V˚age for many interesting and useful comments. Trondheim Helge Holden Oslo Bernt Øksendal Bergen Jan Ubøe Manchester Tusheng Zhang January 2009 ix Preface to the First Edition 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 cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or “noisy”. We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corresponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in reservoir theory and related areas. 3) The theory should be strong and efficient enough to allow us to solve these SPDEs explicitly, or at least provide algorithms or approximations for the solutions. We gradually discovered that the theory that we had developed in an effort to satisfy these three conditions also was applicable to a number of SPDEs other than those related to fluid flow. Moreover, this theory led to a new and useful way of looking at stochastic ordinary differential equations as well. We therefore feel that this approach to SPDEs is of general interest and deserves to be better known. This is our main motivation for writing this book, which gives a detailed presentation of the theory, as well as its application to ordinary and partial stochastic differential equations. We emphasize that our presentation does not make any attempts to give a comprehensive account of the theory of SPDEs in general. There are a number of important contributions that we do not mention at all. We also emphasize that our approach rests on the fundamental work by K. Itôon stochastic calculus, T. Hida’s work on white noise analysis, and J. Walsh’s early papers on SPDEs. Moreover, our work would not have been possible without the inspiration and wisdom of our colleagues, and we are grateful to them all. In particular, we would like to thank Fred Espen Benth, Jon Gjerde, H˚akon Gjessing, Harald Hanche-Olsen, Vagn Lundsgaard Hansen, Takeyuki Hida, Yaozhong Hu, Yuri Kondratiev, Tom Lindstrøm, Paul-André Meyer, Suleyman Ust¨¨ unel and Gjermund V˚age for helpful discussions and comments. xi xii Preface to the First Edition And, most of all, we want to express our gratitude to Jürgen Potthoff.He helped us to get started, taught us patiently about the white noise calculus and kept us going thanks to his continuous encouragement and inspiration. Finally, we would like to thank Tove Christine Møller and Dina Haraldsson for their excellent typing. Bernt Øksendal would like to thank the University of Botswana for its hospitality during parts of this project. Helge Holden acknowledges partial support from Norges forskningsr˚ad. We are all grateful to VISTA for their generous support of this project. April 1996 Helge Holden Bernt Øksendal Jan Ubøe Tusheng Zhang How to use this book This book may be used as a source book for researchers in white noise theory and SPDEs. It can also be used as a textbook for a graduate course or a research seminar on these topics. Depending on the available time, a course outline could, for instance, be as follows: Sections 2.1–2.8, Section 3.1, Section 4.1, and a selection of Sec- tions 4.2–4.8, Sections 5.1–5.7. Contents Preface to the Second Edition ................................ ix Preface to the First Edition .................................. xi 1 Introduction .............................................. 1 1.1 Modeling by Stochastic Differential Equations . 1 2 Framework ............................................... 13 2.1 WhiteNoise........................................... 13 2.1.1 The 1-Dimensional, d-Parameter Smoothed White Noise........................................... 13 2.1.2 The (Smoothed) White Noise Vector . 20 2.2 The Wiener–ItôChaosExpansion ........................ 21 2.2.1 Chaos Expansion in Terms of Hermite Polynomials . 21 2.2.2 Chaos Expansion in Terms of Multiple Itô Integrals . 29 2.3 The Hida Stochastic Test Functions and Stochastic S m;N S m;N Distributions. The Kondratiev Spaces ( )ρ , ( )−ρ ....... 31 2.3.1 The Hida Test Function Space (S) and the Hida Distribution Space (S)∗ ........................... 40 2.3.2 Singular White Noise . 42 2.4 The Wick Product . 43 2.4.1 Some Examples and Counterexamples . 47 2.5 Wick Multiplication and Hitsuda/Skorohod Integration. 50 2.6 The Hermite Transform . 61 S N S 2.7 The ( )ρ,r Spaces and the -Transform................... 75 S N 2.8 The Topology of ( )−1 .................................

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