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Stochastic Partial Differential Equations

A Modeling, White Noise Functional Approach

  • Textbook
  • © 2010
  • Latest edition


  • Focuses on the development of SPDEs and their application both to real-life problems and abstract mathematical topics
  • Includes new discussions of fractional Brownian motion, Lévy processes and Lévy random fields, and applications to finance
  • Provides an excellent introduction to the field and areas of current research
  • Exercises at the end of each chapter
  • Includes supplementary material:

Part of the book series: Universitext (UTX)

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Table of contents (5 chapters)


About this book

The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs driven by space-time Brownian motion noise. In this, the second edition, the authors extend the theory to include SPDEs  driven by space-time Lévy process noise, and introduce new applications of the field.

Because the authors allow the noise to be in both space and time, the solutions to SPDEs are usually of the distribution type, rather than a classical random field. To make this study rigorous and as general as possible, the discussion of SPDEs is therefore placed in the context of Hida white noise theory. The key connection between white noise theory and SPDEs is that integration with respect to Brownian random fields can be expressed as integration with respect to the Lebesgue measure of the Wick product of the integrand with Brownian white noise, and similarly with Lévy processes.

The first part of the book deals with the classical Brownian motion case. The second extends it to the Lévy white noise case. For SPDEs of the Wick type, a general solution method is given by means of the Hermite transform, which turns a given SPDE into a parameterized family of deterministic PDEs. Applications of this theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance.

Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter.

From the reviews of the first edition:

"The authors have made significant contributions to each of the areas. As a whole, the book is well organized and very carefully written and the details of the proofs are basically spelled out... This is a rich and demanding book… It will be of great value for students ofprobability theory or SPDEs with an interest in the subject, and also for professional probabilists."   —Mathematical Reviews

"...a comprehensive introduction to stochastic partial differential equations."   —Zentralblatt MATH

Authors and Affiliations

  • Universitet NTNU, Norges Teknisk Naturvitenskap., Trondheim, Norway

    Helge Holden

  • University of Oslo, Department of Mathematics, Oslo, Norway

    Bernt Øksendal

  • Business Administration, Norwegian School of Economics and, Bergen, Norway

    Jan Ubøe

  • School of Mathematics, University of Manchester, Manchester, United Kingdom

    Tusheng Zhang

About the authors

Helge Holden is a professor of mathematics at the Norwegian University of Science and Technology and an adjunt professor at the Center of Mathematics for Applications, part of the University of Oslo.  He has done extensive research in stochastic analysis, in particular in its application to flow in porous media.

Bernt Øksendal is a professor at the Center of Mathematics for Applications at the University of Oslo.  He is a winner of the Nansen Prize for research in stochastic analysis and its applications.

Jan Ubøe is a professor in the Department of Finance and Management Sciences at the Norwegian School of Economics and Business Administration.  He has written many papers about this subject.

Tusheng Zhang is a professor of probability at the University of Manchester.  His current area of research is stochastic differential and partial differential equations, and he recently published a monograph on fractional Brownian fields with Bernt Øksendal and others.

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