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Stochastic Partial Differential Equations: An Introduction

  • Wei Liu
  • Michael Röckner

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-vi
  2. Wei Liu, Michael Röckner
    Pages 1-8
  3. Wei Liu, Michael Röckner
    Pages 55-68
  4. Wei Liu, Michael Röckner
    Pages 69-121
  5. Wei Liu, Michael Röckner
    Pages 123-178
  6. Wei Liu, Michael Röckner
    Pages 179-207
  7. Back Matter
    Pages 209-266

About this book

Introduction

This book provides an introduction to the theory of stochastic partial differential equations (SPDEs) of evolutionary type. SPDEs are one of the main research directions in probability theory with several wide ranging applications. Many types of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. The theory of SPDEs is based both on the theory of deterministic partial differential equations, as well as on modern stochastic analysis.

Whilst this volume mainly follows the ‘variational approach’, it also contains a short account on the ‘semigroup (or mild solution) approach’. In particular, the volume contains a complete presentation of the main existence and uniqueness results in the case of locally monotone coefficients. Various types of generalized coercivity conditions are shown to guarantee non-explosion, but also a systematic approach to treat SPDEs with explosion in finite time is developed. It is, so far, the only book where the latter and the ‘locally monotone case’ is presented in a detailed and complete way for SPDEs. The extension to this more general framework for SPDEs, for example, in comparison to the well-known case of globally monotone coefficients, substantially widens the applicability of the results. In addition, it leads to a unified approach and to simplified proofs in many classical examples. These include a large number of SPDEs not covered by the ‘globally monotone case’, such as, for exa

mple, stochastic Burgers or stochastic 2D and 3D Navier-Stokes equations, stochastic Cahn-Hilliard equations and stochastic surface growth models.

To keep the book self-contained and prerequisites low, necessary results about SDEs in finite dimensions are also included with complete proofs as well as a chapter on stochastic integration on Hilbert spaces. Further fundamentals (for example, a detailed account on the Yamada-Watanabe theorem in infinite dimensions) used in the book have added proofs in the appendix. The book can be used as a textbook for a one-year graduate course.

Keywords

Explosive Solutions Gelfand Triples Generalized Coercivity Girsanov Theorem on Hilbert Invariant Measures Itô-Formula Locally Monotone Coefficients Markov Property Stochastic 2D and 3D Navier-Stokes Equation Stochastic Cahn-Hilliard Equations Stochastic Evolution Equations Stochastic Integration on Hilbert Spaces Stochastic Partial Differential Equations Stochastic Porous Media Equations Stochastic Surface Growth Models Stochastic p-Laplace Equations Variational Approach Weak and Strong Solutions Yamada-Watanabe Theorem in Infinite Dimensions

Authors and affiliations

  • Wei Liu
    • 1
  • Michael Röckner
    • 2
  1. 1.School of Mathematics and StatisticsJiangsu Normal UniversityXuzhouChina
  2. 2.Faculty of MathematicsBielefeld UniversityBielefeldGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-22354-4
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-22353-7
  • Online ISBN 978-3-319-22354-4
  • Series Print ISSN 0172-5939
  • Series Online ISSN 2191-6675
  • Buy this book on publisher's site