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© 2016

From Lévy-Type Processes to Parabolic SPDEs

  • Frederic Utzet
  • Lluis Quer-Sardanyons
Textbook

Part of the Advanced Courses in Mathematics - CRM Barcelona book series (ACMBIRK)

Table of contents

  1. Front Matter
    Pages i-viii
  2. An Introduction to Lévy and Feller Processes

    1. Front Matter
      Pages 1-6
    2. René Schilling
      Pages 7-11
    3. René Schilling
      Pages 13-16
    4. René Schilling
      Pages 17-25
    5. René Schilling
      Pages 27-33
    6. René Schilling
      Pages 35-40
    7. René Schilling
      Pages 41-48
    8. René Schilling
      Pages 49-54
    9. René Schilling
      Pages 55-61
    10. René Schilling
      Pages 63-72
    11. René Schilling
      Pages 73-85
    12. René Schilling
      Pages 87-97
    13. René Schilling
      Pages 99-107
    14. René Schilling
      Pages 109-125
  3. Invariance and Comparison Principles for Parabolic Stochastic Partial Differential Equations

    1. Front Matter
      Pages 127-132
    2. Davar Khoshnevisan
      Pages 133-149
    3. Davar Khoshnevisan
      Pages 151-165
    4. Davar Khoshnevisan
      Pages 167-181
    5. Davar Khoshnevisan
      Pages 183-193

About this book

Introduction

This volume presents the lecture notes from two courses given by Davar Khoshnevisan and René Schilling, respectively, at the second Barcelona Summer School on Stochastic Analysis.

René Schilling’s notes are an expanded version of his course on Lévy and Lévy-type processes, the purpose of which is two-fold: on the one hand, the course presents in detail selected properties of the Lévy processes, mainly as Markov processes, and their different constructions, eventually leading to the celebrated Lévy-Itô decomposition. On the other, it identifies the infinitesimal generator of the Lévy process as a pseudo-differential operator whose symbol is the characteristic exponent of the process, making it possible to study the properties of Feller processes as space inhomogeneous processes that locally behave like Lévy processes. The presentation is self-contained, and includes dedicated chapters that review Markov processes, operator semigroups, random measures, etc.

In turn, Davar Khoshnevisan’s course investigates selected problems in the field of stochastic partial differential equations of parabolic type. More precisely, the main objective is to establish an Invariance Principle for those equations in a rather general setting, and to deduce, as an application, comparison-type results. The framework in which these problems are addressed goes beyond the classical setting, in the sense that the driving noise is assumed to be a multiplicative space-time white noise on a group, and the underlying elliptic operator corresponds to a generator of a Lévy process on that group. This implies that stochastic integration with respect to the above noise, as well as the existence and uniqueness of a solution for the corresponding equation, become relevant in their own right. These aspects are also developed and supplemented by a wealth of illustrative examples.

Keywords

Stochastic partial differential equations invariance principle comparison principle Lévy processes Feller processes pseudo-differential operator

Authors and affiliations

  1. 1.Department of MathematicsUniversity of UtahSalt Lake CityUSA
  2. 2.Institut für Mathematische StochastikFachrichtung Mathematik TU DresdenDresdenGermany

Editors and affiliations

  • Frederic Utzet
    • 1
  • Lluis Quer-Sardanyons
    • 2
  1. 1.Departament de MatemàtiquesUniversitat Autònoma de Barcelona Departament de MatemàtiquesBellaterraSpain
  2. 2.Departament de MatematiquesUniversitat Autonoma de Barcelona Departament de MatematiquesBellaterraSpain

About the editors

Davar Khoshnevisan is Professor of Mathematics at The University of Utah.

René L. Schilling is Professor of Probability at Technische Universität Dresden.

Bibliographic information

  • Book Title From Lévy-Type Processes to Parabolic SPDEs
  • Authors Davar Khoshnevisan
    René Schilling
  • Editors Frederic Utzet
    Lluis Quer-Sardanyons
  • Series Title Advanced Courses in Mathematics - CRM Barcelona
  • Series Abbreviated Title Advanced Courses in Mathematics -CRM Barcelona (Birkhäuser)
  • DOI https://doi.org/10.1007/978-3-319-34120-0
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-34119-4
  • eBook ISBN 978-3-319-34120-0
  • Series ISSN 2297-0304
  • Series E-ISSN 2297-0312
  • Edition Number 1
  • Number of Pages VIII, 220
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Probability Theory and Stochastic Processes
    Partial Differential Equations
  • Buy this book on publisher's site
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Reviews

“The presentation also includes materials reviewing the classical theory of Markov processes, operator semigroups and random measures, which makes the notes self-contained and an excellent introductory material to the theory of  Lévy and Feller processes. ... It is nice to read and it provides exhaustive treatments of the topics.” (Nikola Sandrić, zbMATH 1382.60005, 2018)