From Lévy-Type Processes to Parabolic SPDEs

  • Davar Khoshnevisan
  • René Schilling
  • Frederic Utzet
  • Lluis Quer-Sardanyons

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
    6. Davar Khoshnevisan
      Pages 195-201
    7. Davar Khoshnevisan
      Pages 203-216
  4. Back Matter
    Pages 217-220

About this book


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.


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

Authors and affiliations

  • Davar Khoshnevisan
    • 1
  • René Schilling
    • 2
  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

Bibliographic information