Potential Analysis of Stable Processes and its Extensions

Stable Lévy processes and related stochastic processes play an important role in stochastic modelling in applied sciences, in particular in financial mathematics. This book is about the potential theory of stable stochastic processes. It also deals with related topics, such as the subordinate Brownian motions (including the relativistic process) and Feynman–Kac semigroups generated by certain Schrödinger operators. The authors focus on classes of stable and related processes that contain the Brownian motion as a special case.

This is the first book devoted to the probabilistic potential theory of stable stochastic processes, and, from the analytical point of view, of the fractional Laplacian. The introduction is accessible to non-specialists and provides a general presentation of the fundamental objects of the theory. Besides recent and deep scientific results the book also provides a didactic approach to its topic, as all chapters have been tested on a wide audience, including young mathematicians at a CNRS/HARP Workshop, Angers 2006.

The reader will gain insight into the modern theory of stable and related processes and their potential analysis with a theoretical motivation for the study of their fine properties.

• Brownian motion
• Lévy process
• Stochastic processes
• alpha-harmonic functions
• fractional Laplacian
• probabilistic potential theory
• stable processes
• stochastic process
• subordinate processes

From the reviews:

“The book is a collection of articles on all aspects of the potential theory of stable processes on R^d, written by well-known experts in this field, thereby summarizing recent results in various papers in a unified presentation. … readers interested in the subject will find this book extremely helpful.” (Wilhelm Stannat, Mathematical Reviews, Issue 2011 i)

“Recently, a lot of progress has been made in the potential theory of stable processes and related Lévy processes. This book is a collection of surveys on some of these recent results made into a book form. This book should be very useful for researchers and graduate students to read the recent progress in the potential theory of stable processes and their generalizations.” (Renming Song, Zentralblatt MATH, Vol. 1203, 2011)

• Labo. Angevin de Recherche en, Universitè Angers, Angers CX 01, France

  Piotr Graczyk

• Labo. Mathématiques, Université Clermont-Ferrand II, Aubière CX, France

  Andrzej Stos

• Inst. Mathematics & Computer Science, Wroclaw University of Technology, Wroclaw, Poland

  Krzysztof Bogdan, Tomasz Byczkowski, Michal Ryznar

• Inst. Mathematics, PAN Wroclaw, Wroclaw, Poland

  Tadeusz Kulczycki

• Dept. Mathematics, University of Illinois, Urbana, Urbana, U.S.A.

  Renming Song

• Dept. Mathematics, University of Zagreb, Zagreb, Croatia

  Zoran Vondracek

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