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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

An Introduction to Lévy and Feller Processes

Frontmatter

2016 | OriginalPaper | Buchkapitel

Chapter 1. Orientation

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 2. Lévy Processes

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 3. Examples

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 4. On the Markov Property

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 5. A Digression: Semigroups

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 6. The Generator of a Lévy Process

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 7. Construction of Lévy Processes

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 8. Two Special Lévy Processes

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 9. Random Measures

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 10. A Digression: Stochastic Integrals

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 11. From Lévy to Feller Processes

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 12. Symbols and Semimartingales

René Schilling

2016 | OriginalPaper | Buchkapitel

Chapter 13. Dénouement

René Schilling

Invariance and Comparison Principles for Parabolic Stochastic Partial Differential Equations

Frontmatter

2016 | OriginalPaper | Buchkapitel

Chapter 14. White Noise

Davar Khoshnevisan

2016 | OriginalPaper | Buchkapitel

Chapter 15. Lévy Processes

Davar Khoshnevisan

2016 | OriginalPaper | Buchkapitel

Chapter 16. SPDEs

Davar Khoshnevisan

2016 | OriginalPaper | Buchkapitel

Chapter 17. An Invariance Principle for Parabolic SPDEs

Davar Khoshnevisan

2016 | OriginalPaper | Buchkapitel

Chapter 18. Comparison Theorems

Davar Khoshnevisan

2016 | OriginalPaper | Buchkapitel

Chapter 19. A Dash of Color

Davar Khoshnevisan

Backmatter

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