Overview
- Studies invariance and comparison principles for parabolic SPDEs in a very general framework beyond the classical setting
- Presents an extensive introduction to Lévy processes, including the different constructions
- Provides properties of Feller processes as space inhomogeneous processes that behave locally like Lévy processes
Part of the book series: Advanced Courses in Mathematics - CRM Barcelona (ACMBIRK)
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Table of contents (19 chapters)
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Invariance and Comparison Principles for Parabolic Stochastic Partial Differential Equations
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Back Matter
Keywords
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.
Inturn, 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.
Reviews
Authors, Editors and Affiliations
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
DOI: https://doi.org/10.1007/978-3-319-34120-0
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2016
Softcover ISBN: 978-3-319-34119-4Published: 05 January 2017
eBook ISBN: 978-3-319-34120-0Published: 22 December 2016
Series ISSN: 2297-0304
Series E-ISSN: 2297-0312
Edition Number: 1
Number of Pages: VIII, 220
Topics: Probability Theory and Stochastic Processes, Partial Differential Equations
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