Overview
- Explores the non-standard geometric view of the Wentzell-Freidlin theory of rare transition events
- The general geometric framework may spawn applications outside of probability theory
- Key results and their explanations are well-separated from the necessary technical proofs, making it easy to quickly use the proven existence criteria in practice
- Includes many intuitive examples with color illustrations
- Only a knowledge of graduate level analysis is required; all non-standard concepts are introduced as needed
- Provides detailed complete proofs that do not require any additional work by the reader to fill the gaps
- Includes supplementary material: sn.pub/extras
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2134)
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Table of contents (7 chapters)
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Front Matter
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Proofs
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Front Matter
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Back Matter
Keywords
About this book
Presenting a study of geometric action functionals (i.e., non-negative functionals on the space of unparameterized oriented rectifiable curves), this monograph focuses on the subclass of those functionals whose local action is a degenerate type of Finsler metric that may vanish in certain directions, allowing for curves with positive Euclidean length but with zero action. For such functionals, criteria are developed under which there exists a minimum action curve leading from one given set to another. Then the properties of this curve are studied, and the non-existence of minimizers is established in some settings.
Applied to a geometric reformulation of the quasipotential of Wentzell-Freidlin theory (a subfield of large deviation theory), these results can yield the existence and properties of maximum likelihood transition curves between two metastable states in a stochastic process with small noise.
The book assumes only standard knowledge in graduate-level analysis; all higher-level mathematical concepts are introduced along the way.
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Authors and Affiliations
Bibliographic Information
Book Title: Minimum Action Curves in Degenerate Finsler Metrics
Book Subtitle: Existence and Properties
Authors: Matthias Heymann
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-17753-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2015
Softcover ISBN: 978-3-319-17752-6Published: 21 July 2015
eBook ISBN: 978-3-319-17753-3Published: 08 July 2015
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: XV, 186
Number of Illustrations: 3 b/w illustrations, 11 illustrations in colour
Topics: Probability Theory and Stochastic Processes, Geometry, Optimization, Mathematics, general