Minimum Action Curves in Degenerate Finsler Metrics

Existence and Properties

  • Matthias Heymann

Part of the Lecture Notes in Mathematics book series (LNM, volume 2134)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Results

    1. Front Matter
      Pages 1-1
    2. Matthias Heymann
      Pages 3-11
    3. Matthias Heymann
      Pages 13-28
    4. Matthias Heymann
      Pages 29-55
    5. Matthias Heymann
      Pages 57-65
    6. Matthias Heymann
      Pages 67-68
  3. Proofs

    1. Front Matter
      Pages 69-69
    2. Matthias Heymann
      Pages 97-142
  4. Back Matter
    Pages 143-186

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.



60F10, 51-02, 49J45 Action functional Large deviation theory Minimizer Quasipotential Wentzell-Freidlin theory

Authors and affiliations

  • Matthias Heymann
    • 1
  1. 1.Mathematics DepartmentDuke UniversityDurhamUSA

Bibliographic information

  • DOI
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-17752-6
  • Online ISBN 978-3-319-17753-3
  • Series Print ISSN 0075-8434
  • Series Online ISSN 1617-9692
  • Buy this book on publisher's site
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