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2015 | Book

Introduction Read chapter Read first chapter

Minimum Action Curves in Degenerate Finsler Metrics

Existence and Properties

Author: Matthias Heymann

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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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.

Table of Contents

Frontmatter

Results

Frontmatter
Chapter 1. Introduction
Abstract
In this chapter we introduce the reader to the problem addressed by this monograph. First we explain the main question at hand and its motivation in the context of the Wentzell-Freidlin theory of rare transition paths. We then summarize the main features of our existence theory, and the various approaches used in the literature. Finally, we explain the structure of this monograph and introduce some notation.
Matthias Heymann
Chapter 2. Geometric Action Functionals
Abstract
In this chapter we begin by teaching the reader all the necessary basics of rectifiable curves and absolutely continuous functions. We then introduce the class of geometric action functionals to which our theory can be applied (and in particular the subclass of Hamiltonian geometric actions), give several examples of geometric actions, and prove a lower semi-continuity property for them. Finally, we define the notion of a “drift” of an action, as a generalization of the drift vector field entering the Wentzell-Freidlin action.
Matthias Heymann
Chapter 3. Existence of Minimum Action Curves
Abstract
In this chapter we begin by stating the problem of the existence of a minimum action curve, and we prove our main existence theorem, which requires all relevant points in the state space to “have local minimizers.” We then list three criteria for proving this property for a given point, each designed to target one out of three different categories of points; the key ingredient here is our newly introduced notion of “admissible manifolds.” After illustrating the use of these criteria with a variety of examples, we conclude with a top-level theorem that can free us from having to invoke these criteria by hand if the drift of the given action is of a certain form. The proofs of the main criteria described in Sect. 3.3 will be postponed to Chap. 6 in Part II.
Matthias Heymann
Chapter 4. Properties of Minimum Action Curves
Abstract
In this chapter we study the properties of minimum action curves, often focusing on a specific subclass of actions. First we show which points minimizing curves can pass “in infinite length.” Then we find for a certain type of Hamiltonian actions that the action of the drift vector field’s flowlines vanishes, and that bending curves into the direction of the drift reduces their action. As a consequence, we then prove the non-existence of minimizers in some situations, and we show that minimizers leading from one attractor of the drift to another have to pass a saddle point on the separatrix between the two basins of attraction.
Matthias Heymann
Chapter 5. Conclusions
Abstract
In this chapter we look back and summarize our main results, and we discuss some open problems.
Matthias Heymann

Proofs

Frontmatter
Chapter 6. Proofs for Sect. 3.3: Finding Points with Local Minimizers
Abstract
This chapter contains the proofs of our three criteria—Propositions 3.16, 3.23, and 3.25—for showing that a given point has local minimizers. In the process we develop some valuable tools for working with admissible manifolds, and we prove a powerful inequality that bounds the length of a curve above by its action.
Matthias Heymann
Chapter 7. Proof of Lemma 6.15
Abstract
This chapter contains the proof of the very technical Lemma 6.​15 in Chap. 6 Some details of this proof will be postponed to Appendix B.
Matthias Heymann
Backmatter
Metadata
Title
Minimum Action Curves in Degenerate Finsler Metrics
Author
Matthias Heymann
Copyright Year
2015
Electronic ISBN
978-3-319-17753-3
Print ISBN
978-3-319-17752-6
DOI
https://doi.org/10.1007/978-3-319-17753-3