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Minimum action curves in degenerate Finsler metrics : existence and properties / Matthias Heymann.

By: Series: Lecture notes in mathematics ; 2134.Publication details: Switzerland : Springer, 2015.Description: xv, 184 p. : illustrations (some color) ; 24 cmISBN:
  • 9783319177526
Subject(s): DDC classification:
  • 516.375 23 H618
Contents:
1. Introduction.- 2. Geometric Action Functionals.- 3. Existence of Minimum Action Curves.- 4. Properties of Minimum Action Curves.- 5. Conclusions.- 6. Proofs.- Finding Points with Local Minimizers.- 7. Proof of Lemma 6.15 -- A. Technical proofs and remarks for part I.- B. Technical proofs and remarks for part II.- Glossary.- References.- Index.
Summary: 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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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 516.375 H618 (Browse shelf(Opens below)) Available 136750
Total holds: 0

Includes bibliographical references and index.

1. Introduction.-
2. Geometric Action Functionals.-
3. Existence of Minimum Action Curves.-
4. Properties of Minimum Action Curves.-
5. Conclusions.-
6. Proofs.- Finding Points with Local Minimizers.-
7. Proof of Lemma 6.15 --
A. Technical proofs and remarks for part I.-
B. Technical proofs and remarks for part II.-
Glossary.-
References.-
Index.

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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