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E-raamat: Minimum Action Curves in Degenerate Finsler Metrics: Existence and Properties

  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2134
  • Ilmumisaeg: 08-Jul-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319177533
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Minimum Action Curves in Degenerate Finsler Metrics: Existence and PropertiesSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Lecture Notes in Mathematics 2134
  • Ilmumisaeg: 08-Jul-2015
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319177533
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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.

 
Part I Results
1 Introduction
3(10)
1.1 Geometric Action Functionals
3(1)
1.2 Example: Large Deviation Theory
4(2)
1.3 Key Features of the Existence Theory
6(1)
1.4 Techniques Used in the Literature
7(3)
1.4.1 Constructive Techniques
7(1)
1.4.2 The Lower Semi-Continuity Technique
8(2)
1.5 Properties of Minimum Action Curves
10(1)
1.6 The Structure of this Monograph
10(1)
1.7 Notation and Assumptions
11(2)
2 Geometric Action Functionals
13(16)
2.1 Curves
13(5)
2.1.1 Rectifiable Curves and Absolutely Continuous Functions
13(3)
2.1.2 Curves that Pass Points in Infinite Length
16(1)
2.1.3 Summary of the Various Classes of Curves
17(1)
2.2 Geometric Actions, Drift Vector Fields
18(3)
2.3 The Subclass of Hamiltonian Geometric Actions
21(8)
3 Existence of Minimum Action Curves
29(28)
3.1 A First Existence Result
29(5)
3.2 Points with Local Minimizers, Existence Theorem
34(5)
3.3 Finding Points with Local Minimizers
39(4)
3.4 Examples in R2
43(9)
3.4.1 Two Basins of Attraction
46(1)
3.4.2 Three Basins of Attraction
46(1)
3.4.3 An Example with Trivial Natural Drift
47(1)
3.4.4 Examples to Which Our Criteria Do Not Apply
48(3)
3.4.5 Modifying the Natural Drift
51(1)
3.5 A Top-Level Theorem
52(5)
4 Properties of Minimum Action Curves
57(10)
4.1 Points that Are Passed in Infinite Length
58(1)
4.2 The Advantage of Going with the Flow
59(1)
4.3 Some Results on the Non-Existence of Minimizers
60(3)
4.4 How to Move from One Attractor to Another
63(4)
5 Conclusions
67(4)
5.1 Recapitulation
67(1)
5.2 Open Problems
68(3)
Part II Proofs
6 Proofs for Sect. 3.3: Finding Points with Local Minimizers
71(26)
6.1 Proof of Proposition 3.16
71(2)
6.2 Proof of Lemma 3.22
73(1)
6.3 Admissible Manifolds
74(5)
6.4 Flowline-Tracing Functions
79(5)
6.5 Proof of Proposition 3.23
84(3)
6.6 Proof of Proposition 3.25
87(10)
7 Proof of Lemma 6.15
97(46)
7.1 Setting Things Up
97(7)
7.2 Modification of the Admissible Manifolds
104(7)
7.3 Definition of the Functions fi; Proof of their Properties
111(32)
7.3.1 Proof of Properties (i)--(iv)
112(3)
7.3.2 Proof of Property (v)
115(11)
7.3.3 Proof of Property (vi)
126(15)
7.3.4 Proof of Property (vii)
141(2)
A Technical Proofs and Remarks for Part I
143(18)
A.1 Proof of Lemma 2.3
143(5)
A.2 Proof of Lemma 2.6
148(4)
A.3 Proof of Lemma 2.13
152(1)
A.4 Proof of Lemma 2.14
152(1)
A.5 Proof of Lemma 2.17
153(1)
A.6 Large Deviations for Killed Diffusion Processes
154(1)
A.7 Some Remarks on the Proof of Lemma 3.20
155(1)
A.8 Proof of Lemma 3.26 (ii)
156(1)
A.9 Proof of Lemma 4.4
156(5)
B Technical Proofs and Remarks for Part II
161(16)
B.1 Proof of Lemma 6.1
161(2)
B.2 Remarks on the Construction of Msloc, Muloc, ps and pu
163(2)
B.3 Proof of Lemma 7.3
165(2)
B.4 Proof of Lemma 7.4
167(1)
B.5 Proof of Lemma 7.5
168(1)
B.6 Proof of Lemma 7.6
168(1)
B.7 Proof of Lemma 7.8
169(1)
B.8 Proof of Lemma 7.9
170(1)
B.9 Proof of Lemma 7.10
171(2)
B.10 Proof of Remark 7.11
173(1)
B.11 Proof of Lemma 7.12
174(3)
Glossary 177(4)
References 181(2)
Index 183
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