Probability and real trees: Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2008
|
| Schriftenreihe: | Lecture notes in mathematics
1920 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XI, 193 S. Ill., graph. Darst. |
| ISBN: | 9783540747970 3540747974 |
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| 100 | 1 | |a Evans, Steven N. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Probability and real trees |b Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 |c Steven N. Evans |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2008 | |
| 300 | |a XI, 193 S. |b Ill., graph. Darst. | ||
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| 490 | 1 | |a Lecture notes in mathematics |v 1920 | |
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| 650 | 4 | |a Mathematisches Modell | |
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| 650 | 4 | |a Génétique évolutive |x Modèles mathématiques |v Congrès | |
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| 650 | 4 | |a Phylogeny |x Mathematical models |v Congresses | |
| 650 | 4 | |a Phylogenèse |x Modèles mathématiques |v Congrès | |
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Datensatz im Suchindex
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| adam_text | Contents
1 Introduction 1
2 Around the Continuum Random Tree 9
2.1 Random Trees from Random Walks 9
2.1.1 Markov Chain Tree Theorem 9
2.1.2 Generating Uniform Random Trees 13
2.2 Random Trees from Conditioned Branching Processes 15
2.3 Finite Trees and Lattice Paths 16
2.4 The Brownian Continuum Random Tree 17
2.5 Trees as Subsets of Ix 18
3 R Trees and O Hyperbolic Spaces 21
3.1 Geodesic and Geodesically Linear Metric Spaces 21
3.2 O Hyperbolic Spaces 23
3.3 R Trees 26
3.3.1 Definition, Examples, and Elementary Properties 26
3.3.2 R Trees are O Hyperbolic 32
3.3.3 Centroids in a O Hyperbolic Space 33
3.3.4 An Alternative Characterization of K Trees 36
3.3.5 Embedding O Hyperbolic Spaces in R Trees 36
3.3.6 Yet another Characterization of R Trees 38
3.4 R Trees without Leaves 39
3.4.1 Ends 39
3.4.2 The Ends Compactification 42
3.4.3 Examples of R Trees without Leaves 44
4 Hausdorff and Gromov Hausdorff Distance 45
4.1 Hausdorff Distance 45
4.2 Gromov Hausdorff Distance 47
4.2.1 Definition and Elementary Properties 47
4.2.2 Correspondences and f Isometries 48
X Contents
4.2.3 Gromov Hausdorff Distance for Compact Spaces 50
4.2.4 Gromov Hausdorff Distance for Geodesic Spaces 52
4.3 Compact R Trees and the Gromov Hausdorff Metric 53
4.3.1 Unrooted R Trees 53
4.3.2 Trees with Four Leaves 53
4.3.3 Rooted R Trees 55
4.3.4 Rooted Subtrees and Trimming 58
4.3.5 Length Measure on R Trees 59
4.4 Weighted R Trees 63
5 Root Growth with Re Grafting 69
5.1 Background and Motivation 69
5.2 Construction of the Root Growth with Re Grafting Process ... 71
5.2.1 Outline of the Construction 71
5.2.2 A Deterministic Construction 72
5.2.3 Putting Randomness into the Construction 76
5.2.4 Feller Property 78
5.3 Ergodicity, Recurrence, and Uniqueness 79
5.3.1 Brownian CRT and Root Growth with Re Grafting 79
5.3.2 Coupling 82
5.3.3 Convergence to Equilibrium 83
5.3.4 Recurrence 83
5.3.5 Uniqueness of the Stationary Distribution 84
5.4 Convergence of the Markov Chain Tree Algorithm 85
6 The Wild Chain and other Bipartite Chains 87
6.1 Background 87
6.2 More Examples of State Spaces 90
6.3 Proof of Theorem 6.4 92
6.4 Bipartite Chains 95
6.5 Quotient Processes 99
6.6 Additive Functionals 100
6.7 Bipartite Chains on the Boundary 101
7 Diffusions on a R Tree without Leaves: Snakes and Spiders 105
7.1 Background 105
7.2 Construction of the Diffusion Process 106
7.3 Symmetry and the Dirichlet Form 108
7.4 Recurrence, Transience, and Regularity of Points 113
7.5 Examples 114
7.6 Triviality of the Tail cr field 115
7.7 Martin Compactification and Excessive Functions 116
7.8 Probabilistic Interpretation of the Martin Compactification ... 122
7.9 Entrance Laws 123
7.10 Local Times and Semimartingale Decompositions 125
Contents XI
8 R Trees from Coalescing Particle Systems 129
8.1 Kingman s Coalescent 129
8.2 Coalescing Brownian Motions 132
9 Subtree Prune and Re Graft 143
9.1 Background 143
9.2 The Weighted Brownian CRT 144
9.3 Campbell Measure Facts 146
9.4 A Symmetric Jump Measure 154
9.5 The Dirichlet Form 157
A Summary of Dirichlet Form Theory 163
A.I Non Negative Definite Symmetric Bilinear Forms 163
A.2 Dirichlet Forms 163
A.3 Semigroups and Resolvents 166
A.4 Generators 167
A.5 Spectral Theory 167
A.6 Dirichlet Form, Generator, Semigroup, Resolvent
Correspondence 168
A.7 Capacities 169
A.8 Dirichlet Forms and Hunt Processes 169
B Some Fractal Notions 171
B.I Hausdorff and Packing Dimensions 171
B.2 Energy and Capacity 172
B.3 Application to Trees from Coalescing Partitions 173
References 177
Index 185
List of Participants 187
List of Short Lectures 191
|
| adam_txt |
Contents
1 Introduction 1
2 Around the Continuum Random Tree 9
2.1 Random Trees from Random Walks 9
2.1.1 Markov Chain Tree Theorem 9
2.1.2 Generating Uniform Random Trees 13
2.2 Random Trees from Conditioned Branching Processes 15
2.3 Finite Trees and Lattice Paths 16
2.4 The Brownian Continuum Random Tree 17
2.5 Trees as Subsets of Ix 18
3 R Trees and O Hyperbolic Spaces 21
3.1 Geodesic and Geodesically Linear Metric Spaces 21
3.2 O Hyperbolic Spaces 23
3.3 R Trees 26
3.3.1 Definition, Examples, and Elementary Properties 26
3.3.2 R Trees are O Hyperbolic 32
3.3.3 Centroids in a O Hyperbolic Space 33
3.3.4 An Alternative Characterization of K Trees 36
3.3.5 Embedding O Hyperbolic Spaces in R Trees 36
3.3.6 Yet another Characterization of R Trees 38
3.4 R Trees without Leaves 39
3.4.1 Ends 39
3.4.2 The Ends Compactification 42
3.4.3 Examples of R Trees without Leaves 44
4 Hausdorff and Gromov Hausdorff Distance 45
4.1 Hausdorff Distance 45
4.2 Gromov Hausdorff Distance 47
4.2.1 Definition and Elementary Properties 47
4.2.2 Correspondences and f Isometries 48
X Contents
4.2.3 Gromov Hausdorff Distance for Compact Spaces 50
4.2.4 Gromov Hausdorff Distance for Geodesic Spaces 52
4.3 Compact R Trees and the Gromov Hausdorff Metric 53
4.3.1 Unrooted R Trees 53
4.3.2 Trees with Four Leaves 53
4.3.3 Rooted R Trees 55
4.3.4 Rooted Subtrees and Trimming 58
4.3.5 Length Measure on R Trees 59
4.4 Weighted R Trees 63
5 Root Growth with Re Grafting 69
5.1 Background and Motivation 69
5.2 Construction of the Root Growth with Re Grafting Process . 71
5.2.1 Outline of the Construction 71
5.2.2 A Deterministic Construction 72
5.2.3 Putting Randomness into the Construction 76
5.2.4 Feller Property 78
5.3 Ergodicity, Recurrence, and Uniqueness 79
5.3.1 Brownian CRT and Root Growth with Re Grafting 79
5.3.2 Coupling 82
5.3.3 Convergence to Equilibrium 83
5.3.4 Recurrence 83
5.3.5 Uniqueness of the Stationary Distribution 84
5.4 Convergence of the Markov Chain Tree Algorithm 85
6 The Wild Chain and other Bipartite Chains 87
6.1 Background 87
6.2 More Examples of State Spaces 90
6.3 Proof of Theorem 6.4 92
6.4 Bipartite Chains 95
6.5 Quotient Processes 99
6.6 Additive Functionals 100
6.7 Bipartite Chains on the Boundary 101
7 Diffusions on a R Tree without Leaves: Snakes and Spiders 105
7.1 Background 105
7.2 Construction of the Diffusion Process 106
7.3 Symmetry and the Dirichlet Form 108
7.4 Recurrence, Transience, and Regularity of Points 113
7.5 Examples 114
7.6 Triviality of the Tail cr field 115
7.7 Martin Compactification and Excessive Functions 116
7.8 Probabilistic Interpretation of the Martin Compactification . 122
7.9 Entrance Laws 123
7.10 Local Times and Semimartingale Decompositions 125
Contents XI
8 R Trees from Coalescing Particle Systems 129
8.1 Kingman's Coalescent 129
8.2 Coalescing Brownian Motions 132
9 Subtree Prune and Re Graft 143
9.1 Background 143
9.2 The Weighted Brownian CRT 144
9.3 Campbell Measure Facts 146
9.4 A Symmetric Jump Measure 154
9.5 The Dirichlet Form 157
A Summary of Dirichlet Form Theory 163
A.I Non Negative Definite Symmetric Bilinear Forms 163
A.2 Dirichlet Forms 163
A.3 Semigroups and Resolvents 166
A.4 Generators 167
A.5 Spectral Theory 167
A.6 Dirichlet Form, Generator, Semigroup, Resolvent
Correspondence 168
A.7 Capacities 169
A.8 Dirichlet Forms and Hunt Processes 169
B Some Fractal Notions 171
B.I Hausdorff and Packing Dimensions 171
B.2 Energy and Capacity 172
B.3 Application to Trees from Coalescing Partitions 173
References 177
Index 185
List of Participants 187
List of Short Lectures 191 |
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| id | DE-604.BV022949138 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:01:22Z |
| indexdate | 2024-07-09T21:08:22Z |
| institution | BVB |
| isbn | 9783540747970 3540747974 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016153635 |
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| physical | XI, 193 S. Ill., graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
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| publisher | Springer |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Evans, Steven N. Verfasser aut Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 Steven N. Evans Berlin [u.a.] Springer 2008 XI, 193 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1920 Stochastischer Prozess - Baum <Mathematik> Mathematisches Modell Dirichlet forms Congresses Evolutionary genetics Mathematical models Congresses Génétique évolutive Modèles mathématiques Congrès Hausdorff measures Congresses Markov processes Congresses Metric spaces Congresses Phylogeny Mathematical models Congresses Phylogenèse Modèles mathématiques Congrès Processus stochastiques Congrès Stochastic processes Congresses Trees (Graph theory) Congresses Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Baum Mathematik (DE-588)4004849-4 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Stochastischer Prozess (DE-588)4057630-9 s Baum Mathematik (DE-588)4004849-4 s DE-604 Markov-Prozess (DE-588)4134948-9 s Metrischer Raum (DE-588)4169745-5 s b DE-604 Lecture notes in mathematics 1920 (DE-604)BV000676446 1920 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016153635&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Evans, Steven N. Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 Lecture notes in mathematics Stochastischer Prozess - Baum <Mathematik> Mathematisches Modell Dirichlet forms Congresses Evolutionary genetics Mathematical models Congresses Génétique évolutive Modèles mathématiques Congrès Hausdorff measures Congresses Markov processes Congresses Metric spaces Congresses Phylogeny Mathematical models Congresses Phylogenèse Modèles mathématiques Congrès Processus stochastiques Congrès Stochastic processes Congresses Trees (Graph theory) Congresses Markov-Prozess (DE-588)4134948-9 gnd Baum Mathematik (DE-588)4004849-4 gnd Metrischer Raum (DE-588)4169745-5 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| subject_GND | (DE-588)4134948-9 (DE-588)4004849-4 (DE-588)4169745-5 (DE-588)4057630-9 (DE-588)1071861417 |
| title | Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 |
| title_auth | Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 |
| title_exact_search | Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 |
| title_exact_search_txtP | Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 |
| title_full | Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 Steven N. Evans |
| title_fullStr | Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 Steven N. Evans |
| title_full_unstemmed | Probability and real trees Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 Steven N. Evans |
| title_short | Probability and real trees |
| title_sort | probability and real trees ecole d ete de probabilites de saint flour xxxv 2005 |
| title_sub | Ecole d'Eté de Probabilités de Saint-Flour XXXV - 2005 |
| topic | Stochastischer Prozess - Baum <Mathematik> Mathematisches Modell Dirichlet forms Congresses Evolutionary genetics Mathematical models Congresses Génétique évolutive Modèles mathématiques Congrès Hausdorff measures Congresses Markov processes Congresses Metric spaces Congresses Phylogeny Mathematical models Congresses Phylogenèse Modèles mathématiques Congrès Processus stochastiques Congrès Stochastic processes Congresses Trees (Graph theory) Congresses Markov-Prozess (DE-588)4134948-9 gnd Baum Mathematik (DE-588)4004849-4 gnd Metrischer Raum (DE-588)4169745-5 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
| topic_facet | Stochastischer Prozess - Baum <Mathematik> Mathematisches Modell Dirichlet forms Congresses Evolutionary genetics Mathematical models Congresses Génétique évolutive Modèles mathématiques Congrès Hausdorff measures Congresses Markov processes Congresses Metric spaces Congresses Phylogeny Mathematical models Congresses Phylogenèse Modèles mathématiques Congrès Processus stochastiques Congrès Stochastic processes Congresses Trees (Graph theory) Congresses Markov-Prozess Baum Mathematik Metrischer Raum Stochastischer Prozess Konferenzschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016153635&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT evansstevenn probabilityandrealtreesecoledetedeprobabilitesdesaintflourxxxv2005 |