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Probability and Real Trees : Ecole D'Ete de Probabilites de Saint-Flour XXXV-2005 :  Ecole D'Ete de Probabilites de Saint-Flour XXXV-2005 - Steven N. Evans

Probability and Real Trees : Ecole D'Ete de Probabilites de Saint-Flour XXXV-2005

Ecole D'Ete de Probabilites de Saint-Flour XXXV-2005


Published: November 2007
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"Random trees and tree-valued stochastic processes are of particular importance in combinatorics, computer science, phylogenetics, and mathematical population genetics. Using the framework of abstract "tree-like" metric spaces (so-called real trees) and ideas from metric geometry such as the Gromov-Hausdorff distance, Evans and his collaborators have recently pioneered an approach to studying the asymptotic behaviour of such objects when the number of vertices goes to infinity. These notes survey the relevant mathematical background and present some selected applications of the theory."--BOOK JACKET.

Introductionp. 1
Around the Continuum Random Treep. 9
Random Trees from Random Walksp. 9
Markov Chain Tree Theoremp. 9
Generating Uniform Random Treesp. 13
Random Trees from Conditioned Branching Processesp. 15
Finite Trees and Lattice Pathsp. 16
The Brownian Continuum Random Treep. 17
Trees as Subsets of l[superscript 1]p. 18
R-Trees and 0-Hyperbolic Spacesp. 21
Geodesic and Geodesically Linear Metric Spacesp. 21
0-Hyperbolic Spacesp. 23
R-Treesp. 26
Definition, Examples, and Elementary Propertiesp. 26
R-Trees are 0-Hyperbolicp. 32
Centroids in a 0-Hyperbolic Spacep. 33
An Alternative Characterization of R-Treesp. 36
Embedding 0-Hyperbolic Spaces in R-Treesp. 36
Yet another Characterization of R-Treesp. 38
R-Trees without Leavesp. 39
Endsp. 39
The Ends Compactificationp. 42
Examples of R-Trees without Leavesp. 44
Hausdorff and Gromov-Hausdorff Distancep. 45
Hausdorff Distancep. 45
Gromov-Hausdorff Distancep. 47
Definition and Elementary Propertiesp. 47
Correspondences and [epsilon]-Isometriesp. 48
Gromov-Hausdorff Distance for Compact Spacesp. 50
Gromov-Hausdorff Distance for Geodesic Spacesp. 52
Compact R-Trees and the Gromov-Hausdorff Metricp. 53
Unrooted R-Treesp. 53
Trees with Four Leavesp. 53
Rooted R-Treesp. 55
Rooted Subtrees and Trimmingp. 58
Length Measure on R-Treesp. 59
Weighted R-Treesp. 63
Root Growth with Re-Graftingp. 69
Background and Motivationp. 69
Construction of the Root Growth with Re-Grafting Processp. 71
Outline of the Constructionp. 71
A Deterministic Constructionp. 72
Putting Randomness into the Constructionp. 76
Feller Propertyp. 78
Ergodicity, Recurrence, and Uniquenessp. 79
Brownian CRT and Root Growth with Re-Graftingp. 79
Couplingp. 82
Convergence to Equilibriump. 83
Recurrencep. 83
Uniqueness of the Stationary Distributionp. 84
Convergence of the Markov Chain Tree Algorithmp. 85
The Wild Chain and other Bipartite Chainsp. 87
Backgroundp. 87
More Examples of State Spacesp. 90
Proof of Theorem 6.4p. 92
Bipartite Chainsp. 95
Quotient Processesp. 99
Additive Functionalsp. 100
Bipartite Chains on the Boundaryp. 101
Diffusions on a R-Tree without Leaves: Snakes and Spidersp. 105
Backgroundp. 105
Construction of the Diffusion Processp. 106
Symmetry and the Dirichlet Formp. 108
Recurrence, Transience, and Regularity of Pointsp. 113
Examplesp. 114
Triviality of the Tail [sigma]-fieldp. 115
Martin Compactification and Excessive Functionsp. 116
Probabilistic Interpretation of the Martin Compactificationp. 122
Entrance Lawsp. 123
Local Times and Semimartingale Decompositionsp. 125
R-Trees from Coalescing Particle Systemsp. 129
Kingman's Coalescentp. 129
Coalescing Brownian Motionsp. 132
Subtree Prune and Re-Graftp. 143
Backgroundp. 143
The Weighted Brownian CRTp. 144
Campbell Measure Factsp. 146
A Symmetric Jump Measurep. 154
The Dirichlet Formp. 157
Summary of Dirichlet Form Theoryp. 163
Non-Negative Definite Symmetric Bilinear Formsp. 163
Dirichlet Formsp. 163
Semigroups and Resolventsp. 166
Generatorsp. 167
Spectral Theoryp. 167
Dirichlet Form, Generator, Semigroup, Resolvent Correspondencep. 168
Capacitiesp. 169
Dirichlet Forms and Hunt Processesp. 169
Some Fractal Notionsp. 171
Hausdorff and Packing Dimensionsp. 171
Energy and Capacityp. 172
Application to Trees from Coalescing Partitionsp. 173
Referencesp. 177
Indexp. 185
List of Participantsp. 187
List of Short Lecturesp. 191
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783540747970
ISBN-10: 3540747974
Series: Lecture Notes in Mathematics
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 201
Published: November 2007
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.5 x 15.5  x 1.27
Weight (kg): 0.34