Random walks and heat kernels on graphs:
This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Po...
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| Main Author: | |
|---|---|
| Format: | Electronic eBook |
| Language: | English |
| Published: |
Cambridge
Cambridge University Press
2017
|
| Series: | London Mathematical Society lecture note series
438 |
| Subjects: | |
| Online Access: | BSB01 FHN01 Volltext Inhaltsverzeichnis |
| Summary: | This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Poincaré inequalities. The book presents rough isometries and looks at the properties of a graph that are stable under these transformations. Applications include the 'type problem': determining whether a graph is transient or recurrent. The final chapters show how geometric properties of the graph can be used to establish heat kernel bounds, that is, bounds on the transition probabilities of the random walk, and it is proved that Gaussian bounds hold for graphs that are roughly isometric to Euclidean space. Aimed at graduate students in mathematics, the book is also useful for researchers as a reference for results that are hard to find elsewhere |
| Item Description: | Title from publisher's bibliographic system (viewed on 20 Mar 2017) |
| Physical Description: | 1 online resource (xi, 226 pages) |
| ISBN: | 9781107415690 |
| DOI: | 10.1017/9781107415690 |
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Record in the Search Index
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| adam_text | Titel: Random walks and heat kernels on graphs
Autor: Barlow, Martin T
Jahr: 2017
London Mathematical Society Lecture Note Series: 438 Random Walks and Heat Kernels on Graphs MARTIN T. BARLOW University of British Columbia, Canada Cambridge UNIVERSITY PRESS
Contents Preface page ix Introduction 1 1.1 Graphs and Weighted Graphs 1 1.2 Random Walks on a Weighted Graph 6 1.3 Transition Densities and the Laplacian 11 1.4 Dirichlet or Energy Form 15 1.5 Killed Process 21 1.6 Green’s Functions 22 1.7 Harmonic Functions, Harnack Inequalities, and the Liouville Property 26 1.8 Strong Liouville Property for W 1 31 1.9 Interpretation of the Liouville Property 32 Random Walks and Electrical Resistance 38 2.1 Basic Concepts 38 2.2 Transience and Recurrence 42 2.3 Energy and Variational Methods 44 2.4 Resistance to Infinity 55 2.5 Traces and Electrical Equivalence 61 2.6 Stability under Rough Isometries 67 2.7 Hitting Times and Resistance 70 2.8 Examples 73 2.9 The Sierpi?ski Gasket Graph 75 Isoperimetric Inequalities and Applications 80 3.1 Isoperimetric Inequalities 80 3.2 Nash Inequality 85 3.3 Poincaré Inequality 91 vu
Contents viii 3.4 Spectral Decomposition for a Finite Graph 97 3.5 Strong Isoperimetric Inequality and Spectral Radius 101 4 Discrete Time Heat Kernel 106 4.1 Basic Properties and Bounds on the Diagonal 106 4.2 Carne- Varopoulos Bound 111 4.3 Gaussian and Sub-Gaussian Heat Kernel Bounds 116 4.4 Off-diagonal Upper Bounds 124 4.5 Lower Bounds 128 5 Continuous Time Random Walks 132 5.1 Introduction to Continuous Time 132 5.2 Heat Kernel Bounds 140 6 Heat Kernel Bounds 149 6.1 Strongly Recurrent Graphs 149 6.2 Gaussian Upper Bounds 155 6.3 Poincare Inequality and Gaussian Lower Bounds 160 6.4 Remarks on Gaussian Bounds 168 7 Potential Theory and Hamack Inequalities 172 7.1 Introduction to Potential Theory 172 7.2 Applications 179 Appendix 183 A.l Martingales and Tail Estimates 183 A.2 Discrete Time Markov Chains and the Strong Markov Property 186 A.3 Continuous Time Random Walk 190 A.4 Invariant and Tail a-fields 197 A.5 Hilbert Space Results 202 A.6 Miscellaneous Estimates 205 A.7 Whitney Type Coverings of a Ball 206 A.8 A Maximal Inequality 211 A.9 Poincare Inequalities 213 References 219 Index 224
|
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| discipline | Mathematik |
| doi_str_mv | 10.1017/9781107415690 |
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| isbn | 9781107415690 |
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| spelling | Barlow, M. T. Verfasser aut Random walks and heat kernels on graphs Martin T. Barlow, University of British Columbia, Canada Cambridge Cambridge University Press 2017 1 online resource (xi, 226 pages) txt rdacontent c rdamedia cr rdacarrier London Mathematical Society lecture note series 438 Title from publisher's bibliographic system (viewed on 20 Mar 2017) This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Poincaré inequalities. The book presents rough isometries and looks at the properties of a graph that are stable under these transformations. Applications include the 'type problem': determining whether a graph is transient or recurrent. The final chapters show how geometric properties of the graph can be used to establish heat kernel bounds, that is, bounds on the transition probabilities of the random walk, and it is proved that Gaussian bounds hold for graphs that are roughly isometric to Euclidean space. Aimed at graduate students in mathematics, the book is also useful for researchers as a reference for results that are hard to find elsewhere Random walks (Mathematics) Graph theory Markov processes Heat equation Irrfahrtsproblem (DE-588)4162442-7 gnd rswk-swf Unendlicher Graph (DE-588)4390888-3 gnd rswk-swf Irrfahrtsproblem (DE-588)4162442-7 s Unendlicher Graph (DE-588)4390888-3 s 1\p DE-604 Erscheint auch als Druck-Ausgabe, paperback 978-1-107-67442-4 https://doi.org/10.1017/9781107415690 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029725615&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Barlow, M. T. Random walks and heat kernels on graphs Random walks (Mathematics) Graph theory Markov processes Heat equation Irrfahrtsproblem (DE-588)4162442-7 gnd Unendlicher Graph (DE-588)4390888-3 gnd |
| subject_GND | (DE-588)4162442-7 (DE-588)4390888-3 |
| title | Random walks and heat kernels on graphs |
| title_auth | Random walks and heat kernels on graphs |
| title_exact_search | Random walks and heat kernels on graphs |
| title_full | Random walks and heat kernels on graphs Martin T. Barlow, University of British Columbia, Canada |
| title_fullStr | Random walks and heat kernels on graphs Martin T. Barlow, University of British Columbia, Canada |
| title_full_unstemmed | Random walks and heat kernels on graphs Martin T. Barlow, University of British Columbia, Canada |
| title_short | Random walks and heat kernels on graphs |
| title_sort | random walks and heat kernels on graphs |
| topic | Random walks (Mathematics) Graph theory Markov processes Heat equation Irrfahrtsproblem (DE-588)4162442-7 gnd Unendlicher Graph (DE-588)4390888-3 gnd |
| topic_facet | Random walks (Mathematics) Graph theory Markov processes Heat equation Irrfahrtsproblem Unendlicher Graph |
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