| Description |
1 online resource |
| Series |
London Mathematical Society lecture note series ; 438 |
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London Mathematical Society lecture note series ; 438.
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| Contents |
Cover; Series page; Title page; Copyright page; Dedication; Contents; Preface; 1 Introduction; 1.1 Graphs and Weighted Graphs; 1.2 Random Walks on a Weighted Graph; 1.3 Transition Densities and the Laplacian; 1.4 Dirichlet or Energy Form; 1.5 Killed Process; 1.6 Green's Functions; 1.7 Harmonic Functions, Harnack Inequalities, and the Liouville Property; 1.8 Strong Liouville Property for R[sup(d)]; 1.9 Interpretation of the Liouville Property; 2 Random Walks and Electrical Resistance; 2.1 Basic Concepts; 2.2 Transience and Recurrence; 2.3 Energy and Variational Methods |
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2.4 Resistance to Infinity2.5 Traces and Electrical Equivalence; 2.6 Stability under Rough Isometries; 2.7 Hitting Times and Resistance; 2.8 Examples; 2.9 The Sierpinski Gasket Graph; 3 Isoperimetric Inequalities and Applications; 3.1 Isoperimetric Inequalities; 3.2 Nash Inequality; 3.3 Poincaré Inequality; 3.4 Spectral Decomposition for a Finite Graph; 3.5 Strong Isoperimetric Inequality and Spectral Radius; 4 Discrete Time Heat Kernel; 4.1 Basic Properties and Bounds on the Diagonal; 4.2 Carne-Varopoulos Bound; 4.3 Gaussian and Sub-Gaussian Heat Kernel Bounds; 4.4 Off-diagonal Upper Bounds |
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A.6 Miscellaneous EstimatesA. 7 Whitney Type Coverings of a Ball; A.8 A Maximal Inequality; A.9 Poincaré Inequalities; References; Index |
| 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 Poincar 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 |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Random walks (Mathematics)
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Graph theory.
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Markov processes.
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Heat equation.
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Mathematics.
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Markov Chains
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Mathematics
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MATHEMATICS -- General.
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Mathematics
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Graph theory
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Heat equation
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Markov processes
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Random walks (Mathematics)
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| Form |
Electronic book
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| ISBN |
9781108125604 |
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1108125603 |
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9781107415690 |
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1107415691 |
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