Graphs and discrete Dirichlet spaces / / Matthias Keller, Daniel Lenz, and Radosław K. Wojciechowski

• Record Nr.: UNINA9910506392903321
• Autore: Keller Matthias
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2021]; ©2021
• ISBN: 3-030-81459-9
• Descrizione fisica: 1 online resource (675 pages)
• Collana: Grundlehren der Mathematischen Wissenschaften ; ; v.358
• Disciplina: 511.5
• Soggetti: Functional analysis; Graph theory; Probabilities; Teoria de grafs; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Acknowledgments -- Contents -- Part 0 Prelude -- Chapter 0 Finite Graphs -- 0.1 Graphs, Laplacians and Dirichlet forms -- 0.2 Characterizing forms associated to graphs -- 0.3 Characterizing Laplacians associated to graphs -- 0.4 Networks and electrostatics -- 0.5 The heat equation and the Markov property -- 0.6 Resolvents and heat semigroups -- 0.7 A Perron-Frobenius theorem and large time behavior -- 0.8 When there is no killing -- 0.9 Turning graphs into other graphs* -- 0.10 Markov processes and the Feynman-Kac formula* -- Exercises -- Notes -- Part 1 Foundations and Fundamental Topics -- Chapter 1 Infinite Graphs - Key Concepts -- 1.1 The setting in a nutshell -- 1.2 Graphs and (regular) Dirichlet forms -- 1.3 Approximation, domain monotonicity and the Markov property -- 1.4 Connectedness, irreducibility and positivity improving operators -- 1.5 Boundedness and compactly supported functions -- 1.6 Graphs with standard weights -- Exercises -- Notes -- Chapter 2 Infinite Graphs - Toolbox -- 2.1 Generators, semigroups and resolvents on p -- 2.2 Forms associated to graphs and restrictions to subsets -- 2.3 The curse of non-locality: Leibniz and chain rules -- 2.4 Creatures from the ; abyss* -- 2.5 Markov processes and the Feynman-Kac formula redux* -- Exercises -- Notes -- Chapter 3 Markov Uniqueness and Essential Self-Adjointness -- 3.1 Uniqueness of associated forms -- 3.2 Essential self-adjointness -- 3.3 Markov uniqueness -- Exercises -- Notes -- Chapter 4 Agmon-Allegretto-Piepenbrink and Persson Theorems -- 4.1 A local Harnack inequality and consequences -- 4.2 The ground state transform -- 4.3 The bottom of the spectrum -- 4.4 The bottom of the essential spectrum -- Exercises -- Notes -- Chapter 5 Large Time Behavior of the Heat Kernel -- 5.1 Positivity improving semigroups and the ground state.; 5.2 Theorems of Chavel-Karp and Li -- 5.3 The Neumann Laplacian and finite measure -- Exercises -- Notes -- Chapter 6 Recurrence -- 6.1 General preliminaries -- 6.2 The form perspective -- 6.3 The superharmonic function perspective -- 6.4 The Green's function perspective -- 6.5 The Green's formula perspective -- 6.6 A probabilistic point of view* -- Exercises -- Notes -- Chapter 7 Stochastic Completeness -- 7.1 The heat equation on l -- 7.2 Stochastic completeness at infinity -- 7.3 The heat equation perspective -- 7.4 The Poisson equation perspective -- 7.5 The form perspective -- 7.6 The Green's formula perspective -- 7.7 The Omori-Yau maximum principle -- 7.8 A stability criterion and Khasminskii's criterion -- 7.9 A probabilistic interpretation* -- Exercises -- Notes -- Part 2 Classes of Graphs -- Chapter 8 Uniformly Positive Measure -- 8.1 A Liouville theorem -- 8.2 Uniqueness of the form and essential self-adjointness -- 8.3 A spectral inclusion -- 8.4 The heat equation on p -- 8.5 Graphs with standard weights -- Exercises -- Notes -- Chapter 9 Weak Spherical Symmetry -- 9.1 Symmetry of the heat kernel -- 9.2 The spectral gap -- 9.3 Recurrence -- 9.4 Stochastic completeness at infinity -- Exercises -- Notes -- Chapter 10 Sparseness and Isoperimetric Inequalities -- 10.1 Notions of sparseness -- 10.2 Co-area formulae -- 10.3 Weak sparseness and the form domain -- 10.4 Approximate sparseness and first order eigenvalue asymptotics -- 10.5 Sparseness and second order eigenvalue asymptotics -- 10.6 Isoperimetric inequalities and Weyl asymptotics -- Exercises -- Notes -- Part 3 Geometry and Intrinsic Metrics -- Chapter 11 Intrinsic Metrics: Definition and Basic Facts -- 11.1 Definition and motivation -- 11.2 Path metrics and a Hopf-Rinow theorem -- 11.3 Examples and relations to other metrics -- 11.4 Geometric assumptions and cutoff functions.; Exercises -- Notes -- Chapter 12 Harmonic Functions and Caccioppoli Theory -- 12.1 Caccioppoli inequalities -- 12.2 Liouville theorems -- 12.3 Applications of the Liouville theorems -- 12.4 Shnol' theorems -- Exercises -- Notes -- Chapter 13 Spectral Bounds -- 13.1 Cheeger constants and lower spectral bounds -- 13.2 Volume growth and upper spectral bounds -- Exercises -- Notes -- Chapter 14 Volume Growth Criterion for Stochastic Completeness and Uniqueness Class -- 14.1 Uniqueness class -- 14.2 Refinements -- 14.3 Volume growth criterion for stochastic completeness -- Exercises -- Notes -- Appendix -- Appendix A The Spectral Theorem -- Appendix B Closed Forms on Hilbert spaces -- Appendix C Dirichlet Forms and Beurling-Deny Criteria -- Appendix D Semigroups, Resolvents and their Generators -- Appendix E Aspects of Operator Theory -- E.1 A characterization of the resolvent -- E.2 The discrete and essential spectrum -- E.3 Reducing subspaces and commuting operators -- E.4 The Riesz-Thorin interpolation theorem -- References -- Index -- Notation Index.