LEADER 06486nam 2200577 450 001 9910506392903321 005 20231110213146.0 010 $a3-030-81459-9 035 $a(CKB)5340000000068506 035 $a(MiAaPQ)EBC6789916 035 $a(Au-PeEL)EBL6789916 035 $a(OCoLC)1280420051 035 $a(PPN)258296828 035 $a(EXLCZ)995340000000068506 100 $a20220714d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aGraphs and discrete Dirichlet spaces /$fMatthias Keller, Daniel Lenz, and Rados?aw K. Wojciechowski 210 1$aCham, Switzerland :$cSpringer,$d[2021] 210 4$d©2021 215 $a1 online resource (675 pages) 225 1 $aGrundlehren der Mathematischen Wissenschaften ;$vv.358 311 $a3-030-81458-0 320 $aIncludes bibliographical references and index. 327 $aIntro -- 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. 327 $a5.2 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. 327 $aExercises -- Notes -- Chapter 12 Harmonic Functions and Caccioppoli Theory -- 12.1 Caccioppoli inequalities -- 12.2 Liouville theorems -- 12.3 Applications of the Liouville theorems -- 12.4 Shnol' theorems -- Exercises -- Notes -- Chapter 13 Spectral Bounds -- 13.1 Cheeger constants and lower spectral bounds -- 13.2 Volume growth and upper spectral bounds -- Exercises -- Notes -- Chapter 14 Volume Growth Criterion for Stochastic Completeness and Uniqueness Class -- 14.1 Uniqueness class -- 14.2 Refinements -- 14.3 Volume growth criterion for stochastic completeness -- Exercises -- Notes -- Appendix -- Appendix A The Spectral Theorem -- Appendix B Closed Forms on Hilbert spaces -- Appendix C Dirichlet Forms and Beurling-Deny Criteria -- Appendix D Semigroups, Resolvents and their Generators -- Appendix E Aspects of Operator Theory -- E.1 A characterization of the resolvent -- E.2 The discrete and essential spectrum -- E.3 Reducing subspaces and commuting operators -- E.4 The Riesz-Thorin interpolation theorem -- References -- Index -- Notation Index. 410 0$aGrundlehren der Mathematischen Wissenschaften 606 $aFunctional analysis 606 $aGraph theory 606 $aProbabilities 606 $aTeoria de grafs$2thub 608 $aLlibres electrònics$2thub 615 0$aFunctional analysis. 615 0$aGraph theory. 615 0$aProbabilities. 615 7$aTeoria de grafs 676 $a511.5 700 $aKeller$b Matthias$01072847 702 $aWojciechowski$b Radoslaw K. 702 $aLenz$b Daniel 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910506392903321 996 $aGraphs and discrete Dirichlet spaces$92899855 997 $aUNINA