The spectral geometry of infinite graphs deals with three major themes and their interplay: the spectral theory of the Laplacian, the geometry of the underlying graph, and the heat flow with its probabilistic aspects. In this book, all three themes are brought together coherently under the perspective of Dirichlet forms, providing a powerful and unified approach.
The book gives a complete account of key topics of infinite graphs, such as essential self-adjointness, Markov uniqueness, spectral estimates, recurrence, and stochastic completeness. A major feature of the book is the use of intrinsic metrics to capture the geometry of graphs. As for manifolds, Dirichlet forms in the graph setting offer a structural understanding of the interaction between spectral theory, geometry and probability. For graphs, however, the presentation is much more accessible and inviting thanks to the discreteness of the underlying space, laying bare the main concepts while preserving the deep insights of the manifold case.
Graphs and Discrete Dirichlet Spaces offers a comprehensive treatment of the spectral geometry of graphs, from the very basics to deep and thorough explorations of advanced topics. With modest prerequisites, the book can serve as a basis for a number of topics courses, starting at the undergraduate level.
Author(s): Matthias Keller, daniel Lenz, Radoslaw Wojciechowski
Series: Grundlehren der mathematischen Wissenschaften 358
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 668
Tags: Graphs, Dirichlet Forms, Spectral Geometry
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
