Graphs and discrete dirichlet spaces:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham, Switzerland
Springer
2021
|
| Series: | Grundlehren der mathematischen Wissenschaften
358 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 668 Seiten |
| ISBN: | 9783030814588 3030814580 |
Staff View
MARC
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| 245 | 1 | 0 | |a Graphs and discrete dirichlet spaces |c Matthias Keller, Daniel Lenz, Radoslaw K. Wojciechowski |
| 264 | 1 | |a Cham, Switzerland |b Springer |c 2021 | |
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Record in the Search Index
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| adam_text | Matthias Keller + Daniel Lenz
Radostaw K Wojciechowski
Graphs and Discrete
Dirichlet Spaces
al Springer
Contents
Part 0 Prelude
Finite Graphs ::-222coeeeneesseeeeaeeesennennnn nun
0 1 Graphs, Laplacians and Dirichlet forms ------- -0000
0 2 Characterizing forms associated to graphs --- rer 0000
0 3 Characterizing Laplacians associated to graphs ---
0 4 Networks and electrostatics 1000 0c c cece e cece eee e eee
0 5 The heat equation and the Markov property ---- -2-0055
0 6 Resolvents and heat semigroups 4 0 0c eee eee e ee eee
07A Perron-Frobenius theorem and large time behavior
0 8 When there is no killing 0 c ccc eee eee eee eens
0 9 ‘Turning graphs into other graphs* -00 eee eee ee eee
0 10 Markov processes and the Feynman—Kac formula*
EX€rciS€S 2 0 cece ee eee eee ene nee ete eee n enter eenees
Notes 22222eeeesseeeeeeeneeeeenenennneer een nennen nennen
Part 1 Foundations and Fundamental Topics
Infinite Graphs — Key Concepts 000 e eee e cece eee eee eens
1 1 The setting ina nutshell 0 0 00 cece eee eee reese eeee
1 2 Graphs and (regular) Dirichlet forms - -----orcseeener
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 -+-- 60s es seer eee eee eee
EX€LciS€S 20 eee eben etn e een eee teneeeaee
NOteS 000 e cece cece nee teen e eens enn en ene eae eenes
xi
N
xii
Contents
Infinite Graphs ~ Toolbox 0 0 14
2 1 Generators, semigroups and resolvents on €? 0 000 n
2 2 Forms associated to graphs and restrictions to subsets 159
2 3 The curse of non-locality: Leibniz and chain rules 165
2 4 Creatures from the abyss* «1 200 eee eee eee eee eens 169
2 5 Markov processes and the Feynman—Kac formula redux* 173
EXxerciseS 0 0 cece ce een enn e rete teen teens IT?
Notes 2 00 ccc cece cece eee ene een e teen renee renege 184
Markov Uniqueness and Essential Self-Adjointness - ------ 185
3 1 Uniqueness of associated forms 0 + 220s eee rere renee 185
3 2 Essential self-adjointness - 0c eee eee e reece erent ete 189
3 3 Markov uniqueness 0 00 c eee eee erect tte 193
Exercises 22 enter eee eens enters sees 24
ce a 214
Agmon-Allegretto-Piepenbrink and Persson Theorems ------+- 215
41A local Harnack inequality and consequences - -- rec 216
42 The ground state transform 22 2 tee
4 3 The bottom of the spectrum 002e eee ee eee eee enter ee ee 228
4 4 The bottom of the essential spectrum - ---- 200s seers 230
Exercises 0 00000 coc c cece c ccc eect eect eee ee eee e eee LF
Notes 2000 eee eee ee ce ence n eee eneee 239
Large Time Behavior of the Heat Kernel 0 2 000- 241
5 1 Positivity improving semigroups and the ground state 24)
5 2 Theoremns of Chavel-Karp and Li 2 20 235
5 3 The Neumann Laplacian and finite measure 2 249
Exercises © +--+ 0-2-0 e ee eer ete 252
Notes - -- +1 seer r reer t errr ttt t tte eee 254
Recurtence 2 20 Heer
6 1 General preliminaries ec nerttreeen 253
6 2 The form perspective oo Treten 261
6 3 The superharmonic function perspective 00 7 ttt ee eee, 268
6 4 The Green’s function perspective ttt eee, 275
6 5 The Green’s formula perspective , Tr 5 ~
66 A probabilistic point of view* tte 281
Exercises Kane N 290
Notes creme Tr 293
elle „300
Contents
7 Stochastic Completeness 0 00 0 e cece cece te cee eee eees
7 1 The heat equation on (° 000 eee cee ence eens
7 2 Stochastic completeness at infinity 00 cece eee eee
7 3 The heat equation perspective 00 0 c eee eee eee eee
74 The Poisson equation perspective 0 e cece eee eee
7 5 The form perspective 00 c cece cece eee e eee eee
7 6 The Green’s formula perspective 00e cece eee eee ee
7 7 The Omori-Yau maximum principle - 2 cece e eee
78A stability criterion and Khasminskii’s criterion
79A probabilistic interpretation* 0 00 c eee eee
EX€MciSeS 2 eect eect et ence neretees
NOteS 00 6 e nee nent teen teentees
Part 2 Classes of Graphs
8 Uniformly Positive Measure 00 00c eee eee eeeeeees
81A Liouville theorem 0 000 c cece eee eee eee eee
8 2 Uniqueness of the form and essential self-adjointness
83A spectral inclusion 0 0 02 e cece cece ee eee eee eee
8 4 The heat equation on €? 000 cece cece ener eee eee
8 5 Graphs with standard weights 00 00 ccc cee eeeeeees
EXerciS€S 20 cece cece en eee e nett een a eeatenes
Notes 2 c ccc cece cece eee eet e cent e neta eeeennee
9 Weak Spherical Symmetry 200020 02 ccc cece cece eneee
9 | Symmetry of the heat kernel 0 0 000 20 e cece eee eee ee
9 2 The spectral gap 0 cece cece eee eee eee
9 3 Recurrence 6 cece cee eee teen cee tener eeene
9 4 Stochastic completeness at infinity 0 20 00- eee
Exercises een eee e cece nee ees e es eneeeeetennnenees
| (6) TV
10 Sparseness and Isoperimetric Inequalities 2 0065
10 1 Notions of sparseness 002 cece eee eee eee eeee ann
10 2 Co-area formulae 00 c cece eee eee nee eens
10 3 Weak sparseness and the form domain - - 0e000-
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 0 6c cee eect ee ee teen eee e ne eeee eens
Notes 202 ccc ccc ce te eee ee een nent reece ereeeees
xiv
Contents
Part 3 Geometry and Intrinsic Metrics
11 Intrinsic Metrics: Definition and Basie Facts ---- ern 443
11 1 Definition and motivation - --eeeeeeee nennen nenne 443
11 2 Path metrics and a Hopf-Rinow theorem - „reset 447
11 3 Examples and relations to other metrics -----+e0r seers 453
11 4 Geometric assumptions and cutoff functions - --++-+++500° 458
EXe€rciseS 0 0 cece cece eee ee ene eee e nent nn erent 463
Notes 2 00 cece ccc cece ce ee renee renee r er eeeeereene ene? 466
12 Harmonic Functions and Caccioppoli Theoty - --- 40 469
12 1 Caccioppoli inequalities -- 2 --ee nern ernennen nen 470
12 2 Liouville theorems 020e ee eee eee eee e reer ener 483
12 3 Applications of the Liouville theorems - +++0++0¢002 000° 489
12 4 Shnol’ theorems 0 000 c cece ee teen eee eens eneeee 493
Exercises © 00 c0cccceccecscccccucsceceeeeeeeseeeeeneees eres 1 500
Notes 2 00 00 ccc cece cee cence eee eee ernennen en nenne 504
13 Spectral Bounds - --- ---seeeeeeeesnneeneneenn ren 507
13 1 Cheeger constants and lower spectral bounds „+ 7 7 507
13 2 Volume growth and upper spectral bounds - ---2+--ee00+ 513
ExerciseS 2 00 cece cece ete etter eee e cree eeeeesenent 32
Notes 000 cccccececeeceeeeeeeeeveetecceeesstenteneereesees S24
14 Volume Growth Criterion for Stochastic Completeness and
Uniqueness Class 00 000 :0 cee cece eect etree nen eneeee 525
14 1 Uniqueness class ve nccceceteceeteeeteenccsenteeetser sense D6
14 2 Refinements 0 0 020 cce cence cece eee e teeters eee 538
14 3 Volume growth criterion for stochastic completeness - ------- 545
Exefcises „22 oeneeeeseeneeeeeenneeeee nennen nnnne nenn nn eer nenn 548
Notes 222@eeeeneeeeseeneennnenenen nennen een nenne nennen en 550
Appendix
A The Spectral Theorem 0000000eeeeeeee cece eee e eee eee SOS
B Closed Forms on Hilbert spaces 2 - 200 e cece reece eens 587
C Dirichlet Forms and Beurling-Deny Criteria 5%9
D Semigroups, Resolvents and their Generators --+ --: 605
Contents xv
E Aspects of Operator Theory weet e eee eee eee 623
E 1 Acharacterization of the resolvent 0 0 cece ee eee 623
E 2 The discrete and essential spectrum 0002 eeee 626
E 3 Reducing subspaces and commuting operators 638
E 4 The Riesz-Thorin interpolation theorem -000005 644
References 2000 cence eee eee een e rece e een eee 647
Index 22 een ene e nee ener rene et eee 663
|
| adam_txt |
Matthias Keller + Daniel Lenz
Radostaw K Wojciechowski
Graphs and Discrete
Dirichlet Spaces
al Springer
Contents
Part 0 Prelude
Finite Graphs ::-222coeeeneesseeeeaeeesennennnn nun
0 1 Graphs, Laplacians and Dirichlet forms ------- -0000
0 2 Characterizing forms associated to graphs --- rer 0000
0 3 Characterizing Laplacians associated to graphs ---
0 4 Networks and electrostatics 1000 0c c cece e cece eee e eee
0 5 The heat equation and the Markov property ---- -2-0055
0 6 Resolvents and heat semigroups 4 0 0c eee eee e ee eee
07A Perron-Frobenius theorem and large time behavior
0 8 When there is no killing 0 c ccc eee eee eee eens
0 9 ‘Turning graphs into other graphs* -00 eee eee ee eee
0 10 Markov processes and the Feynman—Kac formula*
EX€rciS€S 2 0 cece ee eee eee ene nee ete eee n enter eenees
Notes 22222eeeesseeeeeeeneeeeenenennneer een nennen nennen
Part 1 Foundations and Fundamental Topics
Infinite Graphs — Key Concepts 000 e eee e cece eee eee eens
1 1 The setting ina nutshell 0 0 00 cece eee eee reese eeee
1 2 Graphs and (regular) Dirichlet forms - -----orcseeener
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 -+-- 60s es seer eee eee eee
EX€LciS€S 20 eee eben etn e een eee teneeeaee
NOteS 000 e cece cece nee teen e eens enn en ene eae eenes
xi
N
xii
Contents
Infinite Graphs ~ Toolbox 0 0 14
2 1 Generators, semigroups and resolvents on €? 0 000 n
2 2 Forms associated to graphs and restrictions to subsets 159
2 3 The curse of non-locality: Leibniz and chain rules 165
2 4 Creatures from the abyss* «1 200 eee eee eee eee eens 169
2 5 Markov processes and the Feynman—Kac formula redux* 173
EXxerciseS 0 0 cece ce een enn e rete teen teens IT?
Notes 2 00 ccc cece cece eee ene een e teen renee renege 184
Markov Uniqueness and Essential Self-Adjointness - ------ 185
3 1 Uniqueness of associated forms 0 + 220s eee rere renee 185
3 2 Essential self-adjointness - 0c eee eee e reece erent ete 189
3 3 Markov uniqueness 0 00 c eee eee erect tte 193
Exercises 22 enter eee eens enters sees 24
ce a 214
Agmon-Allegretto-Piepenbrink and Persson Theorems ------+- 215
41A local Harnack inequality and consequences - -- rec 216
42 The ground state transform 22 2 tee
4 3 The bottom of the spectrum 002e eee ee eee eee enter ee ee 228
4 4 The bottom of the essential spectrum - ---- 200s seers 230
Exercises 0 00000 coc c cece c ccc eect eect eee ee eee e eee LF
Notes 2000 eee eee ee ce ence n eee eneee 239
Large Time Behavior of the Heat Kernel 0 2 000- 241
5 1 Positivity improving semigroups and the ground state 24)
5 2 Theoremns of Chavel-Karp and Li 2 20 235
5 3 The Neumann Laplacian and finite measure 2 249
Exercises © +--+ 0-2-0 e ee eer ete 252
Notes - -- +1 seer r reer t errr ttt t tte eee 254
Recurtence 2 20 Heer
6 1 General preliminaries ec nerttreeen 253
6 2 The form perspective oo Treten 261
6 3 The superharmonic function perspective 00 7 ttt ee eee, 268
6 4 The Green’s function perspective ttt eee, 275
6 5 The Green’s formula perspective , Tr 5 ~
66 A probabilistic point of view* tte 281
Exercises Kane N 290
Notes creme Tr 293
elle „300
Contents
7 Stochastic Completeness 0 00 0 e cece cece te cee eee eees
7 1 The heat equation on (° 000 eee cee ence eens
7 2 Stochastic completeness at infinity 00 cece eee eee
7 3 The heat equation perspective 00 0 c eee eee eee eee
74 The Poisson equation perspective 0 e cece eee eee
7 5 The form perspective 00 c cece cece eee e eee eee
7 6 The Green’s formula perspective 00e cece eee eee ee
7 7 The Omori-Yau maximum principle - 2 cece e eee
78A stability criterion and Khasminskii’s criterion
79A probabilistic interpretation* 0 00 c eee eee
EX€MciSeS 2 eect eect et ence neretees
NOteS 00 6 e nee nent teen teentees
Part 2 Classes of Graphs
8 Uniformly Positive Measure 00 00c eee eee eeeeeees
81A Liouville theorem 0 000 c cece eee eee eee eee
8 2 Uniqueness of the form and essential self-adjointness
83A spectral inclusion 0 0 02 e cece cece ee eee eee eee
8 4 The heat equation on €? 000 cece cece ener eee eee
8 5 Graphs with standard weights 00 00 ccc cee eeeeeees
EXerciS€S 20 cece cece en eee e nett een a eeatenes
Notes 2 c ccc cece cece eee eet e cent e neta eeeennee
9 Weak Spherical Symmetry 200020 02 ccc cece cece eneee
9 | Symmetry of the heat kernel 0 0 000 20 e cece eee eee ee
9 2 The spectral gap 0 cece cece eee eee eee
9 3 Recurrence 6 cece cee eee teen cee tener eeene
9 4 Stochastic completeness at infinity 0 20 00- eee
Exercises een eee e cece nee ees e es eneeeeetennnenees
| (6) TV
10 Sparseness and Isoperimetric Inequalities 2 0065
10 1 Notions of sparseness 002 cece eee eee eee eeee ann
10 2 Co-area formulae 00 c cece eee eee nee eens
10 3 Weak sparseness and the form domain - - 0e000-
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 0 6c cee eect ee ee teen eee e ne eeee eens
Notes 202 ccc ccc ce te eee ee een nent reece ereeeees
xiv
Contents
Part 3 Geometry and Intrinsic Metrics
11 Intrinsic Metrics: Definition and Basie Facts ---- ern 443
11 1 Definition and motivation - --eeeeeeee nennen nenne 443
11 2 Path metrics and a Hopf-Rinow theorem - „reset 447
11 3 Examples and relations to other metrics -----+e0r seers 453
11 4 Geometric assumptions and cutoff functions - --++-+++500° 458
EXe€rciseS 0 0 cece cece eee ee ene eee e nent nn erent 463
Notes 2 00 cece ccc cece ce ee renee renee r er eeeeereene ene? 466
12 Harmonic Functions and Caccioppoli Theoty - --- 40 469
12 1 Caccioppoli inequalities -- 2 --ee nern ernennen nen 470
12 2 Liouville theorems 020e ee eee eee eee e reer ener 483
12 3 Applications of the Liouville theorems - +++0++0¢002 000° 489
12 4 Shnol’ theorems 0 000 c cece ee teen eee eens eneeee 493
Exercises © 00 c0cccceccecscccccucsceceeeeeeeseeeeeneees eres 1 500
Notes 2 00 00 ccc cece cee cence eee eee ernennen en nenne 504
13 Spectral Bounds - --- ---seeeeeeeesnneeneneenn ren 507
13 1 Cheeger constants and lower spectral bounds „+ 7 7 507
13 2 Volume growth and upper spectral bounds - ---2+--ee00+ 513
ExerciseS 2 00 cece cece ete etter eee e cree eeeeesenent 32
Notes 000 cccccececeeceeeeeeeeeveetecceeesstenteneereesees S24
14 Volume Growth Criterion for Stochastic Completeness and
Uniqueness Class 00 000 :0 cee cece eect etree nen eneeee 525
14 1 Uniqueness class ve nccceceteceeteeeteenccsenteeetser sense D6
14 2 Refinements 0 0 020 cce cence cece eee e teeters eee 538
14 3 Volume growth criterion for stochastic completeness - ------- 545
Exefcises „22 oeneeeeseeneeeeeenneeeee nennen nnnne nenn nn eer nenn 548
Notes 222@eeeeneeeeseeneennnenenen nennen een nenne nennen en 550
Appendix
A The Spectral Theorem 0000000eeeeeeee cece eee e eee eee SOS
B Closed Forms on Hilbert spaces 2 - 200 e cece reece eens 587
C Dirichlet Forms and Beurling-Deny Criteria 5%9
D Semigroups, Resolvents and their Generators --+ --: 605
Contents xv
E Aspects of Operator Theory weet e eee eee eee 623
E 1 Acharacterization of the resolvent 0 0 cece ee eee 623
E 2 The discrete and essential spectrum 0002 eeee 626
E 3 Reducing subspaces and commuting operators 638
E 4 The Riesz-Thorin interpolation theorem -000005 644
References 2000 cence eee eee een e rece e een eee 647
Index 22 een ene e nee ener rene et eee 663 |
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| any_adam_object_boolean | 1 |
| author | Keller, Matthias Lenz, Daniel 1970- Wojciechowski, Radoslaw K. 1980- |
| author_GND | (DE-588)13023172X (DE-588)1208017357 |
| author_facet | Keller, Matthias Lenz, Daniel 1970- Wojciechowski, Radoslaw K. 1980- |
| author_role | aut aut aut |
| author_sort | Keller, Matthias |
| author_variant | m k mk d l dl r k w rk rkw |
| building | Verbundindex |
| bvnumber | BV047600743 |
| classification_rvk | SK 170 SK 890 SK 540 |
| ctrlnum | (OCoLC)1282606237 (DE-599)KXP1776180593 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV047600743 |
| illustrated | Not Illustrated |
| index_date | 2024-07-03T18:36:43Z |
| indexdate | 2024-07-10T09:15:51Z |
| institution | BVB |
| isbn | 9783030814588 3030814580 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032985865 |
| oclc_num | 1282606237 |
| open_access_boolean | |
| owner | DE-20 |
| owner_facet | DE-20 |
| physical | xv, 668 Seiten |
| publishDate | 2021 |
| publishDateSearch | 2021 |
| publishDateSort | 2021 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Keller, Matthias Verfasser aut Graphs and discrete dirichlet spaces Matthias Keller, Daniel Lenz, Radoslaw K. Wojciechowski Cham, Switzerland Springer 2021 xv, 668 Seiten txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 358 Spektraltheorie (DE-588)4116561-5 gnd rswk-swf Dirichletsche Form (DE-588)4150137-8 gnd rswk-swf Dirichletsche Form (DE-588)4150137-8 s Spektraltheorie (DE-588)4116561-5 s DE-604 Lenz, Daniel 1970- Verfasser (DE-588)13023172X aut Wojciechowski, Radoslaw K. 1980- Verfasser (DE-588)1208017357 aut Erscheint auch als Online-Ausgabe 978-3-030-81459-5 Grundlehren der mathematischen Wissenschaften 358 (DE-604)BV000000395 358 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032985865&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Keller, Matthias Lenz, Daniel 1970- Wojciechowski, Radoslaw K. 1980- Graphs and discrete dirichlet spaces Grundlehren der mathematischen Wissenschaften Spektraltheorie (DE-588)4116561-5 gnd Dirichletsche Form (DE-588)4150137-8 gnd |
| subject_GND | (DE-588)4116561-5 (DE-588)4150137-8 |
| title | Graphs and discrete dirichlet spaces |
| title_auth | Graphs and discrete dirichlet spaces |
| title_exact_search | Graphs and discrete dirichlet spaces |
| title_exact_search_txtP | Graphs and discrete dirichlet spaces |
| title_full | Graphs and discrete dirichlet spaces Matthias Keller, Daniel Lenz, Radoslaw K. Wojciechowski |
| title_fullStr | Graphs and discrete dirichlet spaces Matthias Keller, Daniel Lenz, Radoslaw K. Wojciechowski |
| title_full_unstemmed | Graphs and discrete dirichlet spaces Matthias Keller, Daniel Lenz, Radoslaw K. Wojciechowski |
| title_short | Graphs and discrete dirichlet spaces |
| title_sort | graphs and discrete dirichlet spaces |
| topic | Spektraltheorie (DE-588)4116561-5 gnd Dirichletsche Form (DE-588)4150137-8 gnd |
| topic_facet | Spektraltheorie Dirichletsche Form |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032985865&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT kellermatthias graphsanddiscretedirichletspaces AT lenzdaniel graphsanddiscretedirichletspaces AT wojciechowskiradoslawk graphsanddiscretedirichletspaces |