An introduction to the analysis of paths on a Riemannian manifold:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Math. Soc.
2000
|
| Schriftenreihe: | Mathematical surveys and monographs
74 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 269 S. |
| ISBN: | 9780821838396 0821820206 |
Internformat
MARC
| LEADER | 00000nam a2200000 cb4500 | ||
|---|---|---|---|
| 001 | BV013266047 | ||
| 003 | DE-604 | ||
| 005 | 20230911 | ||
| 007 | t | ||
| 008 | 000725s2000 |||| 00||| eng d | ||
| 020 | |a 9780821838396 |9 978-0-8218-3839-6 | ||
| 020 | |a 0821820206 |9 0-8218-2020-6 | ||
| 035 | |a (OCoLC)42289749 | ||
| 035 | |a (DE-599)BVBBV013266047 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-355 |a DE-29T |a DE-384 |a DE-188 | ||
| 050 | 0 | |a QA649 | |
| 082 | 0 | |a 516.3/73 |2 21 | |
| 084 | |a SK 820 |0 (DE-625)143258: |2 rvk | ||
| 100 | 1 | |a Stroock, Daniel W. |d 1940- |e Verfasser |0 (DE-588)130519561 |4 aut | |
| 245 | 1 | 0 | |a An introduction to the analysis of paths on a Riemannian manifold |c Daniel W. Stroock |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2000 | |
| 300 | |a XVII, 269 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical surveys and monographs |v 74 | |
| 650 | 4 | |a Mouvement brownien, Processus de | |
| 650 | 7 | |a Mouvement brownien, Processus de |2 ram | |
| 650 | 7 | |a Processos de difusão |2 larpcal | |
| 650 | 7 | |a Processos estocasticos |2 larpcal | |
| 650 | 4 | |a Riemann, Variétés de | |
| 650 | 7 | |a Riemann, Variétés de |2 ram | |
| 650 | 4 | |a Brownian motion processes | |
| 650 | 4 | |a Riemannian manifolds | |
| 650 | 0 | 7 | |a Brownsche Bewegung |0 (DE-588)4128328-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Riemannscher Raum |0 (DE-588)4128295-4 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Riemannscher Raum |0 (DE-588)4128295-4 |D s |
| 689 | 0 | 1 | |a Brownsche Bewegung |0 (DE-588)4128328-4 |D s |
| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-1301-9 |
| 830 | 0 | |a Mathematical surveys and monographs |v 74 |w (DE-604)BV000018014 |9 74 | |
| 856 | 4 | 2 | |m HBZ Datenaustausch |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009043897&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-009043897 | ||
Datensatz im Suchindex
| _version_ | 1804128031700156416 |
|---|---|
| adam_text | Contents
Preface xv
Chapter 1 Brownian Motion in Euclidean Space 1
1.1. Wiener Measure 1
1.1.1. Deconstructing Brownian Paths 2
1.1.2. Levy s Construction 5
1.1.3. Modulus of Continuity 6
1.1.4. Multi dimensional Brownian Motion 8
1.2. The Infinite Dimensional Sphere and Related Matters 9
1.2.1. Square Variation of Brownian Paths 9
1.2.2. Paley Wiener Integrals 10
1.2.3. Fourier Characterization 11
1.2.4. Extension to Higher Dimensions 11
1.2.5. The Cameron Martin Formula 12
1.2.6. Integration by Parts 14
1.3. Feynman s Picture of Wiener Measure 15
1.3.1. Rescaling Feynman s Picture 16
1.4. Wiener Measure, the Laplacian, and Martingales 18
1.4.1. A Preliminary Manipulation 18
1.4.2. Reinterpretation 20
1.4.3. A Heuristic Interpretation 23
Chapter 2 Diffusions in Euclidean Space 27
2.1. Martingale Problems for Operators in Hormander Form .... 27
2.2. The Abelian Case 28
2.2.1. A Single Vector Field 29
2.2.2. A Single Vector Field Squared 30
2.2.3. Several Commuting Vector Fields 32
2.3. The Non Abelian Case 34
2.3.1. The Scheme for Smooth Paths 35
2.3.2. The Scheme in the Stochastic Case 39
2.3.3. Basic Size Estimates 44
2.3.4. A Continuity Estimate 47
2.4. Derivatives 47
2.4.1. Burkholder s Inequality 48
2.4.2. Estimating Derivatives 49
2.4.3. A Little Bit of Sobolev 51
2.4.4. Existence of a Smooth Choice 53
2.4.5. Loosening Things Up 54
ix
x Contents
2.5. The Flow Property 55
2.5.1. Renewal at Stopping Times 55
2.5.2. The Heat Flow Semigroup for C and Uniqueness 56
Chapter 3 Some Addenda, Extensions, and Refinements ... 59
3.1. Explosion and Non explosion 59
3.1.1. An Example 59
3.2. Localization 61
3.2.1. Random Paths which may Explode 62
3.2.2. Splicing 63
3.2.3. Localizing the Martingale Problem 65
3.2.4. A Non explosion Criterion 66
3.2.5. Well posed Martingale Problems 67
3.3. A Polygonal Approximation Scheme 67
3.3.1. The Bounded Case 68
3.3.2. The General Case 68
3.4. Subordination 69
3.4.1. Time Dependent Vector Fields 70
3.4.2. Subordination for Diffusions 74
3.5. Semigroups of Diffeomorphisms 76
3.5.1. Flowing Backwards 76
3.5.2. Existence of a Continuous Version 77
3.5.3. Non Degenerate Jacobian 79
3.5.4. In General 80
3.6. Invariant and Symmetric Measures 81
3.6.1. Criterion for Invariance 82
3.6.2. Symmetric Measures 84
3.6.3. An Application to the Explosion Problem 86
Chapter 4 Doing it on a Manifold, An Extrinsic Approach . . 89
4.1. Diffusions on a Submanifold of RN 89
4.1.1. The Martingale Problem 89
4.1.2. Invariant and Symmetric Measures 91
4.1.3. Non Explosion Criterion 95
4.2. Brownian Motion on a Submanifold 95
4.2.1. Extrinsic Expressions 97
4.2.2. Extrinsic Brownian Motion 99
4.2.3. Brownian Motion Normal to a Submanifold 100
4.2.4. An Extrinsic Non Explosion Criterion for Brownian Motion . . 103
4.3. A Question of Measurable Interest 104
4.3.1. An Internal Approximation Scheme 104
Contents xi
Chapter 5 More about Extrinsic Riemannian Geometry . . . Ill
5.1. Parallel Transport Ill
5.1.1. Parallel Transport along Smooth Paths Ill
5.1.2. Parallel Transport along Brownian Paths 113
5.1.3. An Internal Description 115
5.2. Riemannian Connection, Covariant Derivatives, Curvature . . 116
5.2.1. Riemannian Connection and Covariant Derivatives 116
5.2.2. The Second Fundamental Form Minimal Submanifolds . . . 117
5.2.3. Riemannian and Ricci Curvature 120
5.3. The Distance Function and Explosion 122
5.3.1. Derivatives of the Distance Function 123
5.3.2. An Intrinsic Non Explosion Criterion for Brownian Motion . . 128
5.3.3. A Comparison of Explosion Criteria 131
5.3.4. Growth Estimate when Ricci Curvature is Bounded Below . . 133
Chapter 6 Bochner s Identity 137
6.1. The Jacobian Process Bochner s Identity 137
6.1.1. The Martingale Characterization of the Jacobian Process . . . 137
6.1.2. A Stochastic Version of Bochner s Identity 139
6.1.3. The Classical Bochner s Identity 145
6.1.4. The Case of Positive Ricci Curvature 146
6.2. Applications of Bochner s Identity 149
6.2.1. A Couple of Important Analytic Facts 149
6.2.2. Integrating Bochner s Identity 153
6.2.3. A Logarithmic Sobolev Inequality 155
6.3. Bismut s Formula 157
6.3.1. Variations on Bochner s Identity 157
6.3.2. The Bismut Factor 158
6.3.3. Measurability Again 161
6.3.4. An Estimate on Logarithmic Gradients 162
Chapter 7 Some Intrinsic Riemannian Geometry 165
7.1. Diffusions on an Abstract Manifold 166
7.1.1. Basic Existence Statement 166
7.2. Riemannian Manifolds 167
7.2.1. Basic Quantities 167
7.2.2. The Levi Civita Connection 168
7.2.3. Parallel Transport 169
7.2.4. An Alternative Expression for the Divergence 171
7.2.5. The Laplacian as the Trace of the Hessian 173
7.3. Brownian Motion on M 174
7.3.1. Localizing the Laplacian 174
7.3.2. Construction of Brownian Motion via Localization 175
xii Contents
Chapter 8 The Bundle of Orthonormal Frames 177
8.1. The Bundle O{M) 178
8.1.1. The Riemannian Connection and the Horizontal Subspace . . 179
8.1.2. Rolling, Geodesies, and Completeness 184
8.1.3. Canonical Vector Fields and the Laplacian 185
8.1.4. A Measure on O(M) 185
8.2. Brownian Motion on M via Projection from O(M) 187
8.2.1. The Basic Construction 187
8.2.2. Parallel Transport along Brownian Paths 188
8.2.3. Measurability Considerations 189
8.3. Curvature Considerations and an Explosion Criterion 192
8.3.1. Cartan s Structural Equations 192
8.3.2. Riemann and Ricci Curvatures 195
8.4. Derivatives of the Distance Function 197
8.4.1. Yau s Non Explosion Criterion 197
8.4.2. An Example of Explosion 200
8.5. Bochner on O(M) 201
8.5.1. Bochner s Identity 202
8.5.2. Integrated Version of Bochner s Identity 203
8.5.3. Bismut s Formula on O{M) 204
8.5.4. A Technical Comment 205
Chapter 9 Local Analysis of Brownian Motion 207
9.1. Normal Coordinates 207
9.1.1. Relationship to the Distance Function 208
9.2. Brownian Motion in Normal Coordinates 210
9.3. Asymptotic Expansion of Metric in Normal Coordinates . . . . 211
9.3.1. Relationship to Jacobi Fields 212
9.3.2. The Laplacian in Non Divergence Form 216
9.4. Coupling 218
9.4.1. Applications 223
Chapter 10 Perturbing Brownian Paths 227
10.1. Heuristic Explanation 227
10.2. Formulation as a Flow 231
10.2.1. Initial Reformulation 231
10.2.2. Formulation as a System of O.D.E. s on Pathspace 232
10.2.3. The State Space and Vector Fields 235
10.2.4. Perturbed Brownian Motion 237
10.3. Bochner via Perturbation of Brownian Paths 240
10.3.1. A Generalization of Bochner s Identity 240
10.3.2. An Application to Coupling 242
10.4. Bismut via Perturbation of Brownian Paths 243
10.4.1. The Perturbation and the Radon Nikodym Factor 244
10.5. Second Derivatives 249
Contents xiii
10.5.1. Derivative of the Bismut Factor 249
10.5.2. An Expression for Second Covariant Derivatives 255
10.5.3. Estimates for Derivatives of the Heat Flow 257
10.5.4. Estimate on Derivatives of the Heat Kernel 258
10.6. An Admission of Defeat 261
10.6.1. Li and Yau for Einstein Manifolds 262
Bibliography 265
Index 267
|
| any_adam_object | 1 |
| author | Stroock, Daniel W. 1940- |
| author_GND | (DE-588)130519561 |
| author_facet | Stroock, Daniel W. 1940- |
| author_role | aut |
| author_sort | Stroock, Daniel W. 1940- |
| author_variant | d w s dw dws |
| building | Verbundindex |
| bvnumber | BV013266047 |
| callnumber-first | Q - Science |
| callnumber-label | QA649 |
| callnumber-raw | QA649 |
| callnumber-search | QA649 |
| callnumber-sort | QA 3649 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)42289749 (DE-599)BVBBV013266047 |
| dewey-full | 516.3/73 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/73 |
| dewey-search | 516.3/73 |
| dewey-sort | 3516.3 273 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02023nam a2200505 cb4500</leader><controlfield tag="001">BV013266047</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20230911 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">000725s2000 |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9780821838396</subfield><subfield code="9">978-0-8218-3839-6</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">0821820206</subfield><subfield code="9">0-8218-2020-6</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)42289749</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV013266047</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-355</subfield><subfield code="a">DE-29T</subfield><subfield code="a">DE-384</subfield><subfield code="a">DE-188</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA649</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516.3/73</subfield><subfield code="2">21</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 820</subfield><subfield code="0">(DE-625)143258:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Stroock, Daniel W.</subfield><subfield code="d">1940-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)130519561</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">An introduction to the analysis of paths on a Riemannian manifold</subfield><subfield code="c">Daniel W. Stroock</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Providence, RI</subfield><subfield code="b">American Math. Soc.</subfield><subfield code="c">2000</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XVII, 269 S.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Mathematical surveys and monographs</subfield><subfield code="v">74</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Mouvement brownien, Processus de</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Mouvement brownien, Processus de</subfield><subfield code="2">ram</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Processos de difusão</subfield><subfield code="2">larpcal</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Processos estocasticos</subfield><subfield code="2">larpcal</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Riemann, Variétés de</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Riemann, Variétés de</subfield><subfield code="2">ram</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Brownian motion processes</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Riemannian manifolds</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Brownsche Bewegung</subfield><subfield code="0">(DE-588)4128328-4</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Riemannscher Raum</subfield><subfield code="0">(DE-588)4128295-4</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Riemannscher Raum</subfield><subfield code="0">(DE-588)4128295-4</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2="1"><subfield code="a">Brownsche Bewegung</subfield><subfield code="0">(DE-588)4128328-4</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-1-4704-1301-9</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Mathematical surveys and monographs</subfield><subfield code="v">74</subfield><subfield code="w">(DE-604)BV000018014</subfield><subfield code="9">74</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HBZ Datenaustausch</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009043897&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-009043897</subfield></datafield></record></collection> |
| id | DE-604.BV013266047 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:42:47Z |
| institution | BVB |
| isbn | 9780821838396 0821820206 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009043897 |
| oclc_num | 42289749 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-29T DE-384 DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-29T DE-384 DE-188 |
| physical | XVII, 269 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Mathematical surveys and monographs |
| series2 | Mathematical surveys and monographs |
| spelling | Stroock, Daniel W. 1940- Verfasser (DE-588)130519561 aut An introduction to the analysis of paths on a Riemannian manifold Daniel W. Stroock Providence, RI American Math. Soc. 2000 XVII, 269 S. txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs 74 Mouvement brownien, Processus de Mouvement brownien, Processus de ram Processos de difusão larpcal Processos estocasticos larpcal Riemann, Variétés de Riemann, Variétés de ram Brownian motion processes Riemannian manifolds Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 s Brownsche Bewegung (DE-588)4128328-4 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1301-9 Mathematical surveys and monographs 74 (DE-604)BV000018014 74 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009043897&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stroock, Daniel W. 1940- An introduction to the analysis of paths on a Riemannian manifold Mathematical surveys and monographs Mouvement brownien, Processus de Mouvement brownien, Processus de ram Processos de difusão larpcal Processos estocasticos larpcal Riemann, Variétés de Riemann, Variétés de ram Brownian motion processes Riemannian manifolds Brownsche Bewegung (DE-588)4128328-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4128328-4 (DE-588)4128295-4 |
| title | An introduction to the analysis of paths on a Riemannian manifold |
| title_auth | An introduction to the analysis of paths on a Riemannian manifold |
| title_exact_search | An introduction to the analysis of paths on a Riemannian manifold |
| title_full | An introduction to the analysis of paths on a Riemannian manifold Daniel W. Stroock |
| title_fullStr | An introduction to the analysis of paths on a Riemannian manifold Daniel W. Stroock |
| title_full_unstemmed | An introduction to the analysis of paths on a Riemannian manifold Daniel W. Stroock |
| title_short | An introduction to the analysis of paths on a Riemannian manifold |
| title_sort | an introduction to the analysis of paths on a riemannian manifold |
| topic | Mouvement brownien, Processus de Mouvement brownien, Processus de ram Processos de difusão larpcal Processos estocasticos larpcal Riemann, Variétés de Riemann, Variétés de ram Brownian motion processes Riemannian manifolds Brownsche Bewegung (DE-588)4128328-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Mouvement brownien, Processus de Processos de difusão Processos estocasticos Riemann, Variétés de Brownian motion processes Riemannian manifolds Brownsche Bewegung Riemannscher Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009043897&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT stroockdanielw anintroductiontotheanalysisofpathsonariemannianmanifold |