Diffusion processes and their sample paths:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1974
|
| Edition: | 2. print., corr. |
| Series: | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen
125 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 321 S. |
| ISBN: | 3540033025 0387033025 |
Staff View
MARC
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| 245 | 1 | 0 | |a Diffusion processes and their sample paths |c K. Ito ; H. P. McKean |
| 250 | |a 2. print., corr. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1974 | |
| 300 | |a XIV, 321 S. | ||
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Record in the Search Index
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| adam_text |
Contents
page
Prerequisites 1
Chapter l. The standard BROWNian motion 5
1.1. The standard random walk 5
1.2. Passage times for the standard random walk 7
1.3. Hincin's proof of the de Moivre Laplace limit theorem . 10
1.4. The standard BROWNian motion 12
1.5. P. Levy's construction 19
1.6. Strict Markov character 22
1.7. Passage times for the standard BROWNian motion 25
Note 1: Homogeneous differential processes with increasing paths 31
1.8. Kolmogorov's test and the law of the iterated logarithm . 33
1.9. P. Levy's Holder condition 36
1.10. Approximating the BROWNian motion by a random walk . 38
Chapter 2. BROWNian local times 40
2.1. The reflecting BROWNian motion 40
2.2. P. Levy's local time 42
2.3. Elastic BROWNian motion 45
2.4. t+ and down crossings 48
2.5. t+ as Haxjsdorff Besicovitch 1/2 dimensional measure SO
Note 1: Submartingales 52
Note 2: Hausdorff measure and dimension 53
2.6. Kac's formula for BROWNian functionals 54
2.7. Bessel processes 59
2.8. Standard BROWNian local time 63
2.9. BROWNian excursions 75
2.10. Application of the Bessel process to BROWNian excursions . . . 79
2.11. A time substitution 81
Chapter 3. The general 1 dimensional diffusion 83
3.1. Definition 83
3.2. Markov times 86
3.3. Matching numbers 89
3.4. Singular points 91
3.5. Decomposing the general diffusion into simple pieces 92
3.6. Green operators and the space D 94
3.7. Generators . . • 98
3.8. Generators continued * 100
3.9. Stopped diffusion 102
Contents XIII
page
Chapter 4. Generators 105
4.1. A general view 105
4.2. © as local differential operator: conservative non singular case . . Ill
4.3. © as local differential operator: general non singular case . 116
4.4. A second proof 119
4.5. © at an isolated singular point 125
4.6. Solving ©*« = (X u 128
4.7. © as global differential operator: non singular case 135
4.8. © on the shunts 136
4.9. © as global differential operator: singular case 142
4.10. Passage times 144
Note 1: Differential processes with increasing paths 146
4.11. Eigen differential expansions for Green functions and transition
densities 149
4.12. Kolmogorov's test 161
Chapter 5. Time changes and killing 164
5.1. Construction of sample paths: a general view 164
5.2. Time changes: Q = R1 167
5.3. Time changes: Q = [0, +00) 171
5.4. Local times 174
5.5. Subordination and chain rule I76
5.6. Killing times 179
5.7. Feller's BROwNian motions 186
5.8. Ikeda's example 18S
5.9. Time substitutions must come from local time integrals 190
5.1U. Shunts 191
5.11. Shunts with killing 196
5.12. Creation of mass 200
5.13. A parabolic equation 201
5.14. Explosions 206
5.15. A non linear parabolic equation 209
Chapter 6. Local and inverse local times 212
6.1. Local and inverse local times 212
6.2. Levy measures 214
6.3. t and the intervals of [0, + 00) — 3 218
6.4. A counter example: t and the intervals of [0, +00) — g 220
6.5 a t and downcrossings 222
6.5b t as Hausdorff measure 223
6.5 c t as diffusion 223
6.5d Excursions 223
6.6. Dimension numbers 224
6.7. Comparison tests 225
Note 1: Dimension numbers and fractional dimensional capacities 227
6.8. An individual ergodic theorem 22S
Chapter 7. BROWNian motion in several dimensions 232
7.1. Diffusion in several dimensions 232
7.2. The standard BROWKian motion in several dimensions 233
7.3. Wandering out to 00 236
XIV Contents
page
7.4. GREENian domains and Gkeen functions 237
7.5. Excessive functions 243
7.6. Application to the spectrum of Aji 245
7.7 Potentials and hitting probabilities 247
7.8. NEWTONian capacities 250
7.9. Gauss's quadratic form 253
7.10. Wiener's test 255
7.11. Applications of Wiener's test 257
7.12. Dirichlet problem 261
7.13. Neumann problem 264
7.14. Space time BROWNian motion 266
7.15. Spherical BROWNian motion and skew products 269
7.16. Spinning 274
7.17 An individual ergodic theorem for the standard 2 dimensional
BROWNian motion 277
7.18. Covering BROwNfan motions 279
7.19. Diffusions with BROWNian hitting probabilities 283
7.20. Right continuous paths 286
7.21. Riesz potentials 288
Chapter 8. A general view of diffusion in several dimensions . . . 291
8.1. Similar diffusions 291
8.2. 3 as differential operator 293
8.3 Time substitutions 295
8.4. Potentials 296
8.5. Boundaries 299
8.6. Elliptic operators 302
8.7. Feller's little boundary and tail algebras 303
Bibliography 306
List of notations 313
Index 315 |
| adam_txt |
Contents
page
Prerequisites 1
Chapter l. The standard BROWNian motion 5
1.1. The standard random walk 5
1.2. Passage times for the standard random walk 7
1.3. Hincin's proof of the de Moivre Laplace limit theorem . 10
1.4. The standard BROWNian motion 12
1.5. P. Levy's construction 19
1.6. Strict Markov character 22
1.7. Passage times for the standard BROWNian motion 25
Note 1: Homogeneous differential processes with increasing paths 31
1.8. Kolmogorov's test and the law of the iterated logarithm . 33
1.9. P. Levy's Holder condition 36
1.10. Approximating the BROWNian motion by a random walk . 38
Chapter 2. BROWNian local times 40
2.1. The reflecting BROWNian motion 40
2.2. P. Levy's local time 42
2.3. Elastic BROWNian motion 45
2.4. t+ and down crossings 48
2.5. t+ as Haxjsdorff Besicovitch 1/2 dimensional measure SO
Note 1: Submartingales 52
Note 2: Hausdorff measure and dimension 53
2.6. Kac's formula for BROWNian functionals 54
2.7. Bessel processes 59
2.8. Standard BROWNian local time 63
2.9. BROWNian excursions 75
2.10. Application of the Bessel process to BROWNian excursions . . . 79
2.11. A time substitution 81
Chapter 3. The general 1 dimensional diffusion 83
3.1. Definition 83
3.2. Markov times 86
3.3. Matching numbers 89
3.4. Singular points 91
3.5. Decomposing the general diffusion into simple pieces 92
3.6. Green operators and the space D 94
3.7. Generators . . • 98
3.8. Generators continued * 100
3.9. Stopped diffusion 102
Contents XIII
page
Chapter 4. Generators 105
4.1. A general view 105
4.2. © as local differential operator: conservative non singular case . . Ill
4.3. © as local differential operator: general non singular case . 116
4.4. A second proof 119
4.5. © at an isolated singular point 125
4.6. Solving ©*« = (X u 128
4.7. © as global differential operator: non singular case 135
4.8. © on the shunts 136
4.9. © as global differential operator: singular case 142
4.10. Passage times 144
Note 1: Differential processes with increasing paths 146
4.11. Eigen differential expansions for Green functions and transition
densities 149
4.12. Kolmogorov's test 161
Chapter 5. Time changes and killing 164
5.1. Construction of sample paths: a general view 164
5.2. Time changes: Q = R1 167
5.3. Time changes: Q = [0, +00) 171
5.4. Local times 174
5.5. Subordination and chain rule I76
5.6. Killing times 179
5.7. Feller's BROwNian motions 186
5.8. Ikeda's example 18S
5.9. Time substitutions must come from local time integrals 190
5.1U. Shunts 191
5.11. Shunts with killing 196
5.12. Creation of mass 200
5.13. A parabolic equation 201
5.14. Explosions 206
5.15. A non linear parabolic equation 209
Chapter 6. Local and inverse local times 212
6.1. Local and inverse local times 212
6.2. Levy measures 214
6.3. t and the intervals of [0, + 00) — 3 218
6.4. A counter example: t and the intervals of [0, +00) — g 220
6.5 a t and downcrossings 222
6.5b t as Hausdorff measure 223
6.5 c t as diffusion 223
6.5d Excursions 223
6.6. Dimension numbers 224
6.7. Comparison tests 225
Note 1: Dimension numbers and fractional dimensional capacities 227
6.8. An individual ergodic theorem 22S
Chapter 7. BROWNian motion in several dimensions 232
7.1. Diffusion in several dimensions 232
7.2. The standard BROWKian motion in several dimensions 233
7.3. Wandering out to 00 236
XIV Contents
page
7.4. GREENian domains and Gkeen functions 237
7.5. Excessive functions 243
7.6. Application to the spectrum of Aji 245
7.7 Potentials and hitting probabilities 247
7.8. NEWTONian capacities 250
7.9. Gauss's quadratic form 253
7.10. Wiener's test 255
7.11. Applications of Wiener's test 257
7.12. Dirichlet problem 261
7.13. Neumann problem 264
7.14. Space time BROWNian motion 266
7.15. Spherical BROWNian motion and skew products 269
7.16. Spinning 274
7.17 An individual ergodic theorem for the standard 2 dimensional
BROWNian motion 277
7.18. Covering BROwNfan motions 279
7.19. Diffusions with BROWNian hitting probabilities 283
7.20. Right continuous paths 286
7.21. Riesz potentials 288
Chapter 8. A general view of diffusion in several dimensions . . . 291
8.1. Similar diffusions 291
8.2. 3 as differential operator 293
8.3 Time substitutions 295
8.4. Potentials 296
8.5. Boundaries 299
8.6. Elliptic operators 302
8.7. Feller's little boundary and tail algebras 303
Bibliography 306
List of notations 313
Index 315 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Itō, Kiyoshi 1915-2008 McKean, Henry P. 1930-2024 |
| author_GND | (DE-588)119388073 (DE-588)130546275 |
| author_facet | Itō, Kiyoshi 1915-2008 McKean, Henry P. 1930-2024 |
| author_role | aut aut |
| author_sort | Itō, Kiyoshi 1915-2008 |
| author_variant | k i ki h p m hp hpm |
| building | Verbundindex |
| bvnumber | BV022118386 |
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| callnumber-label | QA274 |
| callnumber-raw | QA274.75 |
| callnumber-search | QA274.75 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 820 |
| ctrlnum | (OCoLC)490148859 (DE-599)BVBBV022118386 |
| dewey-full | 519.2 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.2 |
| dewey-search | 519.2 |
| dewey-sort | 3519.2 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. print., corr. |
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| id | DE-604.BV022118386 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T16:16:07Z |
| indexdate | 2024-08-20T00:35:13Z |
| institution | BVB |
| isbn | 3540033025 0387033025 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015333252 |
| oclc_num | 490148859 |
| open_access_boolean | |
| owner | DE-706 |
| owner_facet | DE-706 |
| physical | XIV, 321 S. |
| publishDate | 1974 |
| publishDateSearch | 1974 |
| publishDateSort | 1974 |
| publisher | Springer |
| record_format | marc |
| series | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |
| series2 | Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen |
| spelling | Itō, Kiyoshi 1915-2008 Verfasser (DE-588)119388073 aut Diffusion processes and their sample paths K. Ito ; H. P. McKean 2. print., corr. Berlin [u.a.] Springer 1974 XIV, 321 S. txt rdacontent n rdamedia nc rdacarrier Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 125 Capacité Jussieu Diffusion (physique) ram Diffusion Jussieu Mesure Lévy Jussieu Mouvement brownien Jussieu Mouvement brownien ram Potentiel Riesz Jussieu Processus Bessel Jussieu Processus de diffusion ram Processus stochastiques ram Brownian movements Diffusion Stochastic processes Stochastik (DE-588)4121729-9 gnd rswk-swf Differentialoperator (DE-588)4012251-7 gnd rswk-swf Theorie (DE-588)4059787-8 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Brownsche Bewegung (DE-588)4128328-4 gnd rswk-swf Diffusionsprozess (DE-588)4274463-5 gnd rswk-swf Diffusion (DE-588)4012277-3 gnd rswk-swf Differentialoperator (DE-588)4012251-7 s DE-604 Diffusion (DE-588)4012277-3 s Mathematik (DE-588)4037944-9 s Theorie (DE-588)4059787-8 s 1\p DE-604 Brownsche Bewegung (DE-588)4128328-4 s Diffusionsprozess (DE-588)4274463-5 s 2\p DE-604 3\p DE-604 Stochastik (DE-588)4121729-9 s 4\p DE-604 Stochastischer Prozess (DE-588)4057630-9 s 5\p DE-604 McKean, Henry P. 1930-2024 Verfasser (DE-588)130546275 aut Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 125 (DE-604)BV000000395 125 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015333252&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 5\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Itō, Kiyoshi 1915-2008 McKean, Henry P. 1930-2024 Diffusion processes and their sample paths Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Capacité Jussieu Diffusion (physique) ram Diffusion Jussieu Mesure Lévy Jussieu Mouvement brownien Jussieu Mouvement brownien ram Potentiel Riesz Jussieu Processus Bessel Jussieu Processus de diffusion ram Processus stochastiques ram Brownian movements Diffusion Stochastic processes Stochastik (DE-588)4121729-9 gnd Differentialoperator (DE-588)4012251-7 gnd Theorie (DE-588)4059787-8 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Mathematik (DE-588)4037944-9 gnd Brownsche Bewegung (DE-588)4128328-4 gnd Diffusionsprozess (DE-588)4274463-5 gnd Diffusion (DE-588)4012277-3 gnd |
| subject_GND | (DE-588)4121729-9 (DE-588)4012251-7 (DE-588)4059787-8 (DE-588)4057630-9 (DE-588)4037944-9 (DE-588)4128328-4 (DE-588)4274463-5 (DE-588)4012277-3 |
| title | Diffusion processes and their sample paths |
| title_auth | Diffusion processes and their sample paths |
| title_exact_search | Diffusion processes and their sample paths |
| title_exact_search_txtP | Diffusion processes and their sample paths |
| title_full | Diffusion processes and their sample paths K. Ito ; H. P. McKean |
| title_fullStr | Diffusion processes and their sample paths K. Ito ; H. P. McKean |
| title_full_unstemmed | Diffusion processes and their sample paths K. Ito ; H. P. McKean |
| title_short | Diffusion processes and their sample paths |
| title_sort | diffusion processes and their sample paths |
| topic | Capacité Jussieu Diffusion (physique) ram Diffusion Jussieu Mesure Lévy Jussieu Mouvement brownien Jussieu Mouvement brownien ram Potentiel Riesz Jussieu Processus Bessel Jussieu Processus de diffusion ram Processus stochastiques ram Brownian movements Diffusion Stochastic processes Stochastik (DE-588)4121729-9 gnd Differentialoperator (DE-588)4012251-7 gnd Theorie (DE-588)4059787-8 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Mathematik (DE-588)4037944-9 gnd Brownsche Bewegung (DE-588)4128328-4 gnd Diffusionsprozess (DE-588)4274463-5 gnd Diffusion (DE-588)4012277-3 gnd |
| topic_facet | Capacité Diffusion (physique) Diffusion Mesure Lévy Mouvement brownien Potentiel Riesz Processus Bessel Processus de diffusion Processus stochastiques Brownian movements Stochastic processes Stochastik Differentialoperator Theorie Stochastischer Prozess Mathematik Brownsche Bewegung Diffusionsprozess |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015333252&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT itokiyoshi diffusionprocessesandtheirsamplepaths AT mckeanhenryp diffusionprocessesandtheirsamplepaths |