Fluctuations in Markov processes: time symmetry and martingale approximation
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2012
|
| Series: | Grundlehren der mathematischen Wissenschaften
345 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVII, 491 S. 235 mm x 155 mm |
| ISBN: | 9783642298790 |
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MARC
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Record in the Search Index
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IMAGE 1
CONTENTS
PART I GENERAL THEORY 1 A WARMING-UP EXAMPLE 3
1.1 ERGODIC MARKOV CHAINS 4
1.2 ALMOST SURE CENTRAL LIMIT THEOREM FOR ERGODIC MARKOV CHAINS 6
1.3 CENTRAL LIMIT THEOREM FOR MARTINGALES 9
1.4 TIME-VARIANCE IN REVERSIBLE MARKOV CHAINS 13
1.5 CENTRAL LIMIT THEOREM FOR REVERSIBLE MARKOV CHAINS 17
1.6 THE SPACE OF FINITE TIME-VARIANCE FUNCTIONS 21
1.6.1 THE SPACE 21
1.6.2 THE SPACE M - \ 24
1.7 COMMENTS AND REFERENCES 28
REFERENCES 30
2 CENTRAL LIMIT THEOREMS 33
2.1 CENTRAL LIMIT THEOREM FOR CONTINUOUS TIME MARTINGALES 36
2.2 THE SPACES FFL\ AND I 4 0
2.3 THE RESOLVENT EQUATION 45
2.4 DYNKIN'S MARTINGALES 4 6
2.5 ML I ESTIMATES OF THE TIME-VARIANCE 47
2.6 CENTRAL LIMIT THEOREM FOR MARKOV PROCESSES 50
2.7 SOME EXAMPLES 56
2.7.1 REVERSIBILITY 56
2.7.2 SPECTRAL GAP 57
2.7.3 SECTOR CONDITION 57
2.7.4 GRADED SECTOR CONDITION 59
2.7.5 PERTURBATIONS O F NORMAL OPERATORS 68
2.8 INVARIANCE PRINCIPLES IN THE MULTIDIMENSIONAL CASE 70
2.8.1 MARTINGALES WITH STATIONARY INCREMENTS 7 0
2.8.2 ADDITIVE FUNCTIONALS OF MARKOV PROCESSES 73
X I I I
HTTP://D-NB.INFO/1021362263
IMAGE 2
XIV CONTENTS
2.9 COMMENTS AND REFERENCES 76
REFERENCES 78
3 RANDOM WALKS IN RANDOM ENVIRONMENT 81
3.1 RANDOM WALKS WITH RANDOM CONDUCTANCES 82
3.2 DOUBLY STOCHASTIC RANDOM WALKS 89
3.3 CYCLIC RANDOM WALKS 95
3.4 RANDOM WALKS WITH DRIFT IN M - \ 100
3.4.1 THE CORRECTOR FIELD 102
3.4.2 AN ELLIPTIC EQUATION FOR THE CORRECTOR FIELD 107
3.4.3 THE ENERGY IDENTITY 109
3.5 RANDOM WALKS IN MIXING ENVIRONMENTS 115
3.6 DOUBLY STOCHASTIC RANDOM WALKS IN DIMENSION D - 1 121
3.7 SYMMETRIC RANDOM WALKS 126
3.8 COMMENTS AND REFERENCES 133
REFERENCES 134
4 BOUNDS AND VARIATIONAL PRINCIPLES FOR THE ASYMPTOTIC VARIANCE . . . .
137 4.1 QUADRATIC FUNCTIONAL O F THE RESOLVENT 137
4.2 BOUNDS AND VARIATIONAL FORMULAS FOR THE VARIANCE 143
4.3 VARIATIONAL PRINCIPLES IN THE GRADED SECTOR CONTEXT 147
4.4 ESTIMATES OF THE VARIANCE 148
4.5 COMMENTS AND REFERENCES 151
REFERENCES 151
PART II SIMPLE EXCLUSION PROCESSES
5 THE SIMPLE EXCLUSION PROCESS 155
5.1 EXCLUSION PROCESSES 157
5.2 CENTRAL LIMIT THEOREMS FOR ADDITIVE FUNCTIONALS 161
5.3 THE MEAN ZERO ASYMMETRIC CASE 162
5.4 DUALITY 171
5.5 THE ASYMMETRIC CASE, A = 1 / 2 178
5.6 THE ASYMMETRIC CASE, A / 1/2 182
5.7 TRANSIENT MARKOV PROCESSES 188
5.8 COMMENTS AND REFERENCES 192
REFERENCES 195
6 SELF-DIFFUSION 199
6.1 THE EXCLUSION PROCESS AS SEEN FROM A TAGGED PARTICLE 200
6.2 ELEMENTARY MARTINGALES 204
6.3 THE SPACES ANDJ#!_I 208
6.4 LAW O F LARGE NUMBERS 210
6.5 CENTRAL LIMIT THEOREM 210
6.6 THE MEAN ZERO ASYMMETRIC CASE 214
6.7 DUALITY 216
6.8 THE ASYMMETRIC CASE IN DIMENSION D 3 220
IMAGE 3
CONTENTS X V
6.9 THE SELF-DIFFUSION MATRIX 233
6.10 COMMENTS AND REFERENCES 236
REFERENCES 239
7 EQUILIBRIUM FLUCTUATIONS O F THE DENSITY FIELD 241
7.1 DUALITY 248
7.2 APPROXIMATIONS IN 253
7.3 THE FLUCTUATION-DISSIPATION THEOREM 257
7.4 THE SECOND CLASS PARTICLE 263
7.5 ESTIMATES ON THE OPERATORS E, S ,2, *.A AND 3*, 266
7.6 COMMENTS AND REFERENCES 272
REFERENCES 273
8 REGULARITY O F THE ASYMPTOTIC VARIANCE 275
8.1 THE RESOLVENT EQUATION 276
8.2 THE SYMMETRIC CASE 279
8.3 THE MEAN ZERO CASE 280
8.4 THE ASYMMETRIC CASE IN D 3 284
8.5 REGULARITY O F THE DIFFUSION COEFFICIENTS 287
8.6 COMMENTS AND REFERENCES 288
REFERENCES 289
PART III DIFFUSIONS IN RANDOM ENVIRONMENTS
9 DIFFUSIONS IN RANDOM ENVIRONMENTS 293
9.1 DIFFUSIONS WITH PERIODIC COEFFICIENTS 293
9.2 REMARK ABOUT THE QUASI-PERIODIC CASE 299
9.3 DIFFUSIONS WITH STATIONARY COEFFICIENTS 301
9.3.1 PRELIMINARIES ON STATIONARY ENVIRONMENTS 302
9.3.2 SPACES OF SMOOTH FUNCTIONS 304
9.3.3 ITO EQUATIONS WITH STATIONARY COEFFICIENTS 305
9.4 ENVIRONMENT PROCESS AND ITS PROPERTIES 307
9.5 MARTINGALE DECOMPOSITION AND CENTRAL LIMIT THEOREM 310
9.6 HOMOGENIZATION OF SOLUTIONS OF PARABOLIC PARTIAL DIFFERENTIAL
EQUATIONS 313
9.6.1 RANDOM COEFFICIENT CASE 314
9.6.2 PERIODIC CASE 316
9.7 PROOFS OF PROPOSITIONS 9.8 AND 9.9 317
9.7.1 PROOF O F PROPOSITION 9.8 318
9.7.2 PROOF O F PROPOSITION 9.9 319
9.8 ONE-DIMENSIONAL CASE 321
9.9 DIFFUSIONS WITH TIME DEPENDENT COEFFICIENTS 322
9.9.1 SPACE-TIME STATIONARY ENVIRONMENTS 322
9.9.2 CENTRAL LIMIT THEOREM 326
9.10 COMMENTS AND REFERENCES 327
REFERENCES 328
IMAGE 4
XVI CONTENTS
10 VARIATIONAL PRINCIPLES FOR THE LIMITING VARIANCE 331
10.1 SPACES O F VECTOR FIELDS 331
10.2 UPPER BOUND 334
10.3 LOWER BOUND 339
10.4 SPECTRAL REPRESENTATION OF HOMOGENEOUS FIELDS 341
10.5 COMMENTS AND REFERENCES 343
REFERENCES 343
11 DIFFUSIONS WITH DIVERGENCE FREE DRIFTS 345
11.1 PASSIVE TRACER MODEL 345
11.2 PROPERTIES O F THE FLOW AND THE DEFINITION OF THE STREAM MATRIX . .
345 11.3 CENTRAL LIMIT THEOREM FOR A DIFFUSION WITH BOUNDED STREAM
MATRIX 348
11.4 CONVECTION ENHANCED DIFFUSIONS 349
11.5 TIME DEPENDENT FLOWS WITH FINITE PECLET NUMBER 350
11.6 PROOF O F THEOREM 11.4 351
11.6.1 NOTATION 352
11.6.2 STATEMENTS OF SOME TECHNICAL RESULTS 352
11.6.3 PROPERTIES OF THE ENVIRONMENT PROCESS 353
11.6.4 PROPERTIES OF THE -NORM 355
11.6.5 CONSTRUCTION O F THE CORRECTOR FIELD 357
11.6.6 PROOF OF THE ENERGY IDENTITY 363
11.7 PROOFS OF THE TECHNICAL RESULTS 367
11.7.1 PROOF OF PROPOSITION 11.6 367
11.7.2 PROOF O F PROPOSITION 11.7 367
11.7.3 PROOF OF PROPOSITION 11.8 369
11.7.4 PROOF OF PROPOSITION 11.9 369
11.7.5 ERGODIC THEOREM 370
11.8 COMMENTS AND REFERENCES 371
REFERENCES 372
12 DIFFUSIONS WITH GAUSSIAN DRIFTS 375
12.1 STATIONARY GAUSSIAN FIELDS 376
12.2 HERMITE POLYNOMIALS AND GRADED STRUCTURE OF L 2 ( Q ) 376
12.3 ENVIRONMENT PROCESS AND ITS PROPERTIES 377
12.4 CENTRAL LIMIT THEOREM 381
12.5 PROOFS O F TECHNICAL RESULTS 382
12.5.1 PROOF O F ESTIMATE (12.2) 382
12.5.2 PROOFS OF THEOREM 12.3 AND PROPOSITION 12.4 384
12.6 SUPERDIFFUSIVE TRANSPORT IN A FLOW WITH INFINITE PECLET NUMBER 387
12.6.1 HOMOGENEOUS, ISOTROPIC GAUSSIAN FLOWS 387
12.6.2 FLOWS WITH INFINITE PECLET NUMBERS 389
12.7 CENTRAL LIMIT THEOREM FOR DIFFUSIONS IN GAUSSIAN AND MARKOVIAN
FLOWS 394
12.8 MARKOVIAN DYNAMICS OF THE ENVIRONMENT 396
IMAGE 5
CONTENTS XVII
12.8.1 HERMITE POLYNOMIALS 396
12.8.2 DEFINITION O F THE TRANSITION SEMIGROUP 397
12.8.3 PROPERTIES OF THE GENERATOR 398
12.8.4 MORE GENERAL FORMULATION O F THE MARKOV PROPERTY O F THE
ENVIRONMENT PROCESS 399
12.9 PERIODIC APPROXIMATION OF THE FLOW 400
12.10 ENVIRONMENT PROCESS 403
12.11 PROOF OF PART (1) O F THEOREM 12.13 405
12.12 ON SUPERDIFFUSIVE BEHAVIOR O F A TRACER IN AN ISOTROPIC FLOW . . .
409 12.13 PROOF OF PART (2) O F THEOREM 12.13 413
12.14 PROOFS OF THE RESULTS FROM SECT. 12.8 415
12.14.1 CONSTRUCTION OF THE SEMIGROUP 415
12.14.2 PROOF OF PROPOSITION 12.16 420
12.14.3 PROOF O F PROPOSITION 12.17 420
12.15 PROOFS O F THE RESULTS FROM SECT. 12.10 422
12.15.1 PROOF O F PROPOSITION 12.19 422
12.15.2 PROOF O F PROPOSITION 12.20 423
12.16 APPENDIX: SOME AUXILIARY RESULTS ABOUT GAUSSIAN RANDOM FIELDS 427
12.16.1 MULTIPLE STOCHASTIC INTEGRALS 427
12.16.2 SOME PROPERTIES OF HERMITE POLYNOMIALS 430
12.17 COMMENTS AND REFERENCES 431
REFERENCES 433
13 ORNSTEIN-UHLENBECK PROCESS WITH A RANDOM POTENTIAL 437
13.1 RANDOM DIFFUSION OF A PARTICLE WITH INERTIA 437
13.2 PROOF OF THE CENTRAL LIMIT THEOREM 438
13.3 PROOF OF PROPOSITION 13.2 445
13.4 GAUSSIAN BOUNDS ON TRANSITION PROBABILITY DENSITIES 446
13.5 COMMENTS AND REFERENCES 453
REFERENCES 454
14 ANALYTIC METHODS IN HOMOGENIZATION THEORY 455
14.1 G-CONVERGENCE OF OPERATORS 456
14.2 /^-CONVERGENCE O F QUADRATIC FORMS 458
14.3 G-CONVERGENCE OF MATRIX VALUED FUNCTIONS 460
14.4 APPLICATION TO HOMOGENIZATION O F DIFFUSIONS IN RANDOM MEDIA - 466
14.5 APPENDIX: ELLIPTICITY OF THE COEFFICIENT MATRIX O F A COERCIVE FORM
470 14.6 COMMENTS AND REFERENCES 471
REFERENCES 472
REFERENCES 475
NOTATION 487
SUBJECT INDEX 489 |
| any_adam_object | 1 |
| author | Komorowski, Tomasz 1963- Landim, Claudio 1965- Olla, Stefano 1959- |
| author_GND | (DE-588)1025350685 (DE-588)120477815 (DE-588)1025351312 |
| author_facet | Komorowski, Tomasz 1963- Landim, Claudio 1965- Olla, Stefano 1959- |
| author_role | aut aut aut |
| author_sort | Komorowski, Tomasz 1963- |
| author_variant | t k tk c l cl s o so |
| building | Verbundindex |
| bvnumber | BV040373361 |
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| ctrlnum | (OCoLC)809827700 (DE-599)DNB1021362263 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.233 |
| dewey-search | 519.233 |
| dewey-sort | 3519.233 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-08-21T00:06:28Z |
| institution | BVB |
| isbn | 9783642298790 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025226921 |
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| physical | XVII, 491 S. 235 mm x 155 mm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Komorowski, Tomasz 1963- Verfasser (DE-588)1025350685 aut Fluctuations in Markov processes time symmetry and martingale approximation Tomasz Komorowski ; Claudio Landim ; Stefano Olla Berlin [u.a.] Springer 2012 XVII, 491 S. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 345 Markov-Prozess (DE-588)4134948-9 gnd rswk-swf Zentraler Grenzwertsatz (DE-588)4067618-3 gnd rswk-swf Martingal (DE-588)4126466-6 gnd rswk-swf Markov-Prozess (DE-588)4134948-9 s Zentraler Grenzwertsatz (DE-588)4067618-3 s Martingal (DE-588)4126466-6 s DE-604 Landim, Claudio 1965- Verfasser (DE-588)120477815 aut Olla, Stefano 1959- Verfasser (DE-588)1025351312 aut Erscheint auch als Online-Ausgabe 978-3-642-29880-6 Grundlehren der mathematischen Wissenschaften 345 (DE-604)BV000000395 345 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4004378&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025226921&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Komorowski, Tomasz 1963- Landim, Claudio 1965- Olla, Stefano 1959- Fluctuations in Markov processes time symmetry and martingale approximation Grundlehren der mathematischen Wissenschaften Markov-Prozess (DE-588)4134948-9 gnd Zentraler Grenzwertsatz (DE-588)4067618-3 gnd Martingal (DE-588)4126466-6 gnd |
| subject_GND | (DE-588)4134948-9 (DE-588)4067618-3 (DE-588)4126466-6 |
| title | Fluctuations in Markov processes time symmetry and martingale approximation |
| title_auth | Fluctuations in Markov processes time symmetry and martingale approximation |
| title_exact_search | Fluctuations in Markov processes time symmetry and martingale approximation |
| title_full | Fluctuations in Markov processes time symmetry and martingale approximation Tomasz Komorowski ; Claudio Landim ; Stefano Olla |
| title_fullStr | Fluctuations in Markov processes time symmetry and martingale approximation Tomasz Komorowski ; Claudio Landim ; Stefano Olla |
| title_full_unstemmed | Fluctuations in Markov processes time symmetry and martingale approximation Tomasz Komorowski ; Claudio Landim ; Stefano Olla |
| title_short | Fluctuations in Markov processes |
| title_sort | fluctuations in markov processes time symmetry and martingale approximation |
| title_sub | time symmetry and martingale approximation |
| topic | Markov-Prozess (DE-588)4134948-9 gnd Zentraler Grenzwertsatz (DE-588)4067618-3 gnd Martingal (DE-588)4126466-6 gnd |
| topic_facet | Markov-Prozess Zentraler Grenzwertsatz Martingal |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4004378&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025226921&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT komorowskitomasz fluctuationsinmarkovprocessestimesymmetryandmartingaleapproximation AT landimclaudio fluctuationsinmarkovprocessestimesymmetryandmartingaleapproximation AT ollastefano fluctuationsinmarkovprocessestimesymmetryandmartingaleapproximation |