Multiparameter processes: an introduction to random fields
Multi-parameter processes extend the existing one-parameter theory in an elegant way and have many applications to other fields in mathematics. This book on random fields is designed for a second graduate course in probability.. - Multi-parameter processes extend the existing one-parameter theory in...
Gespeichert in:
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| Format: | Buch |
| Sprache: | English |
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New York
Springer
2002
|
| Schriftenreihe: | Springer monographs in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Multi-parameter processes extend the existing one-parameter theory in an elegant way and have many applications to other fields in mathematics. This book on random fields is designed for a second graduate course in probability.. - Multi-parameter processes extend the existing one-parameter theory in an elegant way and have many applications to other fields in mathematics such as real analysis, functional analysis, group theory, and analytic number theory, to name a few. This book on the vast and rapidly developing subject of random fields is designed for a second graduate course in probability. Recent work on random fields has made it possible to make it an expository subject which interacts with several other areas in mathematics and has enough mathematical depth to be of use to pure as well as applied mathematicians of many backgrounds. |
| Beschreibung: | Includes bibliographical references (p. [543]-565) and indexes |
| Beschreibung: | XIX, 584 S. Ill. |
| ISBN: | 0387954597 9781441930095 |
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Datensatz im Suchindex
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| adam_text | I DAVAR KHOSHNEVISAN MULTIPARAMETER PROCESSES AN INTRODUCTION TO -RANDOM
FIELDS *-. .Y- * SPRINGER CONTENTS PREFACE V LIST OF FIGURES XV GENERAL
NOTATION XVII I DISCRETE-PARAMETER RANDOM FIELDS 1 1 DISCRETE-PARAMETER
MARTINGALES 3 1 ONE-PARAMETER MARTINGALES 4 1.1 DEFINITIONS 4 1.2 THE
OPTIONAL STOPPING THEOREM 7 1.3 A WEAK (1,1) INEQUALITY 8 1.4 A STRONG
(P,P) INEQUALITY 9 1.5 _ THE CASE P= 9 1.6 UPCROSSING INEQUALITIES 10
1.7 THE MARTINGALE CONVERGENCE THEOREM 12 2 ORTHOMARTINGALES 15 2.1
DEFINITIONS AND EXAMPLES 16 2.2 EMBEDDED SUBMARTINGALES 18 2.3 CAIROLI S
STRONG (P,P) INEQUALITY 19 2.4 ANOTHER MAXIMAL INEQUALITY 20 2.5 A WEAK
MAXIMAL INEQUALITY 22 VJII CONTENTS 2.6 ORTHOHISTORIES 22 2.7
CONVERGENCE NOTIONS 24 2.8 TOPOLOGICAL CONVERGENCE 26 2.9 REVERSED
ORTHOMARTINGALES 30 3 MARTINGALES 31 3.1 DEFINITIONS 31 3.2 MARGINAL
FILTRATIONS 31 3.3 A COUNTEREXAMPLE 33 3.4 COMMUTATION 35 3.5
MARTINGALES 37 3.6 CONDITIONAL INDEPENDENCE . 38 4 SUPPLEMENTARY
EXERCISES 40 5 NOTES ON CHAPTER 1 44 2 TWO APPLICATIONS IN ANALYSIS 47 1
HAAR SYSTEMS 47 1.1 THE 1-DIMENSIONAL HAAR SYSTEM 48 1.2 THE
IV-DIMENSIONAL HAAR SYSTEM 51 2 DIFFERENTIATION 54 2.1 LEBESGUE S
DIFFERENTIATION THEOREM 54 2.2 A UNIFORM DIFFERENTIATION THEOREM 58 3
SUPPLEMENTARY EXERCISES 61 4 NOTES ON CHAPTER 2 63 3 RANDOM WALKS 65 1
ONE-PARAMETER RANDOM WALKS 66 1.1 TRANSITION OPERATORS 66 1.2 THE STRONG
MARKOV PROPERTY 69 1.3 RECURRENCE 70 1.4 CLASSIFICATION OF RECURRENCE 72
1.5 TRANSIENCE 74 1.6 RECURRENCE OF POSSIBLE POINTS 75 1.7
RECURRENCE-TRANSIENCE DICHOTOMY 78 2 INTERSECTION PROBABILITIES 80 2.1
INTERSECTIONS OF TWO WALKS 80 2.2 AN ESTIMATE FOR TWO WALKS 85 2.3
INTERSECTIONS OF SEVERAL WALKS 86 2.4 AN ESTIMATE FOR N WALKS 89 3 THE
SIMPLE RANDOM WALK 89 3.1 RECURRENCE 90 3.2 INTERSECTIONS OF TWO SIMPLE
WALKS 91 3.3 THREE SIMPLE WALKS 93 3.4 SEVERAL SIMPLE WALKS 97 4
SUPPLEMENTARY EXERCISES 99 5 NOTES ON CHAPTER 3 103 CONTENTS IX I -.
MULTIPARAMETER WALKS 105 1 THE STRONG LAW OF LARGE NUMBERS 106 1.1
DEFINITIONS 106 1.2 COMMUTATION 107 1.3 A REVERSED ORTHOMARTINGALE 109
1.4 SMYTHE S LAW OF LARGE NUMBERS 110 2 THE LAW OF THE ITERATED
LOGARITHM 112 2.1 THE ONE-PARAMETER GAUSSIAN CASE 113 2.2 THE GENERAL
LIL 116 2.3 SUMMABILITY 117 2.4 DIRICHLET S DIVISOR LEMMA 118 2.5
TRUNCATION 119 2.6 BERNSTEIN S INEQUALITY 121 2.7 MAXIMAL INEQUALITIES
123 2.8 A NUMBER-THEORETIC ESTIMATE 125 2.9 PROOF OF THE LIL: THE UPPER
BOUND 127 2.10 A MODERATE DEVIATIONS ESTIMATE 128 2.11 PROOF OF THE LIL:
THE LOWER BOUND 130 3 SUPPLEMENTARY EXERCISES 132 4 NOTES ON CHAPTER 4
135 I GAUSSIAN RANDOM VARIABLES 137 1 THE BASIC CONSTRUCTION 137 1.1
GAUSSIAN RANDOM VECTORS 137 1.2 GAUSSIAN PROCESSES 140 1.3 WHITE NOISE
142 1.4 THE ISONORMAL PROCESS 144 1.5 THE BROWNIAN SHEET 147 2
REGULARITY THEORY 148 2.1 TOTALLY BOUNDED PSEUDOMETRIC SPACES 149 2.2
MODIFICATIONS AND SEPARABILITY 153 2.3 KOLMOGOROV S CONTINUITY THEOREM
158 2.4 CHAINING - 160 2.5 HOLDER-CONTINUOUS MODIFICATIONS 165 2.6 THE
ENTROPY INTEGRAL 167 2.7 DUDLEY S THEOREM 170 3 THE STANDARD BROWNIAN
SHEET 172 3.1 ENTROPY ESTIMATE 172 3.2 MODULUS OF CONTINUITY 173 4
SUPPLEMENTARY EXERCISES 175 5 NOTES ON CHAPTER 5 178 I LIMIT THEOREMS
181 1 RANDOM VARIABLES 181 1.1 DEFINITIONS 182 CONTENTS 1.2
DISTRIBUTIONS 183 1.3 UNIQUENESS 184 WEAK CONVERGENCE 185 2.1 THE
PORTMANTEAU THEOREM 186 2.2 THE CONTINUOUS MAPPING THEOREM 188 2.3 WEAK
CONVERGENCE IN EUCLIDEAN SPACE 188 2.4 TIGHTNESS 189 2.5 PROHOROV S
THEOREM 190 THE SPACE C 193 3.1 UNIFORM CONTINUITY 193 3.2
FINITE-DIMENSIONAL DISTRIBUTIONS 195 3.3 WEAK CONVERGENCE IN C 196 3.4
CONTINUOUS FUNCTIONALS 199 3.5 A SUFFICIENT CONDITION FOR PRETIGHTNESS
200 INVARIANCE PRINCIPLES 201 4.1 PRELIMINARIES 202 4.2
FINITE-DIMENSIONAL DISTRIBUTIONS 204 4.3 PRETIGHTNESS 207 SUPPLEMENTARY
EXERCISES 210 NOTES ON CHAPTER 6 213 II CONTINUOUS-PARAMETER RANDOM
FIELDS 215 7 CONTINUOUS-PARAMETER MARTINGALES 217 1 ONE-PARAMETER
MARTINGALES 217 1.1 FILTRATIONS AND STOPPING TIMES 218 1.2 ENTRANCE
TIMES 221 1.3 SMARTINGALES AND INEQUALITIES 222 1.4 REGULARITY 223 1.5
MEASURABILITY OF ENTRANCE TIMES 226 1.6 THE OPTIONAL STOPPING THEOREM
226 1.7 BROWNIAN MOTION 228 1.8 POISSON PROCESSES 230 2 MULTIPARAMETER
MARTINGALES 233 2.1 FILTRATIONS AND COMMUTATION 233 2.2 MARTINGALES AND
HISTORIES 234 2.3 CAIROLI S MAXIMAL INEQUALITIES 235 2.4 ANOTHER LOOK AT
THE BROWNIAN SHEET 236 3 ONE-PARAMETER STOCHASTIC INTEGRATION 239 3.1
UNBOUNDED VARIATION 239 3.2 QUADRATIC VARIATION 242 3.3 LOCAL
MARTINGALES 245 3.4 ELEMENTARY PROCESSES 246 3.5 SIMPLE PROCESSES 247
CONTENTS XI I 3.6 CONTINUOUS ADAPTED PROCESSES 248 3.7 TWO APPROXIMATION
THEOREMS 250 3.8 ITO S FORMULA 251 3.9 THE BURKHOLDER-DAVIS-GUNDY
INEQUALITY 253 4 STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS 255 4.1
STOCHASTIC INTEGRATION 256 4.2 HYPERBOLIC SPDES 257 4.3 EXISTENCE AND
UNIQUENESS 260 5 SUPPLEMENTARY EXERCISES 263 6 NOTES ON CHAPTER 7 266 8
CONSTRUCTING MARKOV PROCESSES 267 1 DISCRETE MARKOV CHAINS 267 1.1
PRELIMINARIES 267 1.2 THE STRONG MARKOV PROPERTY 272 1.3 KILLING AND
ABSORBING 272 1.4 TRANSITION OPERATORS 275 1.5 RESOLVENTS AND
A-POTENTIALS 277 1.6 DISTRIBUTION OF ENTRANCE TIMES 279 2 MARKOV
SEMIGROUPS 281 2.1 BOUNDED LINEAR OPERATORS 281 2.2 MARKOV SEMIGROUPS
AND RESOLVENTS 282 2.3 TRANSITION AND POTENTIAL DENSITIES 284 2.4 FELLER
SEMIGROUPS 287 3 MARKOV PROCESSES 288 3.1 INITIAL MEASURES 288 3.2
AUGMENTATION 290 3.3 SHIFTS 292 4 FELLER PROCESSES 293 4.1 FELLER
PROCESSES 294 4.2 THE STRONG MARKOV PROPERTY 298 4.3 LEVY PROCESSES 303
5 SUPPLEMENTARY EXERCISES 307 6 NOTES ON CHAPTER 8 311 9 GENERATION OF
MARKOV PROCESSES 313 1 GENERATION 313 1.1 EXISTENCE 314 1.2 THE
HILLE-YOSIDA THEOREM 315 1.3 THE MARTINGALE PROBLEM 317 2 EXPLICIT
COMPUTATIONS 320 2.1 BROWNIAN MOTION 320 2.2 ISOTROPIC STABLE PROCESSES
322 2.3 THE POISSON PROCESS 325 2.4 THE LINEAR UNIFORM MOTION 326 XII
CONTENTS 3 THE FEYNMAN-KAC FORMULA 326 3.1 THE FEYNMAN-KAC SEMIGROUP 326
3.2 THE DOOB-MEYER DECOMPOSITION 328 4 EXIT TIMES AND BROWNIAN MOTION
329 4.1 DIMENSION ONE 330 4.2 SOME FUNDAMENTAL LOCAL MARTINGALES 331 4.3
THE DISTRIBUTION OF EXIT TIMES 335 5 SUPPLEMENTARY EXERCISES 339 6 NOTES
ON CHAPTER 9 340 10 PROBABILISTIC POTENTIAL THEORY 343 1 RECURRENT LEVY
PROCESSES 344 1.1 SOJOURN TIMES 344 1.2 RECURRENCE OF THE ORIGIN 347 1.3
ESCAPE RATES 350 1.4 HITTING PROBABILITIES 353 2 HITTING PROBABILITIES
FOR FELLER PROCESSES 360 2.1 STRONGLY SYMMETRIC FELLER PROCESSES 360 2.2
BALAYAGE 362 2.3 HITTING PROBABILITIES AND CAPACITIES 367 2.4 PROOF OF
THEOREM 2.3.1 368 3 EXPLICIT COMPUTATIONS 373 3.1 BROWNIAN MOTION AND
CAPACITIES 373 3.2 STABLE DENSITIES AND SUBORDINATION 377 3.3
ASYMPTOTICS FOR STABLE DENSITIES 380 3.4 STABLE PROCESSES AND CAPACITIES
382 3.5 RELATION TO HAUSDORFF DIMENSION 385 4 SUPPLEMENTARY EXERCISES
386 5 NOTES ON CHAPTER 10 388 11 MULTIPARAMETER MARKOV PROCESSES 391 1
DEFINITIONS 391 1.1 PRELIMINARIES 392 1.2 COMMUTATION AND SEMIGROUPS 395
1.3 RESOLVENTS 397 1.4, STRONGLY SYMMETRIC FELLER PROCESSES 398 2
EXAMPLES 401 2.1 GENERAL NOTATION 401 2.2 PRODUCT FELLER PROCESSES 402
2.3 ADDITIVE LEVY PROCESSES 405 2.4 PRODUCT PROCESS 407 3 POTENTIAL
THEORY 408 3.1 THE MAIN RESULT 408 3.2 THREE TECHNICAL ESTIMATES 410 3.3
PROOF OF THEOREM 3.1.1: FIRST HALF 413 CONTENTS XIII T 3.4 PROOF OF
THEOREM 3.1.1: SECOND HALF 418 4 APPLICATIONS 419 4.1 ADDITIVE STABLE
PROCESSES 419 4.2 INTERSECTIONS OF INDEPENDENT PROCESSES 424 4.3
DVORETZKY-ERDOS-KAKUTANI THEOREMS 426 4.4 INTERSECTING AN ADDITIVE
STABLE PROCESS 428 4.5 THE RANGE OF A STABLE PROCESS 429 4.6 EXTENSION
TO ADDITIVE STABLE PROCESSES 433 4.7 STOCHASTIC CODIMENSION 435 5
A-REGULAR GAUSSIAN RANDOM FIELDS 438 5.1 STATIONARY GAUSSIAN PROCESSES
438 5.2 A-REGULAR GAUSSIAN FIELDS 441 5.3 PROOF OF THEOREM 5.2.1: FIRST
PART 443 5.4 PROOF OF THEOREM 5.2.1: SECOND PART 448 6 SUPPLEMENTARY
EXERCISES 450 7 NOTES ON CHAPTER 11 453 12 THE BROWNIAN SHEET AND
POTENTIAL THEORY 455 1 POLAR SETS FOR THE RANGE OF THE BROWNIAN SHEET
455 1.1 INTERSECTION PROBABILITIES 456 1.2 PROOF OF THEOREM 1.1.1: LOWER
BOUND 457 1.3 PROOF OF LEMMA 1.2.2 460 1.4 PROOF OF THEOREM 1.1.1: UPPER
BOUND 468 2 THE CODIMENSION OF THE LEVEL SETS 472 2.1 THE MAIN
CALCULATION 473 2.2 PROOF OF THEOREM 2.1.1: THE LOWER BOUND 474 2.3
PROOF OF THEOREM 2.1.1: THE UPPER BOUND 476 3 LOCAL TIMES AS FROSTMAN S
MEASURES 477 3.1 CONSTRUCTION 478 3.2 WARMUP: LINEAR BROWNIAN MOTION 480
3.3 A VARIANCE ESTIMATE 485 3.4 PROOF OF THEOREM 3.1.1: GENERAL CASE 488
4 SUPPLEMENTARY EXERCISES 491 5 NOTES ON CHAPTER 12 493 III APPENDICES
497 A KOLMOGOROV S CONSISTENCY THEOREM 499 B LAPLACE TRANSFORMS 501 1
UNIQUENESS AND CONVERGENCE THEOREMS 501 1.1 THE UNIQUENESS THEOREM 502
1.2 THE CONVERGENCE THEOREM 503 1.3 BERNSTEIN S THEOREM 505 XIV CONTENTS
2 A TAUBERIAN THEOREM 506 C HAUSDORFF DIMENSIONS AND MEASURES 511 1
PRELIMINARIES 511 1.1 DEFINITION 511 1.2 HAUSDORFF DIMENSION 515 2
FROSTMAN S THEOREMS 517 2.1 FROSTMAN S LEMMA 517 2.2 BESSEL-RIESZ
CAPACITIES 520 2.3 TAYLOR S THEOREM 523 3 NOTES ON APPENDIX C 525 D
ENERGY AND CAPACITY 527 1 PRELIMINARIES 527 1.1 GENERAL DEFINITIONS 527
1.2 PHYSICAL INTERPRETATIONS 530 2 CHOQUET CAPACITIES 533 2.1 MAXIMUM
PRINCIPLE AND NATURAL CAPACITIES 533 2.2 ABSOLUTELY CONTINUOUS
CAPACITIES 537 2.3 PROPER GAUGE FUNCTIONS AND BALAYAGE 539 3 NOTES ON
APPENDIX D 540 REFERENCES 543 NAME INDEX 565 SUBJECT INDEX 572
|
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| id | DE-604.BV014420583 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:02:31Z |
| institution | BVB |
| isbn | 0387954597 9781441930095 |
| language | English |
| lccn | 2002022927 |
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| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Khoshnevisan, Davar 1964- Verfasser (DE-588)124139795 aut Multiparameter processes an introduction to random fields Davar Khoshnevisan New York Springer 2002 XIX, 584 S. Ill. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Includes bibliographical references (p. [543]-565) and indexes Multi-parameter processes extend the existing one-parameter theory in an elegant way and have many applications to other fields in mathematics. This book on random fields is designed for a second graduate course in probability.. - Multi-parameter processes extend the existing one-parameter theory in an elegant way and have many applications to other fields in mathematics such as real analysis, functional analysis, group theory, and analytic number theory, to name a few. This book on the vast and rapidly developing subject of random fields is designed for a second graduate course in probability. Recent work on random fields has made it possible to make it an expository subject which interacts with several other areas in mathematics and has enough mathematical depth to be of use to pure as well as applied mathematicians of many backgrounds. Champs aléatoires Random fields Zufälliges Feld (DE-588)4191094-1 gnd rswk-swf Zufälliges Feld (DE-588)4191094-1 s DE-604 HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009858653&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Khoshnevisan, Davar 1964- Multiparameter processes an introduction to random fields Champs aléatoires Random fields Zufälliges Feld (DE-588)4191094-1 gnd |
| subject_GND | (DE-588)4191094-1 |
| title | Multiparameter processes an introduction to random fields |
| title_auth | Multiparameter processes an introduction to random fields |
| title_exact_search | Multiparameter processes an introduction to random fields |
| title_full | Multiparameter processes an introduction to random fields Davar Khoshnevisan |
| title_fullStr | Multiparameter processes an introduction to random fields Davar Khoshnevisan |
| title_full_unstemmed | Multiparameter processes an introduction to random fields Davar Khoshnevisan |
| title_short | Multiparameter processes |
| title_sort | multiparameter processes an introduction to random fields |
| title_sub | an introduction to random fields |
| topic | Champs aléatoires Random fields Zufälliges Feld (DE-588)4191094-1 gnd |
| topic_facet | Champs aléatoires Random fields Zufälliges Feld |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009858653&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT khoshnevisandavar multiparameterprocessesanintroductiontorandomfields |