Multiparameter Processes : An Introduction to Random Fields

An Introduction to Random Fields

By: Davar Khoshnevisan

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Hardcover | 1 June 2010

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608 Pages

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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.

Industry Reviews

From the reviews:

"This book presents an updated and comprehensible account on the theory of multiparameter stochastic processes. ... This book is certainly a basic reference for subjects like multiparameter martingales and potential theory for the Brownian sheet and several Markov processes. It can be useful for researchers who would like to learn the basis and recent developments of these subjects. The book is self-contained ... . In spite of the technical character of the subject, reading this book is a very pleasant and enriching experience." (David Nualart, Mathematical Reviews, Issue 2004 a)

"This book aims to construct a general framework for the analysis of a large class of random fields, also known as multiparameter processes. A great part of one-parameter theory is also included, with the goal to keep the book self-contained. ... The book contains a lot of supplementary exercises, extended theoretical appendices and is useful both for highly qualified specialists and for advanced graduate students." (Yu. S. Mishura, Zentralblatt Math, Vol. 1005, 2003)

"The present book is the first one presenting a general treatment of random fields. ... The book can be recommended not only to probabilists but to anyone interested in the applications of probability within other areas of mathematics, particularly in analysis." (P. Revesz, Internationale Mathematische Nachrichten, Vol. 57 (192), 2003)

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Stochastics Probability & Statistics Calculus & Mathematical Analysis Functional Analysis & Transforms

Preface	p. v
List of Figures	p. xv
General Notation	p. xvii
Discrete-Parameter Random Fields	p. 1
Discrete-Parameter Martingales	p. 3
One-Parameter Martingales	p. 4
Definitions	p. 4
The Optional Stopping Theorem	p. 7
A Weak (1,1) Inequality	p. 8
A Strong (p,p) Inequality	p. 9
The Case p = 1	p. 9
Upcrossing Inequalities	p. 10
The Martingale Convergence Theorem	p. 12
Orthomartingales	p. 15
Definitions and Examples	p. 16
Embedded Submartingales	p. 18
Cairoli's Strong (p,p) Inequality	p. 19
Another Maximal Inequality	p. 20
A Weak Maximal Inequality	p. 22
Orthohistories	p. 22
Convergence Notions	p. 24
Topological Convergence	p. 26
Reversed Orthomartingales	p. 30
Martingales	p. 31
Definitions	p. 31
Marginal Filtrations	p. 31
A Counterexample	p. 33
Commutation	p. 35
Martingales	p. 37
Conditional Independence	p. 38
Supplementary Exercises	p. 40
Notes on Chapter 1	p. 44
Two Applications in Analysis	p. 47
Haar Systems	p. 47
The 1-Dimensional Haar System	p. 48
The N-Dimensional Haar System	p. 51
Differentiation	p. 54
Lebesgue's Differentiation Theorem	p. 54
A Uniform Differentiation Theorem	p. 58
Supplementary Exercises	p. 61
Notes on Chapter 2	p. 63
Random Walks	p. 65
One-Parameter Random Walks	p. 66
Transition Operators	p. 66
The Strong Markov Property	p. 69
Recurrence	p. 70
Classification of Recurrence	p. 72
Transience	p. 74
Recurrence of Possible Points	p. 75
Recurrence-Transience Dichotomy	p. 78
Intersection Probabilities	p. 80
Intersections of Two Walks	p. 80
An Estimate for Two Walks	p. 85
Intersections of Several Walks	p. 86
An Estimate for N Walks	p. 89
The Simple Random Walk	p. 89
Recurrence	p. 90
Intersections of Two Simple Walks	p. 91
Three Simple Walks	p. 93
Several Simple Walks	p. 97
Supplementary Exercises	p. 99
Notes on Chapter 3	p. 103
Multiparameter Walks	p. 105
The Strong Law of Large Numbers	p. 106
Definitions	p. 106
Commutation	p. 107
A Reversed Orthomartingale	p. 109
Smythe's Law of Large Numbers	p. 110
The Law of the Iterated Logarithm	p. 112
The One-Parameter Gaussian Case	p. 113
The General LIL	p. 116
Summability	p. 117
Dirichlet's Divisor Lemma	p. 118
Truncation	p. 119
Bernstein's Inequality	p. 121
Maximal Inequalities	p. 123
A Number-Theoretic Estimate	p. 125
Proof of the LIL: The Upper Bound	p. 127
A Moderate Deviations Estimate	p. 128
Proof of the LIL: The Lower Bound	p. 130
Supplementary Exercises	p. 132
Notes on Chapter 4	p. 135
Gaussian Random Variables	p. 137
The Basic Construction	p. 137
Gaussian Random Vectors	p. 137
Gaussian Processes	p. 140
White Noise	p. 142
The Isonormal Process	p. 144
The Brownian Sheet	p. 147
Regularity Theory	p. 148
Totally Bounded Pseudometric Spaces	p. 149
Modifications and Separability	p. 153
Kolmogorov's Continuity Theorem	p. 158
Chaining	p. 160
Holder-Continuous Modifications	p. 165
The Entropy Integral	p. 167
Dudley's Theorem	p. 170
The Standard Brownian Sheet	p. 172
Entropy Estimate	p. 172
Modulus of Continuity	p. 173
Supplementary Exercises	p. 175
Notes on Chapter 5	p. 178
Limit Theorems	p. 181
Random Variables	p. 181
Definitions	p. 182
Distributions	p. 183
Uniqueness	p. 184
Weak Convergence	p. 185
The Portmanteau Theorem	p. 186
The Continuous Mapping Theorem	p. 188
Weak Convergence in Euclidean Space	p. 188
Tightness	p. 189
Prohorov's Theorem	p. 190
The Space C	p. 193
Uniform Continuity	p. 193
Finite-Dimensional Distributions	p. 195
Weak Convergence in C	p. 196
Continuous Functionals	p. 199
A Sufficient Condition for Pretightness	p. 200
Invariance Principles	p. 201
Preliminaries	p. 202
Finite-Dimensional Distributions	p. 204
Pretightness	p. 207
Supplementary Exercises	p. 210
Notes on Chapter 6	p. 213
Continuous-Parameter Random Fields	p. 215
Continuous-Parameter Martingales	p. 217
One-Parameter Martingales	p. 217
Filtrations and Stopping Times	p. 218
Entrance Times	p. 221
Smartingales and Inequalities	p. 222
Regularity	p. 223
Measurability of Entrance Times	p. 226
The Optional Stopping Theorem	p. 226
Brownian Motion	p. 228
Poisson Processes	p. 230
Multiparameter Martingales	p. 233
Filtrations and Commutation	p. 233
Martingales and Histories	p. 234
Cairoli's Maximal Inequalities	p. 235
Another Look at the Brownian Sheet	p. 236
One-Parameter Stochastic Integration	p. 239
Unbounded Variation	p. 239
Quadratic Variation	p. 242
Local Martingales	p. 245
Elementary Processes	p. 246
Simple Processes	p. 247
Continuous Adapted Processes	p. 248
Two Approximation Theorems	p. 250
Itô's Formula	p. 251
The Burkholder-Davis-Gundy Inequality	p. 253
Stochastic Partial Differential Equations	p. 255
Stochastic Integration	p. 256
Hyperbolic SPDEs	p. 257
Existence and Uniqueness	p. 260
Supplementary Exercises	p. 263
Notes on Chapter 7	p. 266
Constructing Markov Processes	p. 267
Discrete Markov Chains	p. 267
Preliminaries	p. 267
The Strong Markov Property	p. 272
Killing and Absorbing	p. 272
Transition Operators	p. 275
Resolvents and -Potentials	p. 277
Distribution of Entrance Times	p. 279
Markov Semigroups	p. 281
Bounded Linear Operators	p. 281
Markov Semigroups and Resolvents	p. 282
Transition and Potential Densities	p. 284
Feller Semigroups	p. 287
Markov Processes	p. 288
Initial Measures	p. 288
Augmentation	p. 290
Shifts	p. 292
Feller Processes	p. 293
Feller Processes	p. 294
The Strong Markov Property	p. 298
Levy Processes	p. 303
Supplementary Exercises	p. 307
Notes on Chapter 8	p. 311
Generation of Markov Processes	p. 313
Generation	p. 313
Existence	p. 314
The Hille-Yosida Theorem	p. 315
The Martingale Problem	p. 317
Explicit Computations	p. 320
Brownian Motion	p. 320
Isotropic Stable Processes	p. 322
The Poisson Process	p. 325
The Linear Uniform Motion	p. 326
The Feynman-Kac Formula	p. 326
The Feynman-Kac Semigroup	p. 326
The Doob-Meyer Decomposition	p. 328
Exit Times and Brownian Motion	p. 329
Dimension One	p. 330
Some Fundamental Local Martingales	p. 331
The Distribution of Exit Times	p. 335
Supplementary Exercises	p. 339
Notes on Chapter 9	p. 340
Probabilistic Potential Theory	p. 343
Recurrent Levy Processes	p. 344
Sojourn Times	p. 344
Recurrence of the Origin	p. 347
Escape Rates	p. 350
Hitting Probabilities	p. 353
Hitting Probabilities for Feller Processes	p. 360
Strongly Symmetric Feller Processes	p. 360
Balayage	p. 362
Hitting Probabilities and Capacities	p. 367
Proof of Theorem 2.3.1	p. 368
Explicit Computations	p. 373
Brownian Motion and Capacities	p. 373
Stable Densities and Subordination	p. 377
Asymptotics for Stable Densities	p. 380
Stable Processes and Capacities	p. 382
Relation to Hausdorff Dimension	p. 385
Supplementary Exercises	p. 386
Notes on Chapter 10	p. 388
Multiparameter Markov Processes	p. 391
Definitions	p. 391
Preliminaries	p. 392
Commutation and Semigroups	p. 395
Resolvents	p. 397
Strongly Symmetric Feller Processes	p. 398
Examples	p. 401
General Notation	p. 401
Product Feller Processes	p. 402
Additive Levy Processes	p. 405
Product Process	p. 407
Potential Theory	p. 408
The Main Result	p. 408
Three Technical Estimates	p. 410
Proof of Theorem 3.1.1: First Half	p. 413
Proof of Theorem 3.1.1: Second Half	p. 418
Applications	p. 419
Additive Stable Processes	p. 419
Intersections of Independent Processes	p. 424
Dvoretzky-Erdös-Kakutani Theorems	p. 426
Intersecting an Additive Stable Process	p. 428
The Range of a Stable Process	p. 429
Extension to Additive Stable Processes	p. 433
Stochastic Codimension	p. 435
a-Regular Gaussian Random Fields	p. 438
Stationary Gaussian Processes	p. 438
a-Regular Gaussian Fields	p. 441
Proof of Theorem 5.2.1: First Part	p. 443
Proof of Theorem 5.2.1: Second Part	p. 448
Supplementary Exercises	p. 450
Notes on Chapter 11	p. 453
The Brownian Sheet and Potential Theory	p. 455
Polar Sets for the Range of the Brownian Sheet	p. 455
Intersection Probabilities	p. 456
Proof of Theorem 1.1.1: Lower Bound	p. 457
Proof of Lemma 1.2.2	p. 460
Proof of Theorem 1.1.1: Upper Bound	p. 468
The Codimension of the Level Sets	p. 472
The Main Calculation	p. 473
Proof of Theorem 2.1.1: The Lower Bound	p. 474
Proof of Theorem 2.1.1: The Upper Bound	p. 476
Local Times as Frostman's Measures	p. 477
Construction	p. 478
Warmup: Linear Brownian Motion	p. 480
A Variance Estimate	p. 485
Proof of Theorem 3.1.1: General Case	p. 488
Supplementary Exercises	p. 491
Notes on Chapter 12	p. 493
Appendices	p. 497
Kolmogorov's Consistency Theorem	p. 499
Laplace Transforms	p. 501
Uniqueness and Convergence Theorems	p. 501
The Uniqueness Theorem	p. 502
The Convergence Theorem	p. 503
Bernstein's Theorem	p. 505
A Tauberian Theorem	p. 506
Hausdorff Dimensions and Measures	p. 511
Preliminaries	p. 511
Definition	p. 511
Hausdorff Dimension	p. 515
Frostman's Theorems	p. 517
Frostman's Lemma	p. 517
Bessel-Riesz Capacities	p. 520
Taylor's Theorem	p. 523
Notes on Appendix C	p. 525
Energy and Capacity	p. 527
Preliminaries	p. 527
General Definitions	p. 527
Physical Interpretations	p. 530
Choquet Capacities	p. 533
Maximum Principle and Natural Capacities	p. 533
Absolutely Continuous Capacities	p. 537
Proper Gauge Functions and Balayage	p. 539
Notes on Appendix D	p. 540
References	p. 543
Name Index	p. 565
Subject Index	p. 572
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Multiparameter Processes : An Introduction to Random Fields

An Introduction to Random Fields

At a Glance

Hardcover

Industry Reviews

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