Random Fields and Geometry

by Adler, Robert J.; Taylor, Jonathan E.

• ISBN13: 9780387481128
• ISBN10: 0387481125
• Format: Hardcover
• Copyright: 2007-07-01
• Publisher: Springer Nature

Summary

This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities."Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.

Table of Contents

Preface	p. V
Gaussian Processes
Gaussian Fields	p. 7
Random Fields	p. 7
Gaussian Variables and Fields	p. 8
Boundedness and Continuity	p. 11
Examples	p. 20
Fields on R[superscript N]	p. 20
Differentiability on R[superscript N]	p. 22
The Brownian Family of Processes	p. 24
Generalized Fields	p. 30
Set-Indexed Processes	p. 36
Non-Gaussian Processes	p. 40
Majorizing Measures	p. 41
Gaussian Inequalities	p. 49
Borell-TIS Inequality	p. 49
Comparison Inequalities	p. 57
Orthogonal Expansions	p. 65
The General Theory	p. 66
The Karhunen-Loeve Expansion	p. 70
Excursion Probabilities	p. 75
Entropy Bounds	p. 76
Processes with a Unique Point of Maximal Variance	p. 86
Examples	p. 89
Extensions	p. 93
The Double-Sum Method	p. 95
Local Maxima and Excursion Probabilities	p. 96
Stationary Fields	p. 101
Basic Stationarity	p. 101
Stochastic Integration	p. 103
Moving Averages	p. 105
Spectral Representations on R[superscript N]	p. 109
Spectral Moments	p. 112
Constant Variance	p. 114
Isotropy	p. 115
Stationarity over Groups	p. 119
Geometry
Integral Geometry	p. 127
Basic Integral Geometry	p. 127
Excursion Sets Again	p. 134
Intrinsic Volumes	p. 141
Differential Geometry	p. 149
Manifolds	p. 149
Tensor Calculus	p. 154
Riemannian Manifolds	p. 160
Integration on Manifolds	p. 166
Curvature	p. 171
Intrinsic Volumes for Riemannian Manifolds	p. 175
A Euclidean Example	p. 176
Piecewise Smooth Manifolds	p. 183
Whitney Stratified Spaces	p. 184
Locally Convex Spaces	p. 188
Cone Spaces	p. 190
Critical Point Theory	p. 193
Critical Points	p. 193
The Normal Morse Index	p. 195
The Index	p. 195
Generalized Tangent Spaces and Tame Manifolds	p. 196
Regular Stratified Manifolds	p. 198
The Index on Intersections of Sets	p. 198
Morse's Theorem for Stratified Spaces	p. 206
Morse Functions	p. 206
Morse's Theorem	p. 207
The Euclidean Case	p. 210
Volume of Tubes	p. 213
The Volume-of-Tubes Problem	p. 215
Volume of Tubes and Gaussian Processes	p. 216
Local Geometry of Tube(M, [rho])	p. 219
Basic Structure of Tubes	p. 220
Stratifying the Tube	p. 222
Computing the Volume of a Tube	p. 223
First Steps	p. 223
An Intermediate Computation	p. 224
Subsets of R[superscript l]	p. 225
Subsets of Spheres	p. 230
Weyl's Tube Formula	p. 231
Volume of Tubes and Gaussian Processes, Continued	p. 242
Intrinsic Volumes for Whitney Stratified Spaces	p. 244
Alternative Representation of the Curvature Measures	p. 249
Breakdown of Weyl's Tube Formula	p. 249
Generalized Lipschitz-Killing Curvature Measures	p. 250
The Generalized Curvature Measures	p. 251
Surface Measure on the Boundary of a Tube	p. 252
Series Expansions for the Gaussian Measure of Tubes	p. 254
The Geometry of Random Fields
Random Fields on Euclidean Spaces	p. 263
Rice's Formula	p. 263
An Expectation Metatheorem	p. 266
Suitable Regularity and Morse Functions	p. 280
An Alternate Proof of the Metatheorem	p. 283
Higher Moments	p. 284
Preliminary Gaussian Computations	p. 286
The Mean Euler Characteristic	p. 289
Mean Intrinsic Volumes	p. 298
On the Importance of Stationarity	p. 299
Random Fields on Manifolds	p. 301
The Metatheorem on Manifolds	p. 301
Riemannian Structure Induced by Gaussian Fields	p. 305
Connections and Curvatures	p. 306
Some Covariances	p. 308
Gaussian Fields on R[superscript N]	p. 310
Another Gaussian Computation	p. 312
The Mean Euler Characteristic	p. 315
Manifolds without Boundary	p. 315
Manifolds with Boundary	p. 317
Examples	p. 323
Chern-Gauss-Bonnet Theorem	p. 327
Mean Intrinsic Volumes	p. 331
Crofton's Formula	p. 332
Mean Intrinsic Volumes: The Isotropic Case	p. 333
A Gaussian Crofton Formula	p. 334
Mean Intrinsic Volumes: The General Case	p. 342
Two Gaussian Lemmas	p. 343
Excursion Probabilities for Smooth Fields	p. 349
On Global Suprema	p. 351
A First Representation	p. 352
The Problem with the First Representation	p. 354
A Second Representation	p. 354
Random Fields	p. 360
Suprema and Euler Characteristics	p. 362
Some Fine Tuning	p. 365
Gaussian Fields with Constant Variance	p. 368
Examples	p. 372
Stationary Processes on [0, T]	p. 372
Isotropic Fields with Monotone Covariance	p. 374
A Geometric Approach	p. 376
The Cosine Field	p. 382
Non-Gaussian Geometry	p. 387
A Plan of Action	p. 389
A Representation for Mean Intrinsic Volumes	p. 391
Proof of the Representation	p. 392
Poincare's Limit	p. 398
Kinematic Fundamental Formulas	p. 400
The KFF on R[superscript n]	p. 401
The KFF on S[subscript lambda] (R[superscript n])	p. 402
A Model Process on the l-Sphere	p. 402
The Process	p. 403
Mean Curvatures for the Model Process	p. 404
The Canonical Gaussian Field on the l-Sphere	p. 410
Mean Curvatures for Excursion Sets	p. 411
Implications for More General Fields	p. 415
Warped Products of Riemannian Manifolds	p. 416
Warped Products	p. 417
A Second Fundamental Form	p. 419
Non-Gaussian Mean Intrinsic Volumes	p. 421
Examples	p. 425
The Gaussian Case	p. 426
The [chi superscript 2] Case	p. 427
The F Case	p. 430
References	p. 435
Notation Index	p. 443
Subject Index	p. 445
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