Markov Processes, Gaussian Processes, and Local Times

By: Michael B. Marcus, Jay Rosen, B. Bollobas (Editor), W. Fulton (Editor), A. Katok (Editor)

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Hardcover | 24 July 2006

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630 Pages

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22.86 x 15.24 x 3.96

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This book was first published in 2006. Written by two of the foremost researchers in the field, this book studies the local times of Markov processes by employing isomorphism theorems that relate them to certain associated Gaussian processes. It builds to this material through self-contained but harmonized 'mini-courses' on the relevant ingredients, which assume only knowledge of measure-theoretic probability. The streamlined selection of topics creates an easy entrance for students and experts in related fields. The book starts by developing the fundamentals of Markov process theory and then of Gaussian process theory, including sample path properties. It then proceeds to more advanced results, bringing the reader to the heart of contemporary research. It presents the remarkable isomorphism theorems of Dynkin and Eisenbaum and then shows how they can be applied to obtain new properties of Markov processes by using well-established techniques in Gaussian process theory. This original, readable book will appeal to both researchers and advanced graduate students.

Industry Reviews

Review of the hardback: 'This is a masterly written text which should be accessible to advanced graduate students and non-specialists. For the researcher interested in Gaussian processes or local times it will become an indispensable standard resource.' Zentralblatt MATH Review of the hardback: 'The authors make every effort to make the material accessible, and the proofs presented are often much easier than the original ones. The material is well organized and well presented.' Journal of the American Statistical Association

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Probability & Statistics Applied Mathematics Stochastics

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Paperback

Published: 27th October 2011

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Introduction	p. 1
Preliminaries	p. 6
Brownian motion and Ray-Knight Theorems	p. 11
Brownian motion	p. 11
The Markov property	p. 19
Standard augmentation	p. 28
Brownian local time	p. 31
Terminal times	p. 42
The First Ray-Knight Theorem	p. 48
The Second Ray-Knight Theorem	p. 53
Ray's Theorem	p. 56
Applications of the Ray-Knight Theorems	p. 58
Notes and references	p. 61
Markov processes and local times	p. 62
The Markov property	p. 62
The strong Markov property	p. 67
Strongly symmetric Borel right processes	p. 73
Continuous potential densities	p. 78
Killing a process at an exponential time	p. 81
Local times	p. 83
Jointly continuous local times	p. 98
Calculating u[subscript T subscript 0] and u[subscript tau (lambda)]	p. 105
The h-transform	p. 109
Moment generating functions of local times	p. 115
Notes and references	p. 119
Constructing Markov processes	p. 121
Feller processes	p. 121
Levy processes	p. 135
Diffusions	p. 144
Left limits and quasi left continuity	p. 147
Killing at a terminal time	p. 152
Continuous local times and potential densities	p. 162
Constructing Ray semigroups and Ray processes	p. 164
Local Borel right processes	p. 178
Supermedian functions	p. 182
Extension Theorem	p. 184
Notes and references	p. 188
Basic properties of Gaussian processes	p. 189
Definitions and some simple properties	p. 189
Moment generating functions	p. 198
Zero-one laws and the oscillation function	p. 203
Concentration inequalities	p. 214
Comparison theorems	p. 227
Processes with stationary increments	p. 235
Notes and references	p. 240
Continuity and boundedness of Gaussian processes	p. 243
Sufficient conditions in terms of metric entropy	p. 244
Necessary conditions in terms of metric entropy	p. 250
Conditions in terms of majorizing measures	p. 255
Simple criteria for continuity	p. 270
Notes and references	p. 280
Moduli of continuity for Gaussian processes	p. 282
General results	p. 282
Processes on R[superscript n]	p. 297
Processes with spectral densities	p. 317
Local moduli of associated processes	p. 324
Gaussian lacunary series	p. 336
Exact moduli of continuity	p. 347
Squares of Gaussian processes	p. 356
Notes and references	p. 361
Isomorphism Theorems	p. 362
Isomorphism theorems of Eisenbaum and Dynkin	p. 362
The Generalized Second Ray-Knight Theorem	p. 370
Combinatorial proofs	p. 380
Additional proofs	p. 390
Notes and references	p. 394
Sample path properties of local times	p. 396
Bounded discontinuities	p. 396
A necessary condition for unboundedness	p. 403
Sufficient conditions for continuity	p. 406
Continuity and boundedness of local times	p. 410
Moduli of continuity	p. 417
Stable mixtures	p. 437
Local times for certain Markov chains	p. 441
Rate of growth of unbounded local times	p. 447
Notes and references	p. 454
p-variation	p. 456
Quadratic variation of Brownian motion	p. 456
p-variation of Gaussian processes	p. 457
Additional variational results for Gaussian processes	p. 467
p-variation of local times	p. 479
Additional variational results for local times	p. 482
Notes and references	p. 495
Most visited sites of symmetric stable processes	p. 497
Preliminaries	p. 497
Most visited sites of Brownian motion	p. 504
Reproducing kernel Hilbert spaces	p. 511
The Cameron-Martin Formula	p. 516
Fractional Brownian motion	p. 519
Most visited sites of symmetric stable processes	p. 523
Notes and references	p. 526
Local times of diffusions	p. 530
Ray's Theorem for diffusions	p. 530
Eisenbaum's version of Ray's Theorem	p. 534
Ray's original theorem	p. 537
Markov property of local times of diffusions	p. 543
Local limit laws for h-transforms of diffusions	p. 549
Notes and references	p. 550
Associated Gaussian processes	p. 551
Associated Gaussian processes	p. 552
Infinitely divisible squares	p. 560
Infinitely divisible squares and associated processes	p. 570
Additional results about M-matrices	p. 578
Notes and references	p. 579
Appendix	p. 580
Kolmogorov's Theorem for path continuity	p. 580
Bessel processes	p. 581
Analytic sets and the Projection Theorem	p. 583
Hille-Yosida Theorem	p. 587
Stone-Weierstrass Theorems	p. 589
Independent random variables	p. 590
Regularly varying functions	p. 594
Some useful inequalities	p. 596
Some linear algebra	p. 598
References	p. 603
Index of notation	p. 611
Author index	p. 613
Subject index	p. 616
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Markov Processes, Gaussian Processes, and Local Times

At a Glance

Hardcover

Industry Reviews

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How to return your order

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You Can Find This Book In

Other Editions and Formats

Paperback

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