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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000010079714 | QA274.7 M35 2006 | Open Access Book | Book | Searching... |
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Summary
Summary
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.
Table of Contents
1 Introduction | p. 1 |
1.1 Preliminaries | p. 6 |
2 Brownian motion and Ray-Knight Theorems | p. 11 |
2.1 Brownian motion | p. 11 |
2.2 The Markov property | p. 19 |
2.3 Standard augmentation | p. 28 |
2.4 Brownian local time | p. 31 |
2.5 Terminal times | p. 42 |
2.6 The First Ray-Knight Theorem | p. 48 |
2.7 The Second Ray-Knight Theorem | p. 53 |
2.8 Ray's Theorem | p. 56 |
2.9 Applications of the Ray-Knight Theorems | p. 58 |
2.10 Notes and references | p. 61 |
3 Markov processes and local times | p. 62 |
3.1 The Markov property | p. 62 |
3.2 The strong Markov property | p. 67 |
3.3 Strongly symmetric Borel right processes | p. 73 |
3.4 Continuous potential densities | p. 78 |
3.5 Killing a process at an exponential time | p. 81 |
3.6 Local times | p. 83 |
3.7 Jointly continuous local times | p. 98 |
3.8 Calculating u[subscript T subscript 0] and u[subscript tau (lambda)] | p. 105 |
3.9 The h-transform | p. 109 |
3.10 Moment generating functions of local times | p. 115 |
3.11 Notes and references | p. 119 |
4 Constructing Markov processes | p. 121 |
4.1 Feller processes | p. 121 |
4.2 Levy processes | p. 135 |
4.3 Diffusions | p. 144 |
4.4 Left limits and quasi left continuity | p. 147 |
4.5 Killing at a terminal time | p. 152 |
4.6 Continuous local times and potential densities | p. 162 |
4.7 Constructing Ray semigroups and Ray processes | p. 164 |
4.8 Local Borel right processes | p. 178 |
4.9 Supermedian functions | p. 182 |
4.10 Extension Theorem | p. 184 |
4.11 Notes and references | p. 188 |
5 Basic properties of Gaussian processes | p. 189 |
5.1 Definitions and some simple properties | p. 189 |
5.2 Moment generating functions | p. 198 |
5.3 Zero-one laws and the oscillation function | p. 203 |
5.4 Concentration inequalities | p. 214 |
5.5 Comparison theorems | p. 227 |
5.6 Processes with stationary increments | p. 235 |
5.7 Notes and references | p. 240 |
6 Continuity and boundedness of Gaussian processes | p. 243 |
6.1 Sufficient conditions in terms of metric entropy | p. 244 |
6.2 Necessary conditions in terms of metric entropy | p. 250 |
6.3 Conditions in terms of majorizing measures | p. 255 |
6.4 Simple criteria for continuity | p. 270 |
6.5 Notes and references | p. 280 |
7 Moduli of continuity for Gaussian processes | p. 282 |
7.1 General results | p. 282 |
7.2 Processes on R[superscript n] | p. 297 |
7.3 Processes with spectral densities | p. 317 |
7.4 Local moduli of associated processes | p. 324 |
7.5 Gaussian lacunary series | p. 336 |
7.6 Exact moduli of continuity | p. 347 |
7.7 Squares of Gaussian processes | p. 356 |
7.8 Notes and references | p. 361 |
8 Isomorphism Theorems | p. 362 |
8.1 Isomorphism theorems of Eisenbaum and Dynkin | p. 362 |
8.2 The Generalized Second Ray-Knight Theorem | p. 370 |
8.3 Combinatorial proofs | p. 380 |
8.4 Additional proofs | p. 390 |
8.5 Notes and references | p. 394 |
9 Sample path properties of local times | p. 396 |
9.1 Bounded discontinuities | p. 396 |
9.2 A necessary condition for unboundedness | p. 403 |
9.3 Sufficient conditions for continuity | p. 406 |
9.4 Continuity and boundedness of local times | p. 410 |
9.5 Moduli of continuity | p. 417 |
9.6 Stable mixtures | p. 437 |
9.7 Local times for certain Markov chains | p. 441 |
9.8 Rate of growth of unbounded local times | p. 447 |
9.9 Notes and references | p. 454 |
10 p-variation | p. 456 |
10.1 Quadratic variation of Brownian motion | p. 456 |
10.2 p-variation of Gaussian processes | p. 457 |
10.3 Additional variational results for Gaussian processes | p. 467 |
10.4 p-variation of local times | p. 479 |
10.5 Additional variational results for local times | p. 482 |
10.6 Notes and references | p. 495 |
11 Most visited sites of symmetric stable processes | p. 497 |
11.1 Preliminaries | p. 497 |
11.2 Most visited sites of Brownian motion | p. 504 |
11.3 Reproducing kernel Hilbert spaces | p. 511 |
11.4 The Cameron-Martin Formula | p. 516 |
11.5 Fractional Brownian motion | p. 519 |
11.6 Most visited sites of symmetric stable processes | p. 523 |
11.7 Notes and references | p. 526 |
12 Local times of diffusions | p. 530 |
12.1 Ray's Theorem for diffusions | p. 530 |
12.2 Eisenbaum's version of Ray's Theorem | p. 534 |
12.3 Ray's original theorem | p. 537 |
12.4 Markov property of local times of diffusions | p. 543 |
12.5 Local limit laws for h-transforms of diffusions | p. 549 |
12.6 Notes and references | p. 550 |
13 Associated Gaussian processes | p. 551 |
13.1 Associated Gaussian processes | p. 552 |
13.2 Infinitely divisible squares | p. 560 |
13.3 Infinitely divisible squares and associated processes | p. 570 |
13.4 Additional results about M-matrices | p. 578 |
13.5 Notes and references | p. 579 |
14 Appendix | p. 580 |
14.1 Kolmogorov's Theorem for path continuity | p. 580 |
14.2 Bessel processes | p. 581 |
14.3 Analytic sets and the Projection Theorem | p. 583 |
14.4 Hille-Yosida Theorem | p. 587 |
14.5 Stone-Weierstrass Theorems | p. 589 |
14.6 Independent random variables | p. 590 |
14.7 Regularly varying functions | p. 594 |
14.8 Some useful inequalities | p. 596 |
14.9 Some linear algebra | p. 598 |
References | p. 603 |
Index of notation | p. 611 |
Author index | p. 613 |
Subject index | p. 616 |