Markov processes, gaussian processes and local times

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.

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

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