Markov Processes, Gaussian Processes, and Local Times

by Michael B. Marcus , Jay Rosen

• ISBN13: 9780521863001
• ISBN10: 0521863007
• Edition: 1st
• Format: Hardcover
• Copyright: 2006-07-24
• Publisher: Cambridge University Press

Summary

Written by two 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 for 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 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 page	1	(10)
1.1 Preliminaries	6	(5)
2 Brownian motion and Ray–Knight Theorems	11	(51)
2.1 Brownian motion	11	(8)
2.2 The Markov property	19	(9)
2.3 Standard augmentation	28	(3)
2.4 Brownian local time	31	(11)
2.5 Terminal times	42	(6)
2.6 The First Ray—Knight Theorem	48	(5)
2.7 The Second Ray—Knight Theorem	53	(3)
2.8 Ray's Theorem	56	(2)
2.9 Applications of the Ray—Knight Theorems	58	(3)
2.10 Notes and references	61	(1)
3 Markov processes and local times	62	(59)
3.1 The Markov property	62	(5)
3.2 The strong Markov property	67	(6)
3.3 Strongly symmetric Borel right processes	73	(5)
3.4 Continuous potential densities	78	(3)
3.5 Killing a process at an exponential time	81	(2)
3.6 Local times	83	(15)
3.7 Jointly continuous local times	98	(7)
3.8 Calculating uTo and uτ(λ)	105	(4)
3.9 The h-transform	109	(6)
3.10 Moment generating functions of local times	115	(4)
3.11 Notes and references	119	(2)
4 Constructing Markov processes	121	(68)
4.1 Feller processes	121	(14)
4.2 Lévy processes	135	(9)
4.3 Diffusions	144	(3)
4.4 Left limits and quasi left continuity	147	(5)
4.5 Killing at a terminal time	152	(10)
4.6 Continuous local times and potential densities	162	(2)
4.7 Constructing Ray semigroups and Ray processes	164	(14)
4.8 Local Borel right processes	178	(4)
4.9 Supermedian functions	182	(2)
4.10 Extension Theorem	184	(4)
4.11 Notes and references	188	(1)
5 Basic properties of Gaussian processes	189	(54)
5.1 Definitions and some simple properties	189	(9)
5.2 Moment generating functions	198	(5)
5.3 Zero-one laws and the oscillation function	203	(11)
5.4 Concentration inequalities	214	(13)
5.5 Comparison theorems	227	(8)
5.6 Processes with stationary increments	235	(5)
5.7 Notes and references	240	(3)
6 Continuity and boundedness of Gaussian processes	243	(39)
6.1 Sufficient conditions in terms of metric entropy	244	(6)
6.2 Necessary conditions in terms of metric entropy	250	(5)
6.3 Conditions in terms of majorizing measures	255	(15)
6.4 Simple criteria for continuity	270	(10)
6.5 Notes and references	280	(2)
7 Moduli of continuity for Gaussian processes	282	(80)
7.1 General results	282	(15)
7.2 Processes on Rn	297	(20)
7.3 Processes with spectral densities	317	(7)
7.4 Local moduli of associated processes	324	(12)
7.5 Gaussian lacunary series	336	(11)
7.6 Exact moduli of continuity	347	(9)
7.7 Squares of Gaussian processes	356	(5)
7.8 Notes and references	361	(1)
8 Isomorphism Theorems	362	(34)
8.1 Isomorphism theorems of Eisenbaum and Dynkin	362	(8)
8.2 The Generalized Second Ray-Knight Theorem	370	(10)
8.3 Combinatorial proofs	380	(10)
8.4 Additional proofs	390	(4)
8.5 Notes caul references	394	(2)
9 Sample path properties of local times	396	(60)
9.1 Bounded discontinuities	396	(7)
9.2 A necessary condition for unboundedness	403	(3)
9.3 Sufficient conditions for continuity	406	(4)
9.4 Continuity and boundedness of local times	410	(7)
9.5 Moduli of continuity	417	(20)
9.6 Stable mixtures	437	(4)
9.7 Local times for certain Markov chains	441	(6)
9.8 Rate of growth of unbounded local times	447	(7)
9.9 Notes and references	454	(2)
10 p-variation	456	(41)
10.1 Quadratic variation of Brownian motion	456	(1)
10.2 p-variation of Gaussian processes	457	(10)
10.3 Additional variational results for Gaussian processes	467	(12)
10.4 p-variation of local times	479	(3)
10.5 Additional variational results for local times	482	(13)
10.6 Notes and references	495	(2)
11 Most visited sites of symmetric stable processes	497	(33)
11.1 Preliminaries	497	(7)
11.2 Most visited sites of Brownian motion	504	(7)
11.3 Reproducing kernel Hilbert spaces	511	(5)
11.4 The Cameron-Martin Formula	516	(3)
11.5 Fractional Brownian motion	519	(4)
11.6 Most visited sites of symmetric stable processes	523	(3)
11.7 Notes and references	526	(4)
12 Local times of diffusions	530	(21)
12.1 Ray's Theorem for diffusions	530	(4)
12.2 Eisenbaum's version of Ray's Theorem	534	(3)
12.3 Ray's original theorem	537	(6)
12.4 Markov property of local times of diffusions	543	(6)
12.5 Local limit laws for h-transforms of diffusions	549	(1)
12.6 Notes and references	550	(1)
13 Associated Gaussian processes	551	(29)
13.1 Associated Gaussian processes	552	(8)
13.2 Infinitely divisible squares	560	(10)
13.3 Infinitely divisible squares and associated processes	570	(8)
13.4 Additional results about M-matrices	578	(1)
13.5 Notes and references	579	(1)
14 Appendix	580	(23)
14.1 Kolmogorov's Theorem for path continuity	580	(1)
14.2 Bessel processes	581	(2)
14.3 Analytic sets and the Projection Theorem	583	(4)
14.4 Hille-Yosida Theorem	587	(2)
14.5 Stone–Weierstrass Theorems	589	(1)
14.6 Independent random variables	590	(4)
14.7 Regularly varying functions	594	(2)
14.8 Some useful inequalities	596	(2)
14.9 Some linear algebra	598	(5)
References	603	(8)
Index of notation	611	(2)
Author index	613	(3)
Subject index	616