Markov Processes, Gaussian Processes, and Local Times: 100 (Cambridge Studies in Advanced Mathematics, Series Number 100)
Cambridge University Press, 7/24/2006
EAN 9780521863001, ISBN10: 0521863007
Hardcover, 632 pages, 22.9 x 15.2 x 4.1 cm
Language: English
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
2. Brownian motion and Ray-Knight theorems
3. Markov processes and local times
4. Constructing Markov processes
5. Basic properties of Gaussian processes
6. Continuity and boundedness
7. Moduli of continuity
8. Isomorphism theorems
9. Sample path properties of local times
10. p-Variation
11. Most visited site
12. Local times of diffusions
13. Associated Gaussian processes
Appendices
A. Kolmogorov's theorem for path continuity
B. Bessel processes
C. Analytic sets and the projection theorem
D. Hille-Yosida theorem
E. Stone-Weierstrass theorems
F. Independent random variables
G. Regularly varying functions
H. Some useful inequalities
I. Some linear algebra
References
Index.
'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