Markov Processes, Gaussian Processes, and Local Times

A readable 2006 synthesis of three main areas in the modern theory of stochastic processes.

Michael B. Marcus (Author), Jay Rosen (Author)

9781107403758, Cambridge University Press

Paperback / softback, published 27 October 2011

632 pages
22.9 x 15.2 x 3.6 cm, 0.92 kg

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

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.

Subject Areas: Electronics & communications engineering [TJ], Probability & statistics [PBT], Calculus & mathematical analysis [PBK]