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.
Starts with streamlined 'mini-courses that give easy entry to important areas in stochastic process theory
Material on the isomorphism theorems has never before appeared in book form
Authors are acknowledged experts
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. Kolmogorovs 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.
