Markov processes, Gaussian processes, and local times /
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Author / Creator: | Marcus, Michael B. |
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Imprint: | Cambridge ; New York : Cambridge University Press, 2006. |
Description: | x, 620 p. ; 24 cm. |
Language: | English |
Series: | Cambridge studies in advanced mathematics ; 100 |
Subject: | |
Format: | Print Book |
URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/6117539 |
Table of Contents:
- 1. Introduction
- 1.1. Preliminaries
- 2. Brownian motion and Ray-Knight Theorems
- 2.1. Brownian motion
- 2.2. The Markov property
- 2.3. Standard augmentation
- 2.4. Brownian local time
- 2.5. Terminal times
- 2.6. The First Ray-Knight Theorem
- 2.7. The Second Ray-Knight Theorem
- 2.8. Ray's Theorem
- 2.9. Applications of the Ray-Knight Theorems
- 2.10. Notes and references
- 3. Markov processes and local times
- 3.1. The Markov property
- 3.2. The strong Markov property
- 3.3. Strongly symmetric Borel right processes
- 3.4. Continuous potential densities
- 3.5. Killing a process at an exponential time
- 3.6. Local times
- 3.7. Jointly continuous local times
- 3.8. Calculating u[subscript T subscript 0] and u[subscript tau (lambda)]
- 3.9. The h-transform
- 3.10. Moment generating functions of local times
- 3.11. Notes and references
- 4. Constructing Markov processes
- 4.1. Feller processes
- 4.2. Levy processes
- 4.3. Diffusions
- 4.4. Left limits and quasi left continuity
- 4.5. Killing at a terminal time
- 4.6. Continuous local times and potential densities
- 4.7. Constructing Ray semigroups and Ray processes
- 4.8. Local Borel right processes
- 4.9. Supermedian functions
- 4.10. Extension Theorem
- 4.11. Notes and references
- 5. Basic properties of Gaussian processes
- 5.1. Definitions and some simple properties
- 5.2. Moment generating functions
- 5.3. Zero-one laws and the oscillation function
- 5.4. Concentration inequalities
- 5.5. Comparison theorems
- 5.6. Processes with stationary increments
- 5.7. Notes and references
- 6. Continuity and boundedness of Gaussian processes
- 6.1. Sufficient conditions in terms of metric entropy
- 6.2. Necessary conditions in terms of metric entropy
- 6.3. Conditions in terms of majorizing measures
- 6.4. Simple criteria for continuity
- 6.5. Notes and references
- 7. Moduli of continuity for Gaussian processes
- 7.1. General results
- 7.2. Processes on R[superscript n]
- 7.3. Processes with spectral densities
- 7.4. Local moduli of associated processes
- 7.5. Gaussian lacunary series
- 7.6. Exact moduli of continuity
- 7.7. Squares of Gaussian processes
- 7.8. Notes and references
- 8. Isomorphism Theorems
- 8.1. Isomorphism theorems of Eisenbaum and Dynkin
- 8.2. The Generalized Second Ray-Knight Theorem
- 8.3. Combinatorial proofs
- 8.4. Additional proofs
- 8.5. Notes and references
- 9. Sample path properties of local times
- 9.1. Bounded discontinuities
- 9.2. A necessary condition for unboundedness
- 9.3. Sufficient conditions for continuity
- 9.4. Continuity and boundedness of local times
- 9.5. Moduli of continuity
- 9.6. Stable mixtures
- 9.7. Local times for certain Markov chains
- 9.8. Rate of growth of unbounded local times
- 9.9. Notes and references
- 10. p-variation
- 10.1. Quadratic variation of Brownian motion
- 10.2. p-variation of Gaussian processes
- 10.3. Additional variational results for Gaussian processes
- 10.4. p-variation of local times
- 10.5. Additional variational results for local times
- 10.6. Notes and references
- 11. Most visited sites of symmetric stable processes
- 11.1. Preliminaries
- 11.2. Most visited sites of Brownian motion
- 11.3. Reproducing kernel Hilbert spaces
- 11.4. The Cameron-Martin Formula
- 11.5. Fractional Brownian motion
- 11.6. Most visited sites of symmetric stable processes
- 11.7. Notes and references
- 12. Local times of diffusions
- 12.1. Ray's Theorem for diffusions
- 12.2. Eisenbaum's version of Ray's Theorem
- 12.3. Ray's original theorem
- 12.4. Markov property of local times of diffusions
- 12.5. Local limit laws for h-transforms of diffusions
- 12.6. Notes and references
- 13. Associated Gaussian processes
- 13.1. Associated Gaussian processes
- 13.2. Infinitely divisible squares
- 13.3. Infinitely divisible squares and associated processes
- 13.4. Additional results about M-matrices
- 13.5. Notes and references
- 14. Appendix
- 14.1. Kolmogorov's Theorem for path continuity
- 14.2. Bessel processes
- 14.3. Analytic sets and the Projection Theorem
- 14.4. Hille-Yosida Theorem
- 14.5. Stone-Weierstrass Theorems
- 14.6. Independent random variables
- 14.7. Regularly varying functions
- 14.8. Some useful inequalities
- 14.9. Some linear algebra
- References
- Index of notation
- Author index
- Subject index