Markov processes, Gaussian processes, and local times /

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Bibliographic Details
Author / Creator:Marcus, Michael B.
Imprint:Cambridge ; New York : Cambridge University Press, 2006.
Description:x, 620 p. ; 24 cm.
Language:English
Series:Cambridge studies in advanced mathematics ; 100
Subject:Markov processes.
Gaussian processes.
Local times (Stochastic processes)
Gaussian processes.
Local times (Stochastic processes)
Markov processes.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/6117539
Hidden Bibliographic Details
Other authors / contributors:Rosen, Jay, 1948-
ISBN:0521863007 (hardback)
Notes:Includes bibliographical references (p. 603-610) and indexes.
Standard no.:9780521863001
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