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Intersection Local Times, Loop Soups and Permanental Wick Powers.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence : American Mathematical Society, 2017Copyright date: ©2017Edition: 1st edDescription: 1 online resource (92 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470437039
Subject(s): Genre/Form: Additional physical formats: Print version:: Intersection Local Times, Loop Soups and Permanental Wick PowersDDC classification:
  • 519.23
LOC classification:
  • QA274.4.L453 2017
Online resources:
Contents:
Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Loop measures and renormalized intersection local times -- 2.1. Renormalized intersection local times -- 2.2. Bounds for the error terms -- Chapter 3. Continuity of intersection local time processes -- Chapter 4. Loop soup and permanental chaos -- Chapter 5. Isomorphism Theorem I -- Chapter 6. Permanental Wick powers -- Chapter 7. Poisson chaos decomposition, I -- Chapter 8. Loop soup decomposition of permanental Wick powers -- Chapter 9. Poisson chaos decomposition, II -- 9.1. Exponential Poisson chaos -- 9.2. Extensions to martingales -- Chapter 10. Convolutions of regularly varying functions -- References -- Back Cover.
Summary: Several stochastic processes related to transient Lévy processes with potential densities u(x,y)=u(y-x), that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures \mathcal{V} endowed with a metric d. Sufficient conditions are obtained for the continuity of these processes on (\mathcal{V},d). The processes include n-fold self-intersection local times of transient Lévy processes and permanental chaoses, which are `loop soup n-fold self-intersection local times' constructed from the loop soup of the Lévy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of n-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.
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Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Loop measures and renormalized intersection local times -- 2.1. Renormalized intersection local times -- 2.2. Bounds for the error terms -- Chapter 3. Continuity of intersection local time processes -- Chapter 4. Loop soup and permanental chaos -- Chapter 5. Isomorphism Theorem I -- Chapter 6. Permanental Wick powers -- Chapter 7. Poisson chaos decomposition, I -- Chapter 8. Loop soup decomposition of permanental Wick powers -- Chapter 9. Poisson chaos decomposition, II -- 9.1. Exponential Poisson chaos -- 9.2. Extensions to martingales -- Chapter 10. Convolutions of regularly varying functions -- References -- Back Cover.

Several stochastic processes related to transient Lévy processes with potential densities u(x,y)=u(y-x), that need not be symmetric nor bounded on the diagonal, are defined and studied. They are real valued processes on a space of measures \mathcal{V} endowed with a metric d. Sufficient conditions are obtained for the continuity of these processes on (\mathcal{V},d). The processes include n-fold self-intersection local times of transient Lévy processes and permanental chaoses, which are `loop soup n-fold self-intersection local times' constructed from the loop soup of the Lévy process. Loop soups are also used to define permanental Wick powers, which generalizes standard Wick powers, a class of n-th order Gaussian chaoses. Dynkin type isomorphism theorems are obtained that relate the various processes. Poisson chaos processes are defined and permanental Wick powers are shown to have a Poisson chaos decomposition. Additional properties of Poisson chaos processes are studied and a martingale extension is obtained for many of the processes described above.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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