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Stochastic Spectral Theory for Selfadjoint Feller Operators : a functional integration approach / by Michael Demuth, Jan A. Casteren.

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A beautiful interplay between probability theory (Markov processes, martingale theory) on the one hand and operator and spectral theory on the other yields a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and gene...

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Bibliographic Details
Online Access: Full Text (via Springer)
Main Author: Demuth, Michael
Other Authors: Casteren, Jan A.
Format: eBook
Language:English
Published: Basel : Birkhäuser Basel : Imprint : Birkhäuser, 2000.
Series:Probability and its applications.
Subjects:
Mathematics.
Operator theory.
Distribution (Probability theory)
Table of Contents:
  • 1 Basic Assumptions of Stochastic Spectral Analysis:Free Feller Operators
  • A Introduction
  • B Assumptions and Free Feller Generators
  • C Examples
  • D Heat kernels
  • E Summary of Schrödinger semigroup theory
  • 2 Perturbations of Free Feller Operators
  • The framework of stochastic spectral analysis
  • A Regular perturbations
  • B Integral kernels, martingales, pinned measures
  • C Singular perturbations
  • 3 Proof of Continuity and Symmetry of Feynman-Kac Kernels
  • 4 Resolvent and Semigroup Differences for Feller Operators: Operator Norms
  • A Regular perturbations
  • B Singular perturbations
  • 5 Hilbert-Schmidt Properties of Resolvent and Semigroup Differences
  • A Regular perturbations
  • B Singular perturbations
  • 6 Trace Class Properties of Semigroup Differences
  • A General trace class criteria
  • B Regular perturbations
  • C Singular perturbations
  • 7 Convergence of Resolvent Differences
  • 8 Spectral Properties of Self-adjoint Feller Operators
  • A Qualitative spectral results
  • B Quantitative estimates for regular potentials
  • C Quantitative estimates for singular potentials in terms of the weighted Laplace transform of the occupation time (for large coupling parameters)
  • Appendix A Spectral Theory
  • Appendix B Semigroup Theory
  • Appendix C Markov Processes, Martingales and Stopping Times
  • Appendix D Dirichlet Kernels, Harmonic Measures, Capacities
  • Appendix E Dini's Lemma, Scheffé's Theorem, Monotone Class Theorem
  • References
  • Index of Symbols.

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