Stochastic spectral theory for selfadjoint Feller operators : a functional integration approach
著者
書誌事項
Stochastic spectral theory for selfadjoint Feller operators : a functional integration approach
（Probability and its applications）
Birkhäuser, c2000
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注記
Includes bibliography and subject index
内容説明・目次
内容説明
In this book, 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 generators of Ornstein-Uhlenbeck processes. The unified approach provides a new viewpoint of and a deeper insight into the subject.
目次
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 Schroedinger semigroup theory.- E.1 Gaussian processes.- E.2 Brownian motion and related processes.- E.3 Kato-Feller potentials for the Laplace operator.- E.4 Schroedinger semigroups.- E.5 Generalizations and modifications.- 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).- C.1 Estimates for the Laplace transform of the occupation time for Wiener processes.- C.2 Quantitative large coupling estimates for Feller operators in terms of the weighted Laplace transform of the occupation time.- 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, Scheffe's Theorem, Monotone Class Theorem.- References.- Index of Symbols.
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