Stochastic Spectral Theory for Selfadjoint Feller Operators

Dermuth, Michael; Casteren, Jan A.Van

ISBN 10: 3764358874 / ISBN 13: 9783764358877

Editore: Birkhauser Verlag AG, 2000

Explains the interplay between probability theory (Markov processes, martingale theory) and operator and spectral theory. This title provides a uniform treatment of several kinds of Hamiltonians such as the Laplace operator, relativistic Hamiltonian, Laplace-Beltrami operator, and generators of Ornstein-Uhlenbeck processes. Series: Probability and its Applications. Num Pages: 475 pages, biography. BIC Classification: PBWL; PDE; TBJ. Category: (P) Professional & Vocational; (UP) Postgraduate, Research & Scholarly. Dimension: 235 x 155 x 26. Weight in Grams: 1870. . 2000. Hardback. . . . . Books ship from the US and Ireland. Codice articolo V9783764358877

Riassunto:

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.

Contenuti: 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.- E.1 Gaussian processes.- E.2 Brownian motion and related processes.- E.3 Kato-Feller potentials for the Laplace operator.- E.4 Schrödinger 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, Scheffé’s Theorem, Monotone Class Theorem.- References.- Index of Symbols.

Dati bibliografici

Titolo: Stochastic Spectral Theory for Selfadjoint ...
Casa editrice: Birkhauser Verlag AG
Data di pubblicazione: 2000
Legatura: Rilegato
Condizione: New