Non-autonomous Kato classes and Feynman-Kac propagators / Archil Gulisashvili, Jan A. van Casteren.

Includes bibliographical references (pages 331-340) and index.

Print version record.

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Preface -- 1. Transition functions and Markov processes. 1.1. Introduction. 1.2. Markov property. 1.3. Transition functions and backward transition functions. 1.4. Markov processes associated with transition functions. 1.5. Space-time processes. 1.6. Classes of stochastic processes. 1.7. Completions of [symbol]-algebras. 1.8. Path properties of stochastic processes: separability and progressive measurability. 1.9. Path properties of stochastic processes: one-sided continuity and continuity. 1.10. Reciprocal transition functions and reciprocal processes. 1.11. Path properties of reciprocal processes. 1.12. Examples of transition functions and Markov processes. 1.13. Notes and comments -- 2. Propagators: general theory. 2.1. Propagators and backward propagators on Banach spaces. 2.2. Free propagators and free backward propagators. 2.3. Generators of propagators and Kolmogorov's forward and backward equations. 2.4. Howland semigroups. 2.5. Feller-Dynkin propagators and the continuity properties of Markov processes. 2.6. Stopping times and the strong Markov property. 2.7. Strong Markov property with respect to families of measures. 2.8. Feller-Dynkin propagators and completions of [symbol]-algebras. 2.9. Feller-Dynkin propagators and standard processes. 2.10. Hitting times and standard processes. 2.11. Notes and comments -- 3. Non-autonomous Kato classes of measures. 3.1. Additive and multiplicative functionals. 3.2. Potentials of time-dependent measures and non-autonomous Kato classes. 3.3. Backward transition probability functions and non-autonomous Kato classes of functions and measures. 3.4. Weighted non-autonomous Kato classes. 3.5. Examples of functions and measures in non-autonomous Kato classes. 3.6. Transition probability densities and fundamental solutions to parabolic equations in non-divergence form. 3.7. Transition probability densities and fundamental solutions to parabolic equations in divergence form. 3.8. Diffusion processes and stochastic differential equations. 3.9. Additive functionals associated with time-dependent measures. 3.10. Exponential estimates for additive functionals. 3.11. Probabilistic description of non-autonomous Kato classes. 3.12. Notes and comments -- 4. Feynman-Kac propagators. 4.1. Schrödinger semigroups with Kato class potentials. 4.2. Feynman-Kac propagators. 4.3. The behavior of Feynman-Kac propagators in L[symbol]-spaces. 4.4. Feller, Feller-Dynkin, and BUC-property of Feynman-Kac propagators. 4.5. Integral kernels of Feynman-Kac propagators. 4.6. Feynman-Kac propagators and Howland semigroups. 4.7. Duhamel's formula for Feynman-Kac propagators. 4.8. Feynman-Kac propagators and viscosity solutions. 4.9. Notes and comments -- 5. Some theorems of analysis and probability theory. 5.1. Monotone class theorems. 5.2. Kolmogorov's extension theorem. 5.3. Uniform integrability. 5.4. Radon-Nikodym theorem. 5.5. Vitali-Hahn-Saks theorem. 5.6. Doob's inequalities.

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