Non-Autonomous Kato Classes and Feynman-Kac Propagators

Contents......Page 12
Preface......Page 8
1.1.1 Notation......Page 15
1.1.2 Elements of Probability Theory......Page 16
1.1.3 Locally Compact Spaces......Page 18
1.1.4 Stochastic Processes......Page 19
1.1.5 Filtrations......Page 21
1.2 Markov Property......Page 22
1.3 Transition Functions and Backward Transition Functions......Page 24
1.4 Markov Processes Associated with Transition Functions......Page 27
1.5 Space-Time Processes......Page 31
1.6 Classes of Stochastic Processes......Page 39
1.7 Completions of o-Algebras......Page 42
1.8 Path Properties of Stochastic Processes: Separability and Progressive Measurability......Page 47
1.9 Path Properties of Stochastic Processes: One-Sided Continuity and Continuity......Page 58
1.10 Reciprocal Transition Functions and Reciprocal Processes......Page 69
1.11 Path Properties of Reciprocal Processes......Page 93
1.12.1 Brownian motion and Brownian bridge......Page 103
1.12.2 Cauchy process and Cauchy bridge......Page 109
1.12.3 Forward Kolmogorov representation of Brownian bridges......Page 111
1.13 Notes and Comments......Page 112
2.1 Propagators and Backward Propagators on Banach Spaces......Page 115
2.2 Free Propagators and Free Backward Propagators......Page 118
2.3 Generators of Propagators and Kolmogorov's Forward and Backward Equations......Page 120
2.4 Howland Semigroups......Page 135
2.5 Feller-Dynkin Propagators and the Continuity Properties of Markov Processes......Page 138
2.6 Stopping Times and the Strong Markov Property......Page 148
2.7 Strong Markov Property with Respect to Families of Measures......Page 164
2.8 Feller-Dynkin Propagators and Completions of o-Algebras......Page 186
2.9 Feller-Dynkin Propagators and Standard Processes......Page 188
2.10 Hitting Times and Standard Processes......Page 192
2.11 Notes and Comments......Page 204
3.1 Additive and Multiplicative Functionals......Page 207
3.2 Potentials of Time-Dependent Measures and Non-Autonomous Kato Classes......Page 209
3.3 Backward Transition Probability Functions and Non-Autonomous Kato Classes of Functions and Measures......Page 214
3.4 Weighted Non-Autonomous Kato Classes......Page 217
3.5 Examples of Functions and Measures in Non-Autonomous Kato Classes......Page 221
3.6 Transition Probability Densities and Fundamental Solutions to Parabolic Equations in Non-Divergence Form......Page 231
3.7 Transition Probability Densities and Fundamental Solutions to Parabolic Equations in Divergence Form......Page 236
3.8 Diffusion Processes and Stochastic Differential Equations......Page 246
3.9 Additive Functionals Associated with Time-Dependent Measures......Page 269
3.10 Exponential Estimates for Additive Functionals......Page 283
3.11 Probabilistic Description of Non-Autonomous Kato Classes......Page 289
3.12 Notes and Comments......Page 290
4.1 Schrodinger Semigroups with Kato Class Potentials......Page 293
4.2 Feynman-Kac Propagators......Page 297
4.3 The Behavior of Feynman-Kac Propagators in Lp-Spaces......Page 299
4.4 Feller Feller-Dynkin and BUC-Property of Feynman-Kac Propagators......Page 307
4.5 Integral Kernels of Feynman-Kac Propagators......Page 312
4.6 Feynman-Kac Propagators and Howland Semigroups......Page 318
4.7 Duhamel's Formula for Feynman-Kac Propagators......Page 321
4.8 Feynman-Kac Propagators and Viscosity Solutions......Page 325
4.9 Notes and Comments......Page 337
5.1 Monotone Class Theorems......Page 339
5.2 Kolmogorov's Extension Theorem......Page 340
5.3 Uniform Integrability......Page 341
5.4 Radon-Nikodym Theorem......Page 342
5.6 Doob's Inequalities......Page 343
Bibliography......Page 345
Index......Page 355