Non-autonomous Kato Classes And Feynman-kac Propagators

by Gulisashvili, Archil; Casteren, Jan A van

• ISBN13: 9789812565570
• ISBN10: 9812565574
• eBook ISBN(s): 9789812774606
• Format: Hardcover
• Copyright: 2006-09-30
• Publisher: World Scientific Pub Co Inc
• More Book Details
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Summary

"This book provides an introduction to propagator theory. Propagators, or evolution families, are two-parameter analogues of semigroups of operators. Propagators are encountered in analysis, mathematical physics, partial differential equations, and probability theory. They are often used as mathematical models of systems evolving in a changing environment. A unifying theme of the book is the theory of Feynman-Kac propagators associated with time-dependent measures from non-autonomous Kato classes. In applications, a Feynman-Kac propagator describes the evolution of a physical system in the presence of time-dependent absorption and excitation. The book is suitable as an advanced textbook for graduate courses." "Readership: Graduate students and researchers in mathematical analysis, partial differential equations, and probability theory."--BOOK JACKET.

Table of Contents

Preface	vii
1. Transition Functions and Markov Processes	1	(100)
1.1 Introduction	1	(7)
1.1.1 Notation	1	(1)
1.1.2 Elements of Probability Theory	2	(2)
1.1.3 Locally Compact Spaces	4	(1)
1.1.4 Stochastic Processes	5	(2)
1.1.5 Filtrations	7	(1)
1.2 Markov Property	8	(2)
1.3 Transition Functions and Backward Transition Functions	10	(3)
1.4 Markov Processes Associated with Transition Functions	13	(4)
1.5 Space-Time Processes	17	(8)
1.6 Classes of Stochastic Processes	25	(3)
1.7 Completions of σ-Algebras	28	(5)
1.8 Path Properties of Stochastic Processes: Separability and Progressive Measurability	33	(11)
1.9 Path Properties of Stochastic Processes: One-Sided Continuity and Continuity	44	(11)
1.10 Reciprocal Transition Functions and Reciprocal Processes	55	(24)
1.11 Path Properties of Reciprocal Processes	79	(10)
1.12 Examples of Transition Functions and Markov Processes	89	(9)
1.12.1 Brownian motion and Brownian bridge	89	(6)
1.12.2 Cauchy process and Cauchy bridge	95	(2)
1.12.3 Forward Kolmogorov representation of Brownian bridges	97	(1)
1.13 Notes and Comments	98	(3)
2. Propagators: General Theory	101	(92)
2.1 Propagators and Backward Propagators on Banach Spaces	101	(3)
2.2 Free Propagators and Free Backward Propagators	104	(2)
2.3 Generators of Propagators and Kolmogorov's Forward and Backward Equations	106	(15)
2.4 Howland Semigroups	121	(3)
2.5 Feller-Dynkin Propagators and the Continuity Properties of Markov Processes	124	(10)
2.6 Stopping Times and the Strong Markov Property	134	(16)
2.7 Strong Markov Property with Respect to Families of Measures	150	(22)
2.8 Feller-Dynkin Propagators and Completions of σ-Algebras	172	(2)
2.9 Feller-Dynkin Propagators and Standard Processes	174	(4)
2.10 Hitting Times and Standard Processes	178	(12)
2.11 Notes and Comments	190	(3)
3. Non-Autonomous Kato Classes of Measures	193	(86)
3.1 Additive and Multiplicative Functionals	193	(2)
3.2 Potentials of Time-Dependent Measures and Non-Auto-nomous Kato Classes	195	(5)
3.3 Backward Transition Probability Functions and Non-Auto-nomous Kato Classes of Functions and Measures	200	(3)
3.4 Weighted Non-Autonomous Kato Classes	203	(4)
3.5 Examples of Functions and Measures in Non-Autonomous Kato Classes	207	(10)
3.6 Transition Probability Densities and Fundamental Solutions to Parabolic Equations in Non-Divergence Form	217	(5)
3.7 Transition Probability Densities and Fundamental Solutions to Parabolic Equations in Divergence Form	222	(10)
3.8 Diffusion Processes and Stochastic Differential Equations	232	(23)
3.9 Additive Functionals Associated with Time-Dependent Measures	255	(14)
3.10 Exponential Estimates for Additive Functionals	269	(6)
3.11 Probabilistic Description of Non-Autonomous Kato Classes	275	(1)
3.12 Notes and Comments	276	(3)
4. Feynman-Kac Propagators	279	(46)
4.1 Schrödinger Semigroups with Kato Class Potentials	279	(4)
4.2 Feynman-Kac Propagators	283	(2)
4.3 The Behavior of Feynman-Kac Propagators in LP-Spaces	285	(8)
4.4 Feller, Feller-Dynkin, and BUC-Property of Feynman-Kac Propagators	293	(5)
4.5 Integral Kernels of Feynman-Kac Propagators	298	(6)
4.6 Feynman-Kac Propagators and Howland Semigroups	304	(3)
4.7 Duhamel's Formula for Feynman-Kac Propagators	307	(4)
4.8 Feynman-Kac Propagators and Viscosity Solutions	311	(12)
4.9 Notes and Comments	323	(2)
5. Some Theorems of Analysis and Probability Theory	325	(6)
5.1 Monotone Class Theorems	325	(1)
5.2 Kolmogorov's Extension Theorem	326	(1)
5.3 Uniform Integrability	327	(1)
5.4 Radon-Nikodym Theorem	328	(1)
5.5 Vitali-Hahn-Saks Theorem	329	(1)
5.6 Doob's Inequalities	329	(2)
Bibliography	331	(10)
Index	341