Non-autonomous Kato Classes And... | Buy

"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.

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

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