The theory of Markov Processes has become a powerful tool in partial differential equations and potential theory with important applications to physics. Professor Dynkin has made many profound contributions to the subject and in this volume are collected several of his most important expository and survey articles. The content of these articles has not been covered in any monograph as yet. This account is accessible to graduate students in mathematics and operations research and will be welcomed by all those interested in stochastic processes and their applications.
Author(s): E. B. Dynkin
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 1982
Language: English
Pages: 321
Cover......Page 1
Title......Page 4
Copyright......Page 5
CONTENTS......Page 6
Preface......Page 7
1. Introduction......Page 10
2. General problems in the theory of Markov processes......Page 13
3. The form of an infinitesimal operator. Generalized diffusion processes......Page 16
4. Harmonic, subharmonic, and superharmonic functions associated with a Markov process......Page 19
5. Additive functional and associated transformations of Markov processes......Page 21
6. Stochastic integral equations......Page 24
7. Boundary problems in the theory of differential equations and asymptotic behaviour of trajectories......Page 26
8. Concluding remarks......Page 27
References......Page 28
Introduction......Page 34
1. Boundary value problems for Laplace's equation and the Martin boundary......Page 38
2. Cones in linear topological spaces......Page 47
3. The boundary value problem with a directional derivative(Problem 3t)......Page 50
4. Reduction of Problem M?a some particular solutions......Page 52
5. The Minimum Principle and its consequences......Page 62
6. The Green's Function......Page 69
7. Asymptotic behaviour of the Green's function......Page 71
8. The Martin boundary for the boundary value problem 31. A description of the non-negative solutions and the solutions bounded below......Page 81
References......Page 85
Introduction......Page 88
1. Harmonic and excessive functions and measures......Page 90
2. Markov processes......Page 91
3. The Green's function......Page 95
4. Supermartingales......Page 97
5. Excessive functions and supermartingales......Page 100
6. Position of the particle at the last exit time from a set D......Page 101
7. Densities of excessive measures and supermartingales......Page 102
8. Excessive measures with densities of class Ky. The Martin kernel......Page 103
9. Martin compactification......Page 105
10. Distribution of x^......Page 106
11. /z-processes. Martin representation of excessive functions......Page 108
12. The spectral measure of kz. The exit space......Page 109
13. The Uniqueness Theorem......Page 111
15. The operator 6. Final random variables......Page 112
16. The entrance space. Decomposition of excessive measures......Page 116
17. The behaviour of a stationary process as t -> - ????......Page 117
18. Stationary processes with random birth and death times......Page 119
19. Hunt processes......Page 122
Appendix. Measures in spaces of paths......Page 127
References......Page 131
1. Introduction......Page 132
2. Decomposition into ergodic measures......Page 136
3. The construction of the (otf, JTa+) -kernel......Page 138
4. The entrance space and the exit space......Page 142
5. Markov processes with random births and deaths......Page 147
References......Page 152
V Integral representation of excessive measures and excessive functions......Page 154
1. Plan of the paper. Discussion of results......Page 155
2. The construction of a Markov process from an excessive measure and function......Page 166
3. Excessive measures......Page 173
4. Excessive functions......Page 177
5. Homogeneous excessive functions......Page 182
6. The faces of the simplex &cv. Invariant and null excessive measures and functions. The entrance laws......Page 185
7. Excessive functions and supermartingales......Page 187
8. The construction of the adjoint transition function......Page 189
quasi-invariant measures into ergodic measures......Page 190
References......Page 194
VI Regular Markov processes......Page 196
1. Introduction......Page 197
2. Measurable structures in the phase space......Page 204
3. The strong Markov property......Page 210
4. Essential points......Page 212
5. Excessive functions......Page 215
6. The uniqueness of a rigid regular re-construction......Page 218
7. The adjoint process of a regular re-construction .......Page 219
8. Existence of a regular re-construction......Page 223
References......Page 227
1. Introduction......Page 228
2, Regularization of a right Markov representation......Page 239
3. Regular right representations......Page 244
4. Absolutely continuous Markov representations......Page 252
Appendix. S. E. Kuznetsov. Regular topological representations . . .......Page 264
References......Page 267
1. Introduction.......Page 268
2. Convex measurable spaces.......Page 269
3. H-sufficiency and the decomposition into extreme points.......Page 271
4. Asymptotic sufficiency.......Page 276
5. Gibbs states.......Page 278
6. Shifts.......Page 279
7. Symmetric measures.......Page 282
8. Stochastic fields.......Page 284
9. Markov processes with a given transition function.......Page 285
10. Entrance and exit laws.......Page 287
11. Excessive measures and excessive functions.......Page 288
12. Stationary transition functions.......Page 290
REFERENCES......Page 292
1. Introduction.......Page 294
2. Discussion of results.......Page 295
3. Conservative minimal elements.......Page 300
4. Time-dependent excessive measures and functions.......Page 305
5. Markov processes.......Page 307
6. Three lemmas.......Page 309
7. Dissipative minimal elements.......Page 312
Appendix. Decomposition into minimal elements.......Page 315
REFERENCES......Page 320
