Lévy Processes and Infinitely Divisible Distributions

Front Cover......Page 1
Back Cover......Page 2
Title Page......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Remarks on notation......Page 13
1. Definition of Lévy processes......Page 15
2. Characteristic functions......Page 21
3. Poisson processes......Page 28
4. Compound Poisson processes......Page 32
5. Brownian motion......Page 36
6. Exercises 1......Page 42
Notes......Page 44
7. Infinitely divisible distributions and Lévy processes in law......Page 45
8. Representation of infinitely divisible distributions......Page 51
9. Additive processes in law......Page 61
10. Transition functions and the Markov property......Page 68
11. Existence of Lévy and additive processes......Page 73
12. Exercises 2......Page 80
Notes......Page 82
13. Selfsimilar and semi-selfsimilar processes and their exponents......Page 83
14. Representations of stable and semi-stable distributions......Page 91
15. Selfdecomposable and semi-selfdecomposable distributions......Page 104
16. Selfsimilar and semi-selfsimilar additive processes......Page 113
17. Another view of selfdecomposable distributions......Page 118
18. Exercises 3......Page 128
Notes......Page 130
19. Formulation of the Lévy-Itô decomposition......Page 133
20. Proof of the Lévy-Itô decomposition......Page 139
21. Applications to sample function properties......Page 149
22. Exercises 4......Page 156
Notes......Page 158
23. Time dependent distributional properties......Page 159
24. Supports......Page 162
25. Moments......Page 173
26. Lévy measures with bounded supports......Page 182
27. Continuity properties......Page 188
28. Smoothness......Page 203
29. Exercises 5......Page 207
Notes......Page 210
30. Subordination of Lévy processes......Page 211
31. Infinitesimal generators of Lévy processes......Page 219
32. Subordination of semigroups of operators......Page 226
33. Density transformation of Lévy processes......Page 231
34. Exercises 6......Page 247
Notes......Page 250
35. Dichotomy of recurrence and transience......Page 251
36. Laws of large numbers......Page 259
37. Criteria and examples......Page 264
38. The symmetric one-dimensional case......Page 277
39. Exercises 7......Page 284
Notes......Page 286
40. The strong Markov property......Page 287
41. Potential operators......Page 295
42. Capacity......Page 309
43. Hitting probability and regularity of a point......Page 327
44. Exercises 8......Page 342
Notes......Page 345
45. Factorization identities......Page 347
46. Lévy processes without positive jumps......Page 359
47. Short time behavior......Page 365
48. Long time behavior......Page 377
49. Further factorization identities......Page 383
50. Exercises 9......Page 396
Notes......Page 397
51. Infinite divisibility on the half line......Page 399
52. Unimodality and strong unimodality......Page 408
53. Selfdecomposable processes......Page 417
54. Unimodality and multimodality in Lévy processes......Page 430
55. Exercises 10......Page 438
Notes......Page 440
Solutions to exercises......Page 441
References and author index......Page 465
Subject index......Page 493