1. Introduction to Levy Processes .............................. 1
1.1. Notation ............................................... 1
1.2. Poisson Point Processes ................................ 3
1.3. The Levy-Tto Decomposition ............................. 5
1.4. Levy Processes as Markov Processes ..................... 7
2. Subordinators ............................................... 9
2.1. Introduction ........................................... 9
2.2. Basics ................................................. 9
2.3. The Renewal Measure ................................... 10
2.4. Passage Across a Level ................................ 13
2.5. Arc-Sine Laws for Subordinators ....................... 15
2.6. Rates of Growth ....................................... 16
2.7. Killed Subordinators .................................. 17
3. Local Times and Excursions ................................. 19
3.1. Introduction .......................................... 19
3.2. Local Time of a Markov Process ........................ 19
3.3. The Regular, Instantaneous Case ....................... 20
3.4. The Excursion Process ................................. 22
3.5. The Case of Holding and Irregular Points .............. 23
4. Ladder Processes and the Wiener—Hopf Factorisation ......... 25
4.1. Introduction .......................................... 25
4.2. The Random Walk Case .................................. 25
4.3. The Reflected and Ladder Processes .................... 27
4.4. Applications .......................................... 30
4.5. A Stochastic Bound .................................... 35
5. Further Wiener—Hopf Developments ........................... 41
5.1. Introduction .......................................... 41
5.2. Extensions of a Result due to Baxter .................. 41
5.3. Les Equations Amicales of Vigon ....................... 43
5.4. A First Passage Quintuple Identity .................... 49
6. Creeping and Related Questions ............................. 51
6.1. Introduction .......................................... 51
6.2. Notation and Preliminary Results ...................... 52
6.3. The Mean Ladder Height Problem ........................ 53
6.4. Creeping .............................................. 56
6.5. Limit Points of the Supremum Process .................. 59
6.6. Regularity of the Half-Line ........................... 61
6.7. Summary: Four Integral Tests .......................... 64
7. Spitzer's Condition ........................................ 65
7.1. Introduction .......................................... 65
7.2. Proofs ................................................ 65
7.2.1. The Case ρ = 0,1 ............................... 66
7.2.2. A First Proof for the Case 0 < ρ < 1 ........... 66
7.2.3. A Second Proof for the Case 0 < ρ < 1 .......... 68
7.3. Further Results ....................................... 69
7.4. Tailpiece ............................................. 80
8. Levy Processes Conditioned to Stay Positive ................ 81
8.1. Introduction .......................................... 81
8.2. Notation and Preliminaries ............................ 81
8.3. Definition and Path Decomposition ..................... 83
8.4. The Convergence Result ................................ 86
8.5. Pathwise Constructions of (X,  ) ..................... 89
8.5.1. Tanaka's Construction .......................... 89
8.5.2. Bertoin's Construction ......................... 91
9. Spectrally Negative Levy Processes ......................... 95
9.1. Introduction .......................................... 95
9.2. Basics ................................................ 95
9.3. The Random Walk Case .................................. 99
9.4. The Scale Function ................................... 100
9.5. Further Developments ................................. 104
9.6. Exit Problems for the Reflected Process .............. 109
9.7. Addendum ............................................. 112
10. Small-Time Behaviour ...................................... 115
10.1. Introduction ........................................ 115
10.2. Notation and Preliminary Results .................... 115
10.3. Convergence in Probability .......................... 117
10.4. Almost Sure Results ................................. 126
10.5. Summary of Asymptotic Results ....................... 131
10.5.1. Laws of Large Numbers ....................... 131
10.5.2. Central Limit Theorems ...................... 131
10.5.3. Exit from a Symmetric Interval .............. 132
References .................................................... 133
Index ......................................................... 139
List of Participants .......................................... 141
List of Short Lectures ........................................ 145
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