# Subordinators: Examples and Applications

• Jean Bertoin
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 1717)

## Abstract

• 0. Foreword

• 1. Elements on subordinators
• 1.1. Definitions and first properties

• 1.2. The Lévy-Khintchine formula

• 1.3. The renewal measure

• 1.4. The range of a subordinator

• 2. Regenerative property
• 2.1. Regenerative sets

• 2.2. Connection with Markov processes

• 3. Asymptotic behaviour of last passage times
• 3.1. Asymptotic behaviour in distribution
• 3.1.1. The self-similar case

• 3.1.2. The Dynkin-Lamperti theorem

• 3.2. Asymptotic sample path behaviour

• 4. Rates of growth of local time
• 4.1. Law of the iterated logarithm

• 4.2. Modulus of continuity

• 5. Geometric properties of regenerative sets
• 5.1. Fractal dimensions
• 5.1.1. Box-counting dimension

• 5.1.2. Hausdorff and packing dimensions

• 5.2. Intersections with a regenerative set
• 5.2.1. Equilibrium measure and capacity

• 5.2.2. Dimension criteria

• 5.2.3. Intersection of independant regenerative sets

• 6. Burgers equation with Brownian initial velocity
• 6.1. Burgers equation and the Hopf-Cole Solution

• 6.2. Brownian initial velocity

• 6.3. Proof of the theorem

• 7. Random covering
• 7.1. Setting

• 7.2. The Laplace exponent of the uncovered set

• 7.3. Some properties of the uncovered set

• 8. Lévy processes
• 8.1. Local time at a fixed point

• 8.2. Local time at the supremum

• 8.3. The spectrally negative case

• 8.4. Bochner’s subordination for Lévy processes

• 9. Occupation times of a linear Brownian motion
• 9.1. Occupation times and subordinators

• 9.2. Lévy measure and Laplace exponent
• 9.2.1. Lévy measure via excursion theory

• 9.2.2. Laplace exponent via the Sturm-Liouville equation

• 9.2.3. Spectral representation of the Laplace exponent

• 9.3. The zero set of a one-dimensional diffusion

• References

### Mathematics Subject Classification (1991):

60-01 60-06 60D05 60G17 60G18 60J15 60J30 60K35 82,-01 82B20 82B26 82B44 82C05