Date: 28 Aug 2004

Subordinators: Examples and Applications

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Contents.

  • 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