Chapter

Lectures on Probability Theory and Statistics

Volume 1717 of the series Lecture Notes in Mathematics pp 1-91

Date:

Subordinators: Examples and Applications

  • Jean BertoinAffiliated withLaboratoire de Probabilités, Université Pierre et Marie Curie

* Final gross prices may vary according to local VAT.

Get Access

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

Mathematics Subject Classification (1991):

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