Fragmentation associated with Lévy processes using snake

Abstract

We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using Lévy snake tools developed by Le Gall-Le Jan and Duquesne-Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the selfsimilar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is of its own interest.

Dislocation measure of the fragmentation of a general Lévy tree

“…We will see in Section 4.2 that the measure ν nod is the same as the measure ν in the case of a tree without Brownian part (β = 0). Thus the proof of Part 1 of Theorem 1.1 is the same as in [1]. Only Part 2 needs a proof.…”

“…with the same normalization as in [1]. We say that a CRT coded by its associated height process H is a Lévy CRT.…”

Feller Property and Infinitesimal Generator of the Exploration Process

“…Remark 2.8. From [1] (see Theorem 6.1 and Proposition 6.3 in Section 6.1), there is a relationship between the infinitesimal generator of ρ and the infinitesimal generator of the exploration process, ρ (θ) , associated to a Lévy process with exponent ψ (θ) , where…”

“…The strong Markov property of the exploration process is a fundamental property and has been proved in [7]. In [1], the exploration process is used in order to construct a fragmentation process associated to a Lévy process using Markov properties and martingale problems.…”

“…We also give its infinitesimal generator for exponential functionals, Corollary 2.3, as well as the related martingales in Corollaries 2.5 and 2.6. We also recall in Remark 2.8 the relation given in [1] between the infinitesimal generator of the exploration process associated to the Lévy process with Laplace exponent ψ and the infinitesimal generator of the exploration process associated to the Lévy process with Laplace exponent ψ (θ) = ψ(θ + ·) − ψ(θ), θ ≥ 0. Section 3 is devoted to the proof of the Feller property.…”