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The mixing of extremal additive coalescents
In this section, we associate by time-reversal a coalescent process to each Lévy fragmentation. We will show that it is a ranked eternal additive coalescent as described by Evans and Pitman [52] and Aldous and Pitman [12], where the initial random data (at time −∞) depend on X. It is thus a mixing of the extremal coalescents of [12], and we will give the exact law of this mixing.
The most natural way to identify the mixing is to use the representation of the extremal coalescents by Bertoin [22], with the help of Vervaat’s Transforms of some bridges with exchangeable increments and deterministic jumps, by noticing that the bridge of X with length 1 is such a bridge, but with random jumps. We will focus on the case where the total mass is 1, so that we do not have to introduce too many time-changes.
Definition : Recall the definition of Fε1 from section 2.3. Call Lévy coalescent derived from X the process defined on the whole real line by Cε1(t) = Fε1(e−t), t ∈ R.
Unbounded variation case
Let (F0 t )t≥0 the natural filtration on the space of càdlàg functions on R+. Let b P be the law of the spectrally positive Lévy process bX = −X. Without risk of ambiguity, bX will also denote the canonical process on D([0,∞)). Recall the definition of the law Pt 0,0 of the bridge of X with length t > 0 starting and ending at 0 from (2.4). Let Pt be the law of the process X killed at time t, and (Ft)t≥0 be the P-completed filtration.
Let also PJ = PτJc where τJc = inf{s ≥ 0,Xs /∈ J} for any interval J. Recall that n is the excursion measure of the reflected process X − X = bX − bX where bX t = inf0≤s≤t bXs. Since bX oscillates or drifts to −∞, every excursion of the process has a finite lifetime D.
Let nu be the measure associated to the excursion killed at time u ∧ D. Remark that the measure n(·, t ≤ D) is a finite measure, with total mass π((t,∞)) where π is the Lévy measure of the subordinator ( b T−y)y≥0 (where b Tx = inf{s ≥ 0, bXs = x}). We already saw that the inverse local time process of bX − bX is ( b T−y)y≥0, and that it has Lévy measure qv(0)dv/v.
The demonstration that we are giving is close to the method used by Biane [32] for Brownian motion and Chaumont [43] for stable processes. It involves a path decomposition of the trajectories of bX under Pt at its minimum. We will first need the following result (see [42]) which is an application of Maisonneuve’s formula. Chaumont stated the result only for oscillating Lévy processes, but the proof applies without change to processes drifting to −∞.
A Poisson process construction of self-similar fragmentations
In [26], Bertoin shows how to construct an arbitrary partition-valued self-similar fragmentation with characteristics (β, c, ν) from a Poisson process. The conventions we are using here (for labelling partitions, and for taking reduced partitions in property 3 below) are actually those used in [17], but by exchangeability arguments explained therein they do indeed give the same distributional object as the construction in [23, 26].
Let εn be the partition of N into the two blocks {n} and N \ {n}. Given x = (x1, x2, . . . ) ∈ Δ, let Px be the distribution of the random partition obtained by first defining an i.i.d. sequence of random variables (Zi)∞i=1 such that P(Zi = j) = xj and P(Zi = 0) = 1 − P ∞j =1 xj , and then defining to be the partition with the property that i and j are in the same block if and only if Zi = Zj ≥ 1. Let κ be the measure on the set P of partitions of N defined such that for all Borel subsets B of P, we have κ(B) = Z Δ Px(B) ν(dx) + c ∞X i=1 1 {εn∈B} .
Mass of a tagged fragment
Our goal in this section is to prove Proposition 3.1, which pertains to the distribution of the asymptotic frequency of the block containing 1 in a partition-valued self-similar fragmentation or, equivalently, the distribution of a size-biased pick from a self-similar ranked fragmentation. According to [26], the tagged fragment in a self-similar fragmentation with index β has to be the inverse of some increasing semi-stable Markov process of index 1/β started at 1. A semi-stable Markov process with index 1/β > 0 is a real-valued strong Markov process X satisfying the following self-similarity property. If, for x > 0, Px denotes the law of X starting from X0 = x, then for every k > 0, the law of the process (kX(k−βs), s ≥ 0) under Px is the same as the law of (X(s), s ≥ 0) under Pkx.
Table of contents :
1 Introduction
1.1 Objets étudiés
1.2 Aperçu des résultats
2 Ordered additive coalescent and Lévy processes
2.1 Introduction
2.2 Ordered additive coalescent
2.3 The Lévy fragmentation
2.4 The fragmentation semigroup
2.5 The left-most fragment
2.6 The mixing of extremal additive coalescents
2.7 Proof of Vervaat’s Theorem
2.8 Concluding remarks
3 Fragmentations and stable subordinators
3.1 Introduction
3.2 A Poisson process construction of self-similar fragmentations
3.3 Stable subordinators
3.4 Small-time behavior of self-similar fragmentations
3.5 Large-time behavior of self-similar fragmentations
3.6 One-dimensional distributions
3.7 Mass of a tagged fragment
4 Self-similar fragmentations of the stable tree I
4.1 Introduction
4.2 Preliminaries
4.3 Study of F−
4.4 Small-time asymptotics
4.5 On continuous-state branching processes…
5 Self-similar fragmentations of the stable tree II
5.1 Introduction
5.2 Some facts about Lévy processes
5.3 The stable tree
5.4 Study of F+
5.5 Study of F♮
5.6 Asymptotics
6 The genealogy of self-similar fragmentations
6.1 Introduction
6.2 The CRT TF
6.3 Hausdorff dimension of TF
7 The exploration process of the ICRT
7.1 Introduction
7.2 Constructing Xθ and Y θ
7.3 p-trees and associated processes
7.4 Convergence of p-trees to the ICRT
7.5 Height profile
7.6 The exploration process
7.7 Miscellaneous comments
8 Asymptotics for Random p-mappings
8.1 Introduction
8.2 Background
8.3 Proof of Theorem 8.1
8.4 Final comments
Bibliographie
