# Some Aspects of Additive Coalescents

@article{Bertoin2002SomeAO, title={Some Aspects of Additive Coalescents}, author={Jean Bertoin}, journal={arXiv: Probability}, year={2002} }

We present some aspects of the so-called additive coalescence, with a focus on its connections with random trees, Brownian excursion, certain bridges with exchangeable increments, Levy processes, and sticky particle systems.

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#### References

SHOWING 1-10 OF 28 REFERENCES

Eternal additive coalescents and certain bridges with exchangeable increments

- Mathematics
- 2001

Aldous and Pitman have studied the asymptotic behavior of the additive coalescent processes using a nested family random forests derived by logging certain inhomogeneous continuum random trees. Here… Expand

Ordered Additive Coalescent and Fragmentations Associatedto Lévy Processes with No Positive Jumps

- Mathematics
- 2001

We study here the fragmentation processes that can be derived from Levy processes with no positive jumps in the same manner as in the case of a Brownian motion (cf. Bertoin [4]). One of our… Expand

Clustering statistics for sticky particles with Brownian initial velocity

- Mathematics
- 2000

We establish a connection between two different models of clustering: the deterministic model of sticky particles which describes the evolution of a system of infinitesimal particles governed by the… Expand

Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists

- Mathematics
- 1999

Author(s): Aldous, DJ | Abstract: Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at (stochastic) rate K(x, y)/N,… Expand

Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent

- Mathematics
- 2000

Abstract. Regard an element of the set of ranked discrete distributions Δ := {(x1, x2,…):x1≥x2≥…≥ 0, ∑ixi = 1} as a fragmentation of unit mass into clusters of masses xi. The additive coalescent is… Expand

A Relation between Brownian Bridge and Brownian Excursion

- Mathematics
- 1979

It is shown that Brownian excursion is equal in distribution to Brownian bridge with the origin placed at its absolute minimum. This explains why the maximum of Brownian excursion and the range of… Expand

Applications of the continuous-time ballot theorem to Brownian motion and related processes

- Mathematics
- 2001

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the… Expand

A fragmentation process connected to Brownian motion

- Mathematics
- 2000

Abstract. Let (Bs, s≥ 0) be a standard Brownian motion and T1 its first passage time at level 1. For every t≥ 0, we consider ladder time set ℒ(t) of the Brownian motion with drift t, B(t)s = Bs + ts,… Expand

Construction of markovian coalescents

- Mathematics
- 1998

Abstract Partition-valued and measure-valued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of… Expand

Lévy Processes

- 2000

Lévy processes are random processes on Euclidean space that are stochastically continuous and have stationary independent increments. They, and their stochastic integrals, have become useful tools in… Expand