Discretization methods for homogeneous fragmentations

Abstract

Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogs of a certain type of branching random walks, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations. On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, namely, the theory of exchangeable partitions of $\mathbb{N}$. Spatial discretization is especially well-suited to develop directly for continuous times the conceptual method of probability tilting of Lyons, Pemantle and Peres.

21 pages

Cited by in corpus (12)

• Spinal partitions and invariance under re-rooting of continuum random trees
• The Seneta--Heyde scaling for the branching random walk
• Regenerative tree growth: Binary self-similar continuum random trees and Poisson--Dirichlet compositions
• The fragmentation process of an infinite recursive tree and Ornstein-Uhlenbeck type processes
• Infinitely ramified point measures and branching Lévy processes
• The strong Malthusian behavior of growth-fragmentation processes
• Traveling waves and homogeneous fragmentation
• A functional limit theorem for the profile of $b$-ary trees
• Probability tilting of compensated fragmentations
• Growth-fragmentation processes and bifurcators
• A growth-fragmentation model related to Ornstein-Uhlenbeck type processes
• The largest fragment of a homogeneous fragmentation process