The Largest Fragment of a Homogeneous Fragmentation Process

Abstract

We show that in homogeneous fragmentation processes the largest fragment at time t has sizewhere Φ is the Lévy exponent of the fragmentation process, and p is the unique solution of the equation (log Φ) ′ (p) = 1 1+p . We argue that this result is in line with predictions arising from the classification of homogeneous fragmentation processes as logarithmically correlated random fields.

Asymptotics of self-similar growth-fragmentation processes

“…Since (31) is bounded in L q (P 1 ) for every p < q < ω + /ω − , it also converges to 0 in L p (P 1 ) (by Hölder's inequality). Putting everything together yields (30), and thus the first part of the statement. The second part is derived from standard arguments: the space C c ((0, ∞)) of continuous functions on (0, ∞) with compact support being separable, a diagonal extraction procedure easily entails, for every sequence t n → ∞, that there exists an extraction σ : N → N such that, almost surely,…”

“…Remark 2.11. (i) Kyprianou et al [30] recently derived an analogue of (a) for pure homogeneous fragmentations. However the method we employ here (for both statements) is different: basically, we directly transfer the known results on branching random walks to discrete skeletons of the growthfragmentation, and then infer the behavior of the whole process with the help of the…”