The largest fragment in self-similar fragmentation processes of positive index

Abstract

We study a self-similar fragmentation process with dislocation measure $ν$ and self-similarity index $α> 0$. Let $e^{-m_t}$ denote the size of the largest fragment at time $t \geq 0$. For dislocation measures satisfying a regularity condition of the form $ν(1 - s_1 > δ) = δ^{-θ} \ell(1/δ)$ with $θ\in [0,1)$ and slowly varying $\ell$, we prove almost sure convergence \[ \lim_{t \to \infty} (m_t - g(t)) = 0, \] where $g(t) = (\log t - (1 - θ) \log \log t + f(t))/α$, and $f(t) = o(\log \log t)$ is a lower order correction that can be described explicitly in terms of $\ell$ and $θ$. Our results sharpen substantially the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \log (t)/α$.

Figures (3)

• Figure 1: Simulation of a fragmentation process of a unit square. Each rectangle in the system splits into four new rectangles in proportion $U$ of its width and height, where $U$ is uniformly distributed on the unit interval. The dislocation measure $\nu(\mathrm{d}\mathbf{s})$ is therefore given via $\int_{\mathcal{P}_{\rm m}} f(\mathbf{s}) \nu(\mathrm{d}\mathbf{s}) = 2\int^1_{1/2} f\left(x^2, x(1-x), (1-x)x, (1-x)^2 , 0, 0, \ldots\right) \mathrm{d}x$ for $f \colon \mathcal{P}_{\rm m} \to \mathbb{R}$. Both images show simulations of the process at time $t = 5$. On the left, $\alpha = 1/10$; on the right, $\alpha = -1/10$. For simulation purposes, we froze fragments whose area is smaller than $10^{-4}$.
• Figure 2: Simulation of a fragmentation process of a unit square with infinite dislocation measure given via $\int_{\mathcal{P}_{\rm m}} f(\mathbf{s}) \nu(\mathrm{d}\mathbf{s}) = \int_{1/2}^1 2/(x(1-x)) f\left(x^2,x(1-x),x(1-x),(1-x)^2, 0, \ldots \right) \mathrm{d}x$ at time $t=20$. The approximation is made by truncating $\nu$ to the set $\{s_1<1-\delta\}$ and taking $\delta=10^{-15}$. On the left $\alpha=1$ and on the right $\alpha=-1$. Since the dislocation measure is infinite, the fragments lose very small bits of the fragments. These small bits are visible on the simulation as thick segments between larger fragments.
• Figure 3: Simulation of a Lévy subordinator under the time change \ref{['eq:sclock']}. There were simulated $10^4$ trajectories. On the left there is a gamma subordinator under the time change, up to time $t=258.4618$, $\alpha=0.7$ and $h=5.5$. On the right there is a homogeneous Poisson process under the time change, with intensity one up to time $t=72927.76$, $\alpha=1$ and $h=9$. In red we indicated trajectories for which $\xi_{\rho(t)} \leq h$. Note that in both cases initially the path exhibits a typical behaviour and slows down near $t$.

Theorems & Definitions (53)

• Definition 1.1
• Theorem A
• Remark 1: Under \ref{['eq:1:crumbling']}, \ref{['eq:intcond2']} may or may not hold
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.3
• Theorem 3.4
• proof