Martingales and Rates of Presence in Homogeneous Fragmentations

TL;DR

This paper investigates the asymptotic behavior of mass-conservative homogeneous fragmentations, focusing on the probabilities of presence of fragments at specific exponential decay rates, using martingales and spine methods.

Contribution

It extends previous work by analyzing probabilities of presence at exact exponential rates with new martingale techniques and the spine method.

Findings

Characterized effective exponential rates for fragment sizes.

Analyzed probabilities of presence using martingale and spine methods.

Provided new insights into the asymptotic behavior of homogeneous fragmentations.

Abstract

The main focus of this work is the asymptotic behavior of mass-conservative homogeneous fragmentations. Considering the logarithm of masses makes the situation reminiscent of branching random walks. The standard approach is to study {\bf asymptotical} exponential rates. For fixed $v > 0$, either the number of fragments whose sizes at time $t$ are of order $\e^{-vt}$ is exponentially growing with rate $C(v) > 0$, i.e. the rate is effective, or the probability of presence of such fragments is exponentially decreasing with rate $C(v) < 0$, for some concave function $C$. In a recent paper, N. Krell considered fragments whose sizes decrease at {\bf exact} exponential rates, i.e. whose sizes are confined to be of order $\e^{-vs}$ for every $s \leq t$. In that setting, she characterized the effective rates. In the present paper we continue this analysis and focus on probabilities of presence, using the spine method and a suitable martingale.