Martingales and Rates of Presence in Homogeneous Fragmentations

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 ${\text{e}}^{-vt}$ 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 ${\text{e}}^{-vs}$ for every $s\le t$. 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.