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 asymptotical exponential rates (Berestycki (2003)  [3], Bertoin and Rouault (2005) [12]). For fixed v>0v>0, either the number of fragments whose sizes at time tt are of order e−vt is exponentially growing with rate C(v)>0C(v)>0, i.e. the rate is effective, or the probability of the presence of such fragments is exponentially decreasing with rate C(v)<0C(v)<0, for some concave function CC. In a recent paper (Krell (2008) [21]), N. Krell considered fragments whose sizes decrease at exact   exponential rates, i.e. whose sizes are confined to be of order e−vs for every s≤ts≤t. In that setting, she characterized the effective rates. In the present paper we continue this analysis and focus on the probabilities of presence, using the spine method and a suitable martingale. For the sake of completeness, we compare our results with those obtained in the standard approach ( [3] and [12]).