Martingales and rates of presence in homogeneous fragmentations

AbstractThe 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>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 the presence of such fragments is exponentially decreasing with rate C(v)