Martingales and rates of presence in homogeneous fragmentations

• Published in: Stochastic processes and their applications Vol. 121; no. 1; pp. 135 - 154
• Main Authors: Krell, N., Rouault, A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 2011; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Exact sciences and technology; Fragmentation; Fragmentation Lévy process Martingales Probability tilting; Lévy process; Martingales; Mathematics; Probability; Probability and statistics; Probability theory and stochastic processes; Probability tilting; Sciences and techniques of general use; Stochastic analysis; Stochastic processes; 60J85; Lévy process; 60G09; 65J25; Probability tilting; Martingales; Fragmentation; Random walk; Asymptotic behavior; Martingale; Probability; Probability distribution; Stochastic process; Concave function; Completeness; Levy process; fragmentation; probability tilting; martingales
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2010.09.005

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 > 0 , either the number of fragments whose sizes at time t are of order e − v t 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 ) < 0 , for some concave function C . 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 − v s for every s ≤ 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 ( [3,12]).