Loss of mass in deterministic and random fragmentations

We consider a linear rate equation, depending on three parameters, that model fragmentation. For each of these fragmentation equations, there is a corresponding stochastic model, from which we construct an explicit solution to the equation. This solution is proved unique. We then use this solution to obtain criteria for the presence or absence of loss of mass in the fragmentation equation, as a function of the equation parameters. Next, we investigate small and large times asymptotic behavior of the total mass for a wide class of parameters. Finally, we study the loss of mass in the stochastic models.

[1]  Jean Bertoin,et al.  Self-similar fragmentations , 2002 .

[2]  Intae Jeon,et al.  Existence of Gelling Solutions for Coagulation-Fragmentation Equations , 1998 .

[3]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[4]  Marc Yor,et al.  On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes , 2002 .

[5]  D. Aldous.  Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists , 1999 .

[6]  On small masses in self-similar fragmentations , 2004 .

[7]  A. F. Filippov.  On the Distribution of the Sizes of Particles which Undergo Splitting , 1961 .

[8]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[9]  STOCHASTIC FRAGMENTATION AND SOME SUFFICIENT CONDITIONS FOR SHATTERING TRANSITION , 2002 .

[10]  Edwards,et al.  Rate equation and scaling for fragmentation with mass loss. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[11]  L. Rogers,et al.  Diffusions, Markov processes, and martingales , 1979 .

[12]  M. Bartlett,et al.  Markov Processes and Potential Theory , 1972, The Mathematical Gazette.

[13]  J. Norris,et al.  Smoluchowski's coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic coalescent , 1998, math/9801145.

[14]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[15]  N. Kôno.  Tail probabilities for positive random variables satisfying some moment conditions , 1977 .

[16]  D. P. Kennedy,et al.  The distribution of the maximum Brownian excursion , 1976, Journal of Applied Probability.

[17]  N J Kokholm.  On Smoluchowski's coagulation equation , 1988 .

[18]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[19]  J. Bertoin.  Subordinators: Examples and Applications , 1999 .

[20]  J. Bertoin.  Homogeneous fragmentation processes , 2001 .

[21]  V. Rivero.  A law of iterated logarithm for increasing self-similar Markov processes , 2003 .

[22]  J. Bertoin.  The asymptotic behavior of fragmentation processes , 2003 .

[23]  N. Fournier,et al.  On Small Particles in Coagulation-Fragmentation Equations , 2003 .