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 Fragmentation processes and Stochastic Shattering transition Bull. Korean Math. Soc. 2005 Vol. 42, No. 4, 855-867 Published online December 1, 2005 Intae Jeon Catholic University of Korea Abstract : Shattering or disintegration of mass is a well known phenomenon in fragmentation processes first introduced by Kolmogorov and Filippop and extensively studied by many physicists. Though the mass is conserved in each break-up, the total mass decreases in finite time. We investigate this phenomenon in the $n$ particle system. In this system, shattering can be interpreted such that, in uniformly bounded time on $n$, order $n$ of mass is located in order $o(n)$ of clusters. It turns out that the tagged particle processes associated with the systems are useful tools to analyze the phenomenon. For the newly defined stochastic shattering based on the above ideas, we derive far sharper conditions of fragmentation kernels which guarantee the occurrence of such a phenomenon than our previous work [9]. Keywords : fragmentation process, stochastic shattering, tagged particle process MSC numbers : Primary 60K35; Secondary 82C22 Full-Text :