Stochastic Fragmentation and Some Sufficient Conditions for Shattering Transition

Endogenous Downward Jump Diffusion and Blow Up Phenomena Before Crash

“…At this point, we do not have clear intuition as to why ln x is the borderline of the phase transition, but we have a similar result in a different model of Jeon [11]. Now we add a Brownian noise to (3.1), that is, consider the following SDE:…”

Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation equations with mass loss

“…However, there are numerous cases when this is not the case: the mass can be consumed by an external reaction causing the surface to recede continuously, finally destroying the bridges between different parts of a particle and triggering the break up, and also, in the case of heterogeneous particles, due to explosive chemical reactions whenever the reactant is exposed either by fragmentation or by surface recession [12,17]. In both, conservative and mass loss, cases it has been observed [11,12,14,17,18,21,24] that the process can be dishonest. In most papers the analysis was carried out only for coefficients of a special form and the phenomenon was termed the "shattering fragmentation" and attributed to the phase transition and formation of a "dust" of zero-size particles (the corresponding "converse" phenomenon in the coagulation processes is called gelation and is attributed to the formation of a gel consisting of infinitely many particles).…”

“…Such Markov processes are well-known in the probability theory and are referred to as dishonest [2] or explosive [22]. This phenomenon is much less understood from the functional-analytic point of view and though a number of scattered results, often limited to particular applications, can be found in the literature [1,3,4,15,[18][19][20]23], the systematic study has been initiated quite recently in a series of papers [6,7,9,10,16]. In many cases in the models there is a mechanism that allows Q to decrease.…”

On two fragmentation schemes with algebraic splitting probability

“…However, in the subsequent years, this model received considerable attention as it was responsible for the shattering transition (formation of dust), a phenomenon considered by Ziff as the counterpart of gelation phenomenon (infinite cluster formation) in coagulation processes. See McGrady and Ziff (1987) and Jeon (2002), for example. In contrast with our model, size-dependence of successive fragmentations appears here in the rates, not in the splitting probability.…”