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The stable fragmentation with index of self-similarity $alpha in [-1/2,0)$ is derived by looking at the masses of the subtrees formed by discarding the parts of a $(1 + alpha)^{-1}$--stable continuum random tree below height $t$, for $t geq 0$. We give a detailed limiting description of the distribution of such a fragmentation, $(F(t), t geq 0)$, as it approaches its time of extinction, $zeta$. In particular, we show that $t^{1/alpha}F((zeta - t)^+)$ converges in distribution as $t to 0$ to a non-trivial limit. In order to prove this, we go further and describe the limiting behavior of (a) an excursion of the stable height process (conditioned to have length 1) as it approaches its maximum; (b) the collection of open intervals where the excursion is above a certain level and (c) the ranked sequence of lengths of these intervals. Our principal tool is excursion theory. We also consider the last fragment to disappear and show that, with the same time and space scalings, it has a limiting distribution given in terms of a certain size-biased version of the law of $zeta$.
We consider weighted random balls in $real^d$ distributed according to a random Poisson measure with heavy-tailed intensity and study the asymptotic behaviour of the total weight of some configurations in $real^d$. This procedure amounts to be very r
In this paper, we consider the explicit wave-breaking mechanism and its dynamical behavior near this singularity for the generalized b-equation. This generalized b-equation arises from the shallow water theory, which includes the Camassa-Holm equatio
The logistic birth and death process is perhaps the simplest stochastic population model that has both density-dependent reproduction, and a phase transition, and a lot can be learned about the process by studying its extinction time, $tau_n$, as a f
We consider the extinction time of the contact process on increasing sequences of finite graphs obtained from a variety of random graph models. Under the assumption that the infection rate is above the critical value for the process on the integer li
We consider the heat flow of corotational harmonic maps from $mathbb R^3$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic