53 research outputs found
Asymptotic behavior of solutions of the fragmentation equation with shattering: An approach via self-similar Markov processes
The subject of this paper is a fragmentation equation with nonconservative
solutions, some mass being lost to a dust of zero-mass particles as a
consequence of an intensive splitting. Under some assumptions of regular
variation on the fragmentation rate, we describe the large time behavior of
solutions. Our approach is based on probabilistic tools: the solutions to the
fragmentation equation are constructed via nonincreasing self-similar Markov
processes that continuously reach 0 in finite time. Our main probabilistic
result describes the asymptotic behavior of these processes conditioned on
nonextinction and is then used for the solutions to the fragmentation equation.
We note that two parameters significantly influence these large time behaviors:
the rate of formation of "nearly-1 relative masses" (this rate is related to
the behavior near 0 of the L\'evy measure associated with the corresponding
self-similar Markov process) and the distribution of large initial particles.
Correctly rescaled, the solutions then converge to a nontrivial limit which is
related to the quasi-stationary solutions of the equation. Besides, these
quasi-stationary solutions, or, equivalently, the quasi-stationary
distributions of the self-similar Markov processes, are fully described.Comment: Published in at http://dx.doi.org/10.1214/09-AAP622 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Asymptotics of heights in random trees constructed by aggregation
To each sequence of positive real numbers we associate a growing
sequence of continuous trees built recursively by gluing at step a
segment of length on a uniform point of the pre-existing tree, starting
from a segment of length . Previous works on that model focus on the
influence of on the compactness and Hausdorff dimension of the limiting
tree. Here we consider the cases where the sequence is regularly
varying with a non-negative index, so that the sequence exploses. We
determine the asymptotics of the height of and of the subtrees of
spanned by the root and points picked uniformly at random and
independently in , for all
Self-similar scaling limits of non-increasing Markov chains
We study scaling limits of non-increasing Markov chains with values in the
set of non-negative integers, under the assumption that the large jump events
are rare and happen at rates that behave like a negative power of the current
state. We show that the chain starting from and appropriately rescaled,
converges in distribution, as , to a non-increasing
self-similar Markov process. This convergence holds jointly with that of the
rescaled absorption time to the time at which the self-similar Markov process
reaches first 0. We discuss various applications to the study of random walks
with a barrier, of the number of collisions in -coalescents that do
not descend from infinity and of non-consistent regenerative compositions.
Further applications to the scaling limits of Markov branching trees are
developed in our paper, Scaling limits of Markov branching trees, with
applications to Galton--Watson and random unordered trees (2010).Comment: Published in at http://dx.doi.org/10.3150/10-BEJ312 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Scaling limits of k-ary growing trees
For each integer , we introduce a sequence of -ary discrete
trees constructed recursively by choosing at each step an edge uniformly among
the present edges and grafting on "its middle" new edges. When ,
this corresponds to a well-known algorithm which was first introduced by
R\'emy. Our main result concerns the asymptotic behavior of these trees as
becomes large: for all , the sequence of -ary trees grows at speed
towards a -ary random real tree that belongs to the family of
self-similar fragmentation trees. This convergence is proved with respect to
the Gromov-Hausdorff-Prokhorov topology. We also study embeddings of the
limiting trees when varies
Scaling limits of Markov branching trees with applications to Galton-Watson and random unordered trees
We consider a family of random trees satisfying a Markov branching property.
Roughly, this property says that the subtrees above some given height are
independent with a law that depends only on their total size, the latter being
either the number of leaves or vertices. Such families are parameterized by
sequences of distributions on partitions of the integers that determine how the
size of a tree is distributed in its different subtrees. Under some natural
assumption on these distributions, stipulating that "macroscopic" splitting
events are rare, we show that Markov branching trees admit the so-called
self-similar fragmentation trees as scaling limits in the
Gromov-Hausdorff-Prokhorov topology. The main application of these results is
that the scaling limit of random uniform unordered trees is the Brownian
continuum random tree. This extends a result by Marckert-Miermont and fully
proves a conjecture by Aldous. We also recover, and occasionally extend,
results on scaling limits of consistent Markov branching models and known
convergence results of Galton-Watson trees toward the Brownian and stable
continuum random trees.Comment: Published in at http://dx.doi.org/10.1214/11-AOP686 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Random trees constructed by aggregation
We study a general procedure that builds random -trees by gluing
recursively a new branch on a uniform point of the pre-existing tree. The aim
of this paper is to see how the asymptotic behavior of the sequence of lengths
of branches influences some geometric properties of the limiting tree, such as
compactness and Hausdorff dimension. In particular, when the sequence of
lengths of branches behaves roughly like for some , we show that the limiting tree is a compact random tree of Hausdorff
dimension . This encompasses the famous construction of the
Brownian tree of Aldous. When , the limiting tree is thinner and its
Hausdorff dimension is always 1. In that case, we show that
corresponds to the dimension of the set of leaves of the tree.Comment: To appear in Annales de l'Institut Fourie
Regularity of formation of dust in self-similar fragmentations
Dans les fragmentations auto-similaires d'indice négatif, les fragments se brisent d'autant plus vite que leur masse est petite, de telle sorte que la fragmentation s'emballe et réduit de la masse à l'état de poussière. On s'intéresse ici à la régularité de la formation de la poussière. Soit M(t) la masse de la poussière au temps t. On donne des conditions suffisantes et des conditions nécessaires pour que la mesure dM soit absolument continue par rapport à la mesure de Lebesgue. Lorsque c'est le cas, on approxime la densité par des fonctions dépendant des petits fragments. On étudie également la dimension de Hausdorff de la mesure dM et de son support, ainsi que la continuité Hölderienne de la masse de la poussière M.In self-similar fragmentations with a negative index, fragments split even faster as their mass is smaller, so that the fragmentation runs away and some mass is reduced to dust. Our purpose is to investigate the regularity of this formation of dust. Let M(t) denote the mass of dust at time t. We give some sufficient and some necessary conditions for the measure dM to be absolutely continuous. In case of absolute continuity, we obtain an approximation of the density by functions of small fragments. We also study the Hausdorff dimension of dM and of its support, as well as the Hölder-continuity of the dust's mass M.ou
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