Trees within trees II: Nested fragmentations

Duchamps, Jean-Jil

doi:10.1214/19-aihp999

Cited by 6 publications

(7 citation statements)

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“…From a mathematical point of view as well, SNEC processes open up the door to many possible new investigations. For example some of us are currently studying the speed of coming down from infinity of SNEC processes [8,28] as well as similar extensions [15] to fragmentation processes [4]. It will be interesting to investigate how the nested trees generated by SNEC processes can be cast in the frameworks of multilevel measure-valued processes [7,11] and flows of bridges [5,6] as well as of exchangeable combs [20,26].…”

Section: Introductionmentioning

confidence: 99%

Trees within trees: simple nested coalescents

Blancas¹,

Duchamps²,

Lambert³

et al. 2018

Electron. J. Probab.

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We consider the compact space of pairs of nested partitions of N, where by analogy with models used in molecular evolution, we call "gene partition" the finer partition and "species partition" the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of Λ-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo Λ-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. Λ-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure ν s on (0, 1] × M 1 ([0, 1]) providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity.

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Section: Introductionmentioning

confidence: 99%

Trees within trees: simple nested coalescents

Blancas¹,

Duchamps²,

Lambert³

et al. 2018

Electron. J. Probab.

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“…As a first result about ESSF processes, let us show a projective Markov property. It is very analogous to [9, Lemma 3.2] and [13, Proposition 2], but we need another statement in the present context.…”

Section: Extended Self-similar Fragmentationsmentioning

confidence: 68%

Fragmentations with self-similar branching speeds

Duchamps

2021

Adv. Appl. Probab.

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We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincaré Prob. Statist.38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy–Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.

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“…Remark 3.5. It seems plausible to generalize this construction to species Λ-coalescents and even to more general nested coalescents (see Duchamps, 2020) considering the Poisson processes governing the migration to be exchangeable instead of independent.…”

Section: Moment Dualitymentioning

confidence: 99%