Trees within trees II: Nested Fragmentations

Abstract

Similarly as in [4] where nested coalescent processes are studied, we generalize the definition of partition-valued homogeneous Markov fragmentation processes to the setting of nested partitions, i.e. pairs of partitions (ζ, ξ) where ζ is finer than ξ. As in the classical univariate setting, under exchangeability and branching assumptions, we characterize the jump measure of nested fragmentation processes, in terms of erosion coefficients and dislocation measures. Among the possible jumps of a nested fragmentation, three forms of erosion and two forms of dislocation are identified -one of which being specific to the nested setting and relating to a bivariate paintbox process.

Trees within trees: simple nested coalescents

“…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].…”

Fragmentations with self-similar branching speeds

“…By construction, the joint distribution of ( ξn , ξn+1 ) is equal to the one we get from the original process X, and it should now be clear that the point process of jumps of ξ n is equal in distribution to the point process of jumps of ξ n+1 with additional jumps distributed as J = log 1 , arising at rate (J n+1 − J n ). Note that by construction, J has distribution η n+1 , so finally we have proven (12). The fact that (λ n , n ≥ 1) is a nonincreasing sequence of σ-finite measures ensures the existence of a limiting measure λ ∞ on \ {0} such that for all n ∈ ,…”

“….) ∈ M ⋆ ∞ , (c) D • ∩{π 1 and |π| ↓ 1} = ∫ Z ↓ ̺ z (•) Λ ′ (dz),We use similar arguments as in [5, Theorem 3.1], as we have already done in the context of nested fragmentations[12, Proposition 19]. First note that by (16), D-a.e.…”

Particle systems with coordination

“…The nested coalescent is an object introduced recently [4], which has already received some attention [5,8,20]. Its purpose is to integrate speciation events and individual reproduction in the same model, in order to be able to trace ancestry at the level of species.…”

“…(3.17) Remark 1. It seems plausible to generalize this construction to species Λ-coalescents and even to more general nested coalescents (see [8]) considering the Poisson processes governing the migration to be exchangeable instead of independent.…”