Existence of mark functions in marked metric measure spaces

Kliem, Sandra; Loehr, Wolfgang

doi:10.1214/ejp.v20-3969

Cited by 16 publications

(16 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, t ∈ R + ) may be called an M-valued Ξ-Cannings process. In the case without simultaneous multiple reproduction events, it coincides with the tree-valued Λ-Cannings process discussed in [27,Section 4.2], and in the case without multiple reproduction events with the tree-valued Moran process from [19,Definition 2.19]. This can be seen by an application of [37, Theorem 2] similarly to the proof of Lemma 6.7 below, see also Section 2 of [8].…”

Section: Stochastic Processes 41 the Case Without Dustmentioning

confidence: 53%

Pathwise construction of tree-valued Fleming-Viot processes

Gufler¹

2018

Electron. J. Probab.

View full text Add to dashboard Cite

In a random complete and separable metric space that we call the lookdown space, we encode the genealogical distances between all individuals ever alive in a lookdown model with simultaneous multiple reproduction events. We construct families of probability measures on the lookdown space and on an extension of it that allows to include the case with dust. From this construction, we read off the tree-valued Ξ-Fleming-Viot processes and deduce path properties. For instance, these processes usually have a. s. càdlàg paths with jumps at the times of large reproduction events. In the case of coming down from infinity, the construction on the lookdown space also allows to read off a process with values in the space of measure-preserving isometry classes of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prohorov metric. This process has a. s. càdlàg paths with additional jumps at the extinction times of parts of the population.

show abstract

Section: Stochastic Processes 41 the Case Without Dustmentioning

confidence: 53%

Pathwise construction of tree-valued Fleming-Viot processes

Gufler¹

2018

Electron. J. Probab.

View full text Add to dashboard Cite

show abstract

“…In fact we are looking at the particular case where the mark distribution comes from a mark function. A criterion for the existence of the latter is the subject of [43].…”

Section: Definition 23 (Length Measure)mentioning

confidence: 99%

Invariance principles for random walks in random environment on trees

Andriopoulos

2021

Electron. J. Probab.

View full text Add to dashboard Cite

In [24] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the Gromov-Hausdorff-vague topology, and a certain uniform recurrence condition is satisfied. Such a theorem finds particularly nice applications if the resistance metric measure space is a metric measure tree. To illustrate this, we state functional limit theorems in old and new examples of suitably rescaled random walks in random environment on trees.First, we take a critical Galton-Watson tree conditioned on its total progeny and a non-lattice branching random walk on R d indexed by it. Then, conditional on that, we consider a biased random walk on the range of the preceding. Here, by non-lattice we mean that distinct branches of the tree do not intersect once mapped in R d . This excludes the possibility that the random walk on the range may jump from one branch to the other without returning to the most recent common ancestor. We prove, after introducing the bias parameter β n −1/4, for some β > 1, that the biased random walk on the range of a large critical non-lattice branching random walk converges to a Brownian motion in a random Gaussian potential on Aldous' continuum random tree (CRT).Our second new result introduces the scaling limit of the edge-reinforced random walk on a size-conditioned Galton-Watson tree with finite variance as a Brownian motion in a random Gaussian potential on the CRT with a drift proportional to the distance to the root.

show abstract

“…The function f is called mark function in the sense of e.g. [26]. The Z-component of m (0) s is m s given by ( 13) , and the I-component of m (0) s is the type distribution µ (0) s under the neutral transport, which in view of (39) and ( 6) obeys a.s.…”

Section: Neutral Lookdown Space and Marked Sampling Measuresmentioning

confidence: 99%

Evolving genealogies for branching populations under selection and competition

Blancas¹,

Gufler²,

Kliem³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

For a continuous state branching process with two types of individuals which are subject to selection and density dependent competition, we characterize the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE's. Our construction is achieved in the lookdown framework and provides a synthesis as well as a generalization of cases considered separately in two seminal papers by , namely fluctuating population sizes under neutrality, and selection with constant population size. As a conceptual core in our approach, we introduce the selective lookdown space which is obtained from its neutral counterpart through a state-dependent thinning of "potential" selection/competition events whose rates interact with the evolution of the type densities. The updates of the genealogical distance matrix at the "active" selection/competition events are obtained through an appropriate sampling from the selective lookdown space. The solution of the above mentioned system of SDE's is then mapped into the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions. By means of Kurtz's Markov mapping theorem, we characterize the latter process as the unique solution of a martingale problem. For the sake of transparency we restrict the main part of our presentation to a prototypical example with two types, which contains the essential features. In the final section we outline an extension to processes with multiple types including mutation.

show abstract

Existence of mark functions in marked metric measure spaces

Cited by 16 publications

References 12 publications

Pathwise construction of tree-valued Fleming-Viot processes

Pathwise construction of tree-valued Fleming-Viot processes

Invariance principles for random walks in random environment on trees

Evolving genealogies for branching populations under selection and competition

Contact Info

Product

Resources

About