We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov's idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows-provided the sequence is tight-from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultra-metric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra-)metric measure spaces given by the random genealogies of the -coalescents. We show that the -coalescent defines an infinite (random) metric measure space if and only if the so-called "dust-free"-property holds.
Abstract. The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N → ∞ of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous-Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N → ∞ of the Aldous-Broder chain. A key technical ingredient in this work is the use of a pointed Gromov-Hausdorff distance to metrize the space of rooted compact real trees.
The measure-valued Fleming-Viot process is a diffusion which models the evolution of allele frequencies in a multi-type population. In the neutral setting the Kingman coalescent is known to generate the genealogies of the "individuals" in the population at a fixed time. The goal of the present paper is to replace this static point of view on the genealogies by an analysis of the evolution of genealogies.We encode the genealogy of the population as an (isometry class of an) ultra-metric space which is equipped with a probability measure. The space of ultra-metric measure spaces together with the Gromovweak topology serves as state space for tree-valued processes. We use well-posed martingale problems to construct the tree-valued resampling dynamics of the evolving genealogies for both the finite population Moran model and the infinite population Fleming-Viot diffusion.We show that sufficient information about any ultra-metric measure space is contained in the distribution of the vector of subtree lengths obtained by sequentially sampled "individuals". We give explicit formulas for the evolution of the Laplace transform of the distribution of finite subtrees under the tree-valued Fleming-Viot dynamics.
We use Dirichlet form methods to construct and analyze a reversible Markov process, the stationary distribution of which is the Brownian continuum random tree. This process is inspired by the subtree prune and regraft (SPR) Markov chains that appear in phylogenetic analysis.A key technical ingredient in this work is the use of a novel Gromov-Hausdorff type distance to metrize the space whose elements are compact real trees equipped with a probability measure. Also, the investigation of the Dirichlet form hinges on a new path decomposition of the Brownian excursion.
We study the evolution of genealogies of a population of individuals, whose type frequencies result in an interacting Fleming-Viot process on Z. We construct and analyze the genealogical structure of the population in this genealogy-valued Fleming-Viot process as a marked metric measure space, with each individual carrying its spatial location as a mark. We then show that its time evolution converges to that of the genealogy of a continuum-sites stepping stone model on R, if space and time are scaled diffusively. We construct the genealogies of the continuum-sites stepping stone model as functionals of the Brownian web, and furthermore, we show that its evolution solves a martingale problem. The generator for the continuum-sites stepping stone model has a singular feature: at each time, the resampling of genealogies only affects a set of individuals of measure 0. Along the way, we prove some negative correlation inequalities for coalescing Brownian motions, as well as extend the theory of marked metric measure spaces (developed recently by Depperschmidt, Greven and Pfaffelhuber [DGP12]) from the case of probability measures to measures that are finite on bounded sets.AMS 2010 subject classification: 60K35, 60J65, 60J70, 92D25. 1(1.6) (X n , r n , ψ k · µ n ) =⇒ n→∞ (X, r, ψ k · µ) in the Gromov-weak topology for each k ∈ N.When V = R d , we may choose ψ k to be infinitely differentiable. Remark 1.3 (Dependence on o and (ψ k ) k∈N ). Note that the V -marked Gromov-weak # topology does not depend on the choice o ∈ V and the sequence (ψ k ) k∈N , as long as ψ k has bounded support and A k := {v : ψ k (v) = 1} increases to V as k → ∞.Remark 1.4 (M V as a subspace of (M V f ) N ). Let M V f denote the space of (equivalent classes of) V -mmm spaces with finite measures, equipped with the V -marked Gromovweak topology as introduced in [DGP11, Def. 2.4]. Then it is a well-known fact that each element (X, r, µ) ∈ M V can be identified with a sequence ((X, r, ψ 1 · µ), (X, r, ψ 2 · µ), . . .) in the product space (M V f ) N , equipped with the product topology. This identification allows us to easily deduce many properties of M V from properties of M V f that were established in [DGP11]. In particular, we can metrize the V -marked Gromov-weak # topology on M V by introducing a metric (which can be called V -marked Gromov-Prohorov # metric) (1.7)where d M GP is the marked Gromov-Prohorov metric on M V f , which was introduced in [DGP11, Def. 3.1] and metrizes the marked Gromov-weak topology.The proof of the following result is in Appendix A.Theorem 1.5 (Polish space). The space M V , equipped with the V -marked Gromov-weak # topology, is a Polish space.
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