Continuum space limit of the genealogies of interacting Fleming-Viot processes on $\mathbb{Z}$

Abstract

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.

Stochastic Evolution of Genealogies of Spatial Populations: State Description, Characterization of Dynamics and Properties

“…This means that in the recurrent case we have mono-ancestor and mono-type populations developing in any finite spatial window whereas in the transient case, locally we have coexistence of descendants of different ancestors and, if θ is not the point measure we also have coexistence of different types. This is proved in [GKW18] for the more general case of Λ-Fleming-Viot models, see [GSW16] for information on spatial tree-valued Fleming-Viot.…”

“…Mutation and selection have been included in [DGP12] leading to Í Á 1 -valued processes. Spatial models giving Í -valued Fleming-Viot processes were considered in [GSW16]. Both the latter are successively explained next in two paragraphs.…”

The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

“…whereẆ is space-time white noise and σ 2 = m. A related convergence result was proved for the long range voter model in [MT95], and for the spatial Λ-Fleming-Viot process in [EVY18]. A corresponding convergence result regarding the underlying genealogies was also proved in [GSW16].…”

“…This is done by applying the ergodic theorem to an auxiliary process which is defined in Subsection 3.2. Durrett and Restrepo [DR08] already showed that, in the stepping stone model with uniform population sizes and in the long range voter model, the time spent together by two lineages before they coalesce is asymptotically exponential (see also [Mar71] and [GSW16]). Theorem 2.8 extends this result to the stepping stone model in a random environment.…”

Branching trees I: concatenation and infinite divisibility

“…Topology of the state space for spatial models In order to incorporate genealogies in spatial models it is necessary to generalize the concept of ultrametric measure spaces to marked ultrametric measure spaces. How to do this has been developed in [DGP11] and for infinite total population size in [GSW16]. We recall the idea.…”