The Brownian Spatial Coalescent

TL;DR

We introduce the Brownian spatial coalescent, a universal class of spatial coalescents on the $d$-torus in which lineages follow Brownian bridges backwards in time along the coalescent tree and are governed by finite transition measures. A complete characterization is given: sampling consistency holds if and only if these transition measures are Lebesgue multiples and reproduce the rates of a non-spatial $Ξ$- or $Λ$-coalescent, yielding the Brownian spatial $Ξ$-coalescent that describes genealogies of the $Ξ$-Fleming-Viot process at stationarity and, with resampling, its full time reversal. The framework reveals that many spatial population models with Brownian motion and nontrivial spatially dependent branching have non-Markovian genealogies, while providing explicit Wright–Malécot type formulas and a drift representation for lineages. Byproduct results include explicit stationary samples for the Fleming-Viot class, extensions to general spatial motions, and a rigorous link between forward neutral models and their backward genealogies in continuous space.

Abstract

We introduce the Brownian spatial coalescent, a class of Markov spatial coalescent processes on the $d$-dimensional torus in continuous space, that is axiomatically defined by the following property: conditional on the times and locations of all coalescence events, lineages follow independent Brownian bridges backwards in time along the branches of the coalescence tree. We prove that a Brownian spatial coalescent is characterised by a set of finite "transition measures" on the torus, in analogy to the transition rates that characterise a non-spatial coalescent. We prove that a Brownian spatial coalescent is sampling consistent, in a novel sense, if and only if all transition measures are constant multiples of Lebesgue measure, and the multiples are the transition rates of a $Ξ$-coalescent, or a $Λ$-coalescent if simultaneous mergers are not allowed. This defines the "Brownian spatial $Ξ$-coalescent", which we prove describes the genealogies of the $Ξ$-Fleming-Viot process - a generalisation of the well known Fleming-Viot process - at stationarity, and in fact describes its full time reversal if augmented with "resampling" steps at the times of coalescence events. An important consequence of our results is that all spatial population models in which individuals follow independent Brownian motions and the branching mechanism depends non-trivially on the spatial distribution, for example through local regulation, have non-Markovian genealogies. Byproducts of our results include explicit formulas for samples from the stationary distribution of a $Ξ$-Fleming-Viot process, Wright-Malecot type formulas, and a representation of the backward dynamics of lineages in terms of Brownian motions with coupled drift. This includes calculations of the drift that leads to multiple or even simultaneous mergers in any dimension.

Figures (6)

• Figure 1: Illustration of the finite measure on the space and time decorations of a particular genealogical tree shape. Spatial factors associated with branches of the tree are only shown for three of the seven branches.
• Figure 2: Illustration of the notation used for forests. In this example, $\mathop{\mathrm{ch}}\nolimits_F(\left\{ 1,3,4 \right\} ) = \left\{ \left\{ 1 \right\} ,\left\{ 3,4 \right\} \right\}$, and $\mathop{\mathrm{pr}}\nolimits_F(\left\{ 7 \right\}) = \left\{ 1,3,4,7 \right\}$, and $\mathop{\mathrm{ch}}\nolimits_F(\left\{ 9 \right\} ) = \mathop{\mathrm{pr}}\nolimits_F(\left\{ 9 \right\} ) = \emptyset$.
• Figure 3: At the top is an illustration of an element $\omega\in \Omega$. The map $\mathop{\mathrm{Dec}}\nolimits$ (bottom left) extracts the abstract coalescence forest, and times and spatial locations of the merge events. The $\mathrm{Path}$ maps (bottom right) extract the motion of particles along branches of the coalescence forest.
• Figure 4: There are three ways to extend a fixed tree to an additional leaf. Either the leaf $v_{\oplus}$ merges binary with another node $u_{\oplus}$, at a time at which no merge happens in the original tree (left), or simultaneously with an existing merge event (middle); or $v_{\oplus}$ joins an existing merge event (right). The additional degrees of freedom in the tree decoration given that of the underlying tree are, from right to left, none, the location $\xi_{\oplus}$, and the time $\tau_{\oplus}$ and location $\xi_{\oplus}$ of the additional merge event.
• Figure 5: Illustrations for the proof of \ref{['lem:nuconst2']}.

Theorems & Definitions (143)

• Definition 1.1
• Theorem 1.2: DK99pitmanlambdasagitovlambda
• Theorem 1.3: schweinsbergxihaploid
• Definition 1.4
• Definition 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8
• Example 1.9
• Remark 1.10