The Brownian Spatial Coalescent

We introduce a class of Markov coalescent processes on the continuous $d$-dimensional torus, in the most general setting of simultaneous multiple mergers, called the Brownian spatial coalescent. It is axiomatically defined through a property that is satisfied by the genealogies of any population model in which individuals follow independent Brownian motions forwards in time, regardless of the branching mechanism. We prove that a Brownian spatial coalescent is characterised by a set of "transition measures", reminiscent of the transition rates that characterise a non-spatial coalescent. We prove that it is sampling consistent in a suitable sense if and only if all transition measures are uniform with intensity given by the transition rates of a $Ξ$-coalescent. This defines the "Brownian spatial $Ξ$-coalescent", which we show describes the genealogies of neutral population models with Brownian movement in the limit of large population size, and in particular those of the $Ξ$-Fleming-Viot process - a generalisation of the well-known Fleming-Viot process - at stationarity. An important consequence of our results is that all spatial population models in which individuals follow independent Brownian motions and the branching mechanism is not neutral, that is, 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, 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.