# SuperBrownian motion and the spatial Lambda-Fleming-Viot process

## TL;DR

This paper establishes that the spatial Lambda-Fleming-Viot process can be approximated by superBrownian motion for rare subpopulations, extending classical results and showing how to recover stable branching behaviors through appropriate dynamics.

## Contribution

It introduces a spatial analogue of classical diffusion approximations, demonstrating convergence of the SLFV to superBrownian motion and stable branching processes under suitable conditions.

## Key findings

- Subpopulation dynamics approximated by superBrownian motion
- SLFV can be tuned to recover stable branching processes
- Results extend classical diffusion approximations to spatial models

## Abstract

It is well known that the dynamics of a subpopulation of individuals of a rare type in a Wright-Fisher diffusion can be approximated by a Feller branching process. Here we establish an analogue of that result for a spatially distributed population whose dynamics are described by a spatial Lambda-Fleming-Viot process (SLFV). The subpopulation of rare individuals is then approximated by a superBrownian motion. This result mirrors Cox et al. (2000), where it is shown that when suitably rescaled, sparse voter models converge to superBrownian motion. We also prove the somewhat more surprising result, that by choosing the dynamics of the SLFV appropriately we can recover superBrownian motion with stable branching in an analogous way. This is a spatial analogue of (a special case of) results of Bertoin and Le Gall (2006), who show that the generalised Fleming-Viot process that is dual to the beta-coalescent, when suitably rescaled, converges to a continuous state branching process with stable branching mechanism.