Stochastic partial differential equations describing neutral genetic diversity under short range and long range dispersal

In this paper, we consider a mathematical model for the evolution of neutral genetic diversity in a spatial continuum including mutations, genetic drift and either short range or long range dispersal. The model we consider is the spatial $\Lambda$-Fleming-Viot process introduced by Barton, Etheridge and V\'eber, which describes the state of the population at any time by a measure on $\R^d \times [0,1]$, where $\R^d$ is the geographical space and $[0,1]$ is the space of genetic types. In both cases (short range and long range dispersal), we prove a functional central limit theorem for the process as the population density becomes large and under some space-time rescaling. We then deduce from these two central limit theorems a formula for the asymptotic probability of identity of two individuals picked at random from two given spatial locations. In the case of short range dispersal, we recover the classical Wright-Mal\'ecot formula, which is widely used in demographic inference for spatially structured populations. In the case of long range dispersal we obtain a new formula which could open the way for a better appraisal of long range dispersal in inference methods.