  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.
