On a Voter model on R^d: Cluster growth in the Spatial Lambda-Fleming-Viot Process

The spatial Lambda-Fleming-Viot (SLFV) process (Barton, Etheridge and V\'eber, 2010) can be seen as a generalised Voter Model with configuration space $M^{R^d}$, where M is the set of probability measures on some space K. Such processes are usually studied thanks to a dual process that describes the genealogy of a sample of particles. In this paper, we study the growth of a cluster, and our surprising result is that with probability one, every bounded cluster stops growing in finite time. Because the traditional (backward in time) duality methods fail, we develop an original forward in time approach that exploits a martingale property of the process. To make it feasible, we construct adequate objects that allow to handle the complex geometry of the problem. We are able to prove the result in any dimension $d$.