Hydrodynamic limit for a Fleming–Viot type system

We consider a system of N Brownian particles evolving independently in a domain D. As soon as one particle reaches the boundary it is killed and one of the other particles is chosen uniformly and splits into two independent particles resuming a new cycle of independent motion until the next boundary hit. We prove the hydrodynamic limit for the joint law of the empirical measure process and the average number of visits to the boundary as N approaches infinity.