Size-biased diffusion limits and the inclusion process

Abstract

We consider the inclusion process on the complete graph with vanishing diffusivity, which leads to condensation of particles in the thermodynamic limit. Describing particle configurations in terms of size-biased and appropriately scaled empirical measures of mass distribution, we establish convergence in law of the inclusion process to a measure-valued Markov process on the space of probability measures. In the case where the diffusivity vanishes like the inverse of the system size, the derived scaling limit is equivalent to the well known Poisson-Dirichlet diffusion, offering an alternative viewpoint on these well-established dynamics. Moreover, our novel size-biased approach provides a robust description of the dynamics, which covers all scaling regimes of the system parameters and yields a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. We also discuss in detail connections to known results on related Fleming-Viot processes.