Size-biased diffusion limits and the inclusion process

We study the Inclusion Process with vanishing diffusion coefficient, which is a stochastic particle system known to exhibit condensation and metastable dynamics for cluster locations. We focus on the dynamics of the empirical mass distribution and consider the process on the complete graph in the thermodynamic limit with fixed particle density. Describing a given configuration by a measure on a suitably scaled mass space, we establish convergence to a limiting measure-valued process. When the diffusion coefficient scales like the inverse of the system size, the scaling limit is equivalent to the well known Poisson-Dirichlet diffusion. Our approach can be generalized to other scaling regimes, providing a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. Considering size-biased mass distributions, our approach yields an interesting characterization of the limiting dynamics via duality.
This is joint work with Simon Gabriel (Münster) and Paul Chleboun (both Warwick).