Diffusions on a space of interval partitions: the two-parameter model

Abstract

We introduce and study interval partition diffusions with Poisson–Dirichlet$(\mathit{\alpha },\mathit{\theta })$ stationary distribution for parameters $\mathit{\alpha }\in (0,1)$ and $\mathit{\theta }\ge 0$. This extends previous work on the cases $(\mathit{\alpha },0)$ and $(\mathit{\alpha },\mathit{\alpha })$ and builds on our recent work on measure-valued diffusions. Our methods for dealing with general $\mathit{\theta }\ge 0$ allow us to strengthen previous work on the special cases to include initial interval partitions with dust. In contrast to the measure-valued setting, we can show that this extended process is a Feller process improving on the Hunt property established in that setting. These processes can be viewed as diffusions on the boundary of a branching graph of integer compositions. Indeed, by studying their infinitesimal generator on suitable quasi-symmetric functions, we relate them to diffusions obtained as scaling limits of composition-valued up-down chains.