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

TL;DR

This paper introduces and analyzes a new class of interval partition diffusions with Poisson-Dirichlet stationary distribution, extending previous models to include a broader parameter range and initial conditions, and establishing their mathematical properties.

Contribution

It extends existing diffusion models on interval partitions to include general  and dust initial conditions, and proves these processes are Feller, linking them to boundary diffusions of branching graphs.

Findings

The process has Poisson-Dirichlet(,) stationary distribution.

The extended process is a Feller process, improving on Hunt properties.

Connections are established between these diffusions and scaling limits of composition chains.

Abstract

We introduce and study interval partition diffusions with Poisson--Dirichlet$(\alpha,\theta)$ stationary distribution for parameters $\alpha\in(0,1)$ and $\theta\ge 0$. This extends previous work on the cases $(\alpha,0)$ and $(\alpha,\alpha)$ and builds on our recent work on measure-valued diffusions. Our methods for dealing with general $\theta\ge 0$ 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.