Dynamical models for the two-parameter Poisson-Dirichlet distribution and the Pitman-Yor process

Abstract

In this paper, we introduce and study a family of diffusion processes associated with the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution. The diffusion coefficients indexed by a non-negative parameter $\gamma$ are smaller than the corresponding one-parameter models in terms of quadratic forms or bilinear forms when $\gamma$ is positive. The well known Petrov's diffusion corresponds to $\gamma=0$ among the unlabelled diffusions. If $\gamma$ is the same as the stable parameter $\alpha$ in the Pitman-Yor process, we obtain both labelled and unlabelled reversible diffusion processes with the Pitman-Yor process and the two-parameter Poisson-Dirichlet distribution as the corresponding reversible measures. We construct these processes analytically through Dirichlet forms. In comparison with existing models in the literature, our models possess two fundamental new features. Firstly, our labelled model is characterized by an explicit generator, which is the first among all models studied so far. This makes it possible to establish the crucial integration-by-parts formula. Secondly, a novel foundational structure of the two-parameter distributions is the existence of a diversity index (a multiple of local time) for the positive stable parameter. By slowing down the diffusion in our model, the essential role of the diversity index is revealed in the evolution of the population. Additionally, we also obtain properties including ergodicity, path behaviour, and finite dimensional approximations.