Alpha-diversity processes and normalized inverse-Gaussian diffusions

TL;DR

This paper introduces a new class of infinite-dimensional diffusions linked to normalized inverse-Gaussian measures, modeling the evolution of type frequencies and mutation rates in complex population dynamics.

Contribution

It develops a novel family of diffusions associated with Gibbs partitions induced by normalized inverse-Gaussian measures, extending existing models and providing explicit population dynamics interpretations.

Findings

Derived the infinitesimal generator of the new diffusion process.

Showed the diffusion as a limit of Feller diffusions with finitely-many types.

Provided a discrete particle process representation.

Abstract

The infinitely-many-neutral-alleles model has recently been extended to a class of diffusion processes associated with Gibbs partitions of two-parameter Poisson-Dirichlet type. This paper introduces a family of infinite-dimensional diffusions associated with a different subclass of Gibbs partitions, induced by normalized inverse-Gaussian random probability measures. Such diffusions describe the evolution of the frequencies of infinitely-many types together with the dynamics of the time-varying mutation rate, which is driven by an alpha-diversity diffusion. Constructed as a dynamic version, relative to this framework, of the corresponding notion for Gibbs partitions, the latter is explicitly derived from an underlying population model and shown to coincide, in a special case, with the diffusion approximation of a critical Galton-Watson branching process. The class of infinite-dimensional processes is characterized in terms of its infinitesimal generator on an appropriate domain, and shown to be the limit in distribution of a certain sequence of Feller diffusions with finitely-many types. Moreover, a discrete representation is provided by means of appropriately transformed Moran-type particle processes, where the particles are samples from a normalized inverse-Gaussian random probability measure. The relationship between the limit diffusion and the two-parameter model is also discussed.