Alpha-diversity processes and normalized inverse-Gaussian diffusions

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 finitelymany

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.