The Two-parameter Poisson-Dirichlet Diffusion Process

Abstract:

The two-parameter Poisson-Dirichlet point process, ${\cal PD}(\alpha,\theta)$, has random points

$x_{1}>x_{2} > \cdots >0$ with $\sum_{i=1}^\infty x_{i}=1$. The parameters $0 \leq \alpha <1$ and $\theta > - \alpha$. If $\alpha=0$ this is a Dirichet point process which appears in many settings, particularly as the distribution of frequencies of gene types in Mathematical Population Genetics. Thinking of $\{x_{i}\}_{i=1}^\infty$ as a random probability distribution, the sampling distribution partitions of $n\geq 1$ characterize the distribution of the process.

Petrov (2009) introduced the two-parameter Poisson-Dirichlet infinite-dimensional diffusion process with generator

${\cal L}_{\alpha,\theta} = \frac{1}{2}\sum_{i,j=1}^\infty x_i(\delta_{ij}-x_j)\frac{\partial^2}{\partial x_i\partial x_j}-\frac{1}{2}\sum_{i=1}^\infty (\theta x_i+\alpha)\frac{\partial}{\partial x_i},$

which has a stationary ${\cal PD}(\alpha,\theta)$ distribution. Feng et. al (2011) obtain a transition density expansion in terms of orthogonal polynomials in $\{x_{i}\}_{i=1}^\infty$. Rearrangement of the expansion by Zhou (2015) suggests a line-of-descent dual process back in time. Remarkably the process does not depend on $\alpha$. If $\alpha=0$ the dual process has an interpretation as the number of non-mutant lineages in a Kingman coalescent. In this death process lineages are lost by coalescence or mutation at rate $\frac{1}{2}n(n-1) + \frac{1}{2}n\theta$ while $n$ lineages. If $\alpha=0$ Hoppé's urn model generates a partition distribution in a sample of $n$. When $\alpha\ne 0$ there is a connection with a generalized Blackwell and MacQueen Pólya urn scheme. The colloquium will discuss ${\cal PD}(\alpha,\theta)$ background and how the dual process works in the diffusion. This research is joint with Matteo Ruggiero, Dario Spanó, and Youzhou Zhou.

Bio:

Robert Griffiths is an Adjunct Professor of Mathematics at Monash University. He was an academic at Monash University for 25 years from 1973, then at the University of Oxford from 1998 for 20 years, returning to Australia in 2018. His main research interest is in Stochastic Processes in Mathematical Population Genetics. He was elected as a Fellow of the Royal Society in 2010 for his contributions to Mathematical Genetics.