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Register a new userDual process in the Two-parameter Poisson-Dirichlet diffusion
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The two-parameter Poisson-Dirichlet diffusion is an infinite-dimensional diffusion on the ordered simplex with a two-parameter Poisson-Dirichlet (alpha, theta) stationary distribution. We derive a dual process representation for the diffusion, suggested by Feng et al. (2011)s spectral expansion of the transition density, and its rearrangement by Zhou (2015). The dual process is in terms of a line-of-descent process which tracks the evolution of non-mutant frequencies from time zero. Remarkably the line-of-descent process does not depend on alpha. Methods of proof use the sampling distribution of n points in the two-parameter Poisson-Dirichlet diffusion. We connect the sampling distribution with a generalized Blackwell and MacQueen Polya urn model.
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The two parameter Poisson-Dirichlet distribution $PD(alpha,theta)$ is the distribution of an infinite dimensional random discrete probability. It is a generalization of Kingmans Poisson-Dirichlet distribution. The two parameter Dirichlet process $Pi_{alpha,theta, u_0}$ is the law of a pure atomic random measure with masses following the two parameter Poisson-Dirichlet distribution. In this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures $PD(alpha,theta)$ and $Pi_{alpha,theta, u_0}$. The methods used come from the theory of Dirichlet forms.
The two-parameter Poisson--Dirichlet diffusion, introduced in 2009 by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingmans one-parameter Poisson--Dirichlet distribution and to certain Fleming--Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here we shed some light on these dynamics by formulating a $K$-allele Wright--Fisher model for a population of size $N$, involving a uniform mutation pattern and a specific state-dependent migration mechanism. Suitably scaled, this process converges in distribution to a $K$-dimensional diffusion process as $Ntoinfty$. Moreover, the descending order statistics of the $K$-dimensional diffusion converge in distribution to the two-parameter Poisson--Dirichlet diffusion as $Ktoinfty$. The choice of the migration mechanism depends on a delicate balance between reinforcement and redistributive effects. The proof of convergence to the infinite-dimensional diffusion is nontrivial because the generators do not converge on a core. Our strategy for overcoming this complication is to prove textit{a priori} that in the limit there is no loss of mass, i.e., that, for each limit point of the sequence of finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one.
The two-parameter Poisson-Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and Gamma subordinators with the two-parameters, $alpha$ and $theta$, corresponding to the stable component and Gamma component respectively. The moderate deviation principles are established for the two-parameter Poisson-Dirichlet distribution and the corresponding homozygosity when $theta$ approaches infinity, and the large deviation principle is established for the two-parameter Poisson-Dirichlet distribution when both $alpha$ and $theta$ approach zero.
Let $alpha=1/2$, $theta>-1/2$, and $ u_0$ be a probability measure on a type space $S$. In this paper, we investigate the stochastic dynamic model for the two-parameter Dirichlet process $Pi_{alpha,theta, u_0}$. If $S=mathbb{N}$, we show that the bilinear form begin{eqnarray*} left{ begin{array}{l} {cal E}(F,G)=frac{1}{2}int_{{cal P}_1(mathbb{N})}langle abla F(mu), abla G(mu)rangle_{mu} Pi_{alpha,theta, u_0}(dmu), F,Gin {cal F}, {cal F}={F(mu)=f(mu(1),dots,mu(d)):fin C^{infty}(mathbb{R}^d), dge 1} end{array} right. end{eqnarray*} is closable on $L^2({cal P}_1(mathbb{N});Pi_{alpha,theta, u_0})$ and its closure $({cal E}, D({cal E}))$ is a quasi-regular Dirichlet form. Hence $({cal E}, D({cal E}))$ is associated with a diffusion process in ${cal P}_1(mathbb{N})$ which is time-reversible with the stationary distribution $Pi_{alpha,theta, u_0}$. If $S$ is a general locally compact, separable metric space, we discuss properties of the model begin{eqnarray*} left{ begin{array}{l} {cal E}(F,G)=frac{1}{2}int_{{cal P}_1(S)}langle abla F(mu), abla G(mu)rangle_{mu} Pi_{alpha,theta, u_0}(dmu), F,Gin {cal F}, {cal F}={F(mu)=f(langle phi_1,murangle,dots,langle phi_d,murangle): phi_iin B_b(S),1le ile d,fin C^{infty}(mathbb{R}^d),dge 1}. end{array} right. end{eqnarray*} In particular, we prove the Mosco convergence of its projection forms.
We give a pathwise construction of a two-parameter family of purely-atomic-measure-valued diffusions in which ranked masses of atoms are stationary with the Poisson-Dirichlet$(alpha,theta)$ distributions, for $alphain (0,1)$ and $thetage 0$. This resolves a conjecture of Feng and Sun (2010). We build on our previous work on $(alpha,0)$- and $(alpha,alpha)$-interval partition evolutions. Indeed, we first extract a self-similar superprocess from the levels of stable processes whose jumps are decorated with squared Bessel excursions and distinct allelic types. We complete our construction by time-change and normalisation to unit mass. In a companion paper, we show that the ranked masses of the measure-valued processes evolve according to a two-parameter family of diffusions introduced by Petrov (2009), extending work of Ethier and Kurtz (1981). These ranked-mass diffusions arise as continuum limits of up-down Markov chains on Chinese restaurant processes.
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