Transition Density of an Infinite-dimensional diffusion with the Jack Parameter

TL;DR

This paper demonstrates that the transition densities of Z-measure diffusions in infinite-dimensional spaces can be expressed as mixtures of probability measures on the Thoma simplex, linked to the Kingman coalescent process.

Contribution

It introduces a novel representation of Z-measure diffusion transition densities as mixtures involving the Kingman coalescent, expanding understanding of their structure.

Findings

Transition densities expressed as mixtures on the Thoma simplex

Coefficients linked to Kingman coalescent probabilities

Dual process method applied in a special case

Abstract

From the Poisson-Dirichlet diffusions to the $Z$-measure diffusions, they all have explicit transition densities. In this paper, we will show that the transition densities of the $Z$-measure diffusions can also be expressed as a mixture of a sequence of probability measures on the Thoma simplex. The coefficients are still the transition probabilities of the Kingman coalescent stopped at state $1$. This fact will be uncovered by a dual process method in a special case where the $Z$-measure diffusions is established through up-down chain in the Young graph.