Reversible measure-valued processes associated with the Poisson-Dirichlet distribution

The Poisson-Dirichlet distribution is a probability on the space of infinite dimensional simplex. It is the distribution of a discrete-probability-valued random variable.The closely associated random measures include the GEM distribution, the Dirichlet process and their two-parameter generalizations.One of the active areas for research in modern probability is concerned with the construction of reversible measure-valued processes associated with these random measures.This article will survey recent development in this area and list a few interesting problems for further consideration.