Kenji Handa

The two-parameter Poisson–Dirichlet point process

The two-parameter Poisson–Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (that is, the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correlation functions. Using this, we apply the theory of point processes to reveal the mathematical structure of the two-parameter Poisson– Dirichlet distribution. Also, developing the Laplace transform approach due to Pitman and Yor, we are able to extend several results previously known for the one-parameter case. The Markov–Krein identity for the generalized Dirichlet process is discussed from the point of view of functional analysis based on the two-parameter Poisson–Dirichlet distribution.

Stationary distributions for a class of generalized Fleming–Viot processes

We identify stationary distributions of generalized Fleming-Viot processes with jump mechanisms specified by certain beta laws together with a parameter measure. Each of these distributions is obtained from normalized stable random measures after a suitable biased transformation followed by mixing by the law of a Dirichlet random measure with the same parameter measure. The calculations are based primarily on the well-known relationship to measure-valued branching processes with immigration.

Quasi-invariant measures and their characterization by conditional probabilities

Based on quasi-invariance properties of the Gaussian process and the gamma process, we give certain formal expressions of their laws on infinite-dimensional spaces. They are also consistent with some identities of conditional probabilities which are shown to be equivalent to the quasi-invariance properties.

On a Stochastic PDE Related to Burgers’ Equation with Noise

A rigorous treatment for an SPDE describing some physical processes is given. The equation is also related to Burgers’ equation with noise via the so-called Hopf-Cole Transformation.

Entropy Production per Site in (Nonreversible) Spin-Flip Processes

Entropy production per site in a (nonreversible) spin-flip process is studied. We give it a useful expression, from which a property stronger than affinity of the entropy production per site follows. Furthermore, quasi-invariance of nonequilibrium measures in the spin-flip processes is discussed via entropy production.