Transition density of an infinite-dimensional diffusion with the jack parameter

From the Poisson-Dirichlet diffusions to the Z-measure diffusions, they all have explicit transition densities. We 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 the same as the coefficients in the Poisson-Dirichlet diffusions. This fact will be uncovered by a dual process method in a special case where the Z-measure diffusions are established through an up-down chain in the Young graph.