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The present paper originated from our previous study of the problem ofharmonic analysis on the infinite symmetric group. This problem leads to afamily {P_z} of probability measures, the z-measures, which depend on thecomplex parameter z. The z-measures live on the Thoma simplex, aninfinite-dimensional compact space which is a kind of dual object to theinfinite symmetric group. The aim of the paper is to introduce stochasticdynamics related to the z-measures. Namely, we construct a family of diffusionprocesses in the Toma simplex indexed by the same parameter z. Our diffusionsare obtained from certain Markov chains on partitions of natural numbers n in ascaling limit as n goes to infinity. These Markov chains arise in a naturalway, due to the approximation of the infinite symmetric group by the increasingchain of the finite symmetric groups. Each z-measure P_z serves as a uniqueinvariant distribution for the corresponding diffusion process, and the processis ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, sothat the process is reversible. We describe the spectrum of its generator andcompute the associated (pre)Dirichlet form.

Infinite-dimensional diffusions as limits of random walks on partitions