Infinitedimensional diffusions as limits of random walks on partitions
Abstract
The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the zmeasures, which depend on the complex parameter z. The zmeasures live on the Thoma simplex, an infinitedimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the zmeasures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each zmeasure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.
 Publication:

arXiv eprints
 Pub Date:
 June 2007
 DOI:
 10.48550/arXiv.0706.1034
 arXiv:
 arXiv:0706.1034
 Bibcode:
 2007arXiv0706.1034B
 Keywords:

 Mathematics  Probability;
 Mathematics  Representation Theory;
 60J60;
 60C05
 EPrint:
 AMSTex, 33 pages. Version 2: minor changes, typos corrected, to appear in Prob. Theor. Rel. Fields