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The paper deals with a three-dimensional family of diffusion processes on aninfinite-dimensional simplex. These processes were constructed by Borodin andOlshanski (arXiv:0706.1034; arXiv:0902.3395), and they include, as limitobjects, the Ethier-Kurtz's infinitely-many-neutral-allels diffusion model(1981) and its extension found by Petrov (arXiv:0708.1930).  Each process X from our family possesses a symmetrising measure M. Our mainresult is that the transition function of X has continuous density with respectto M. This is a generalization of earlier results due to Ethier (1992) and toFeng, Sun, Wang, and Xu (2011). Our proof substantially uses a special basis inthe algebra of symmetric functions related to Laguerre polynomials.

Transition functions of diffusion processes with the Jack parameter on the Thoma simplex