The paper deals with a three-dimensional family of diffusion processes on an infinite-dimensional simplex. These processes were constructed by Borodin and Olshanski in 2009 and 2010, and they include, as limit objects, the infinitely-many-neutral-allels diffusion model constructed by Ethier and Kurtz in 1981 and its extension found by Petrov in 2009.

Each process X in our family possesses a unique symmetrizing measure M, called the z-measure. Our main result is that the transition function of X has a continuous density with respect to M. This is a generalization of earlier results due to Ethier (1992) and to Feng, Sun, Wang, and Xu (2011). Our proof substantially uses a special basis in the algebra of symmetric functions, which is related to the Laguerre polynomials.

本文讨论了无穷维单纯形上的三维扩散过程族。这些过程是由Borodin和Olshanski在2009年和2010年构建的，它们包括作为极限对象的Ethier和Kurtz在1981年构建的无限多中性等位基因扩散模型，以及Petrov在2009年发现的扩展模型。

我们家庭中的每个过程X都有一个唯一的对称度量M，称为z-度量。我们的主要结果是X的跃迁函数相对于M具有连续的密度。这是由于Ethier（1992）和Feng，Sun，Wang和Xu（2011）导致的早期结果的概括。我们的证明在与Laguerre多项式有关的对称函数的代数中基本上使用了特殊的基础。