{
  "id": "2408.03553",
  "version": "v1",
  "published": "2024-08-07T05:29:14.000Z",
  "updated": "2024-08-07T05:29:14.000Z",
  "title": "Dirichlet forms of diffusion processes on Thoma simplex",
  "authors": [
    "Sergei Korotkikh"
  ],
  "comment": "27 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a prominent two-parametric family of diffusion processes $X_{z,z'}$ on an infinite-dimensional Thoma simplex. The family was constructed by Borodin and Olshanski in 2007 and it closely resembles Ethier-Kurtz's infinitely-many-neutral-allels diffusion model on Kingman simplex (1981) and Petrov's extension of Ethier-Kurtz's model (2007). The processes $X_{z,z'}$ have unique symmetrizing measures, namely, the boundary $z$-measures, which play the role of Poisson-Dirichlet measures in our context. We establish the following behavior of diffusions $X_{z,z'}$: immediately after the initial moment they jump into a dense face of Thoma simplex and then always stay there. In other words, the face acts as a natural state space for the diffusions, while other points of the simplex act like an entrance boundary for our process. As a key intermediate step we study the Dirichlet forms of the diffusions $X_{z,z}$ and find a new description for them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-07T05:29:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60"
    ],
    "keywords": [
      "thoma simplex",
      "dirichlet forms",
      "diffusion processes",
      "resembles ethier-kurtzs infinitely-many-neutral-allels diffusion model",
      "closely resembles ethier-kurtzs infinitely-many-neutral-allels diffusion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}