Dirichlet forms of diffusion processes on Thoma simplex
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917742903296000 |
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| author | Korotkikh, Sergei |
| author_facet | Korotkikh, Sergei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03553 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirichlet forms of diffusion processes on Thoma simplex Korotkikh, Sergei Probability 60J60 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. |
| title | Dirichlet forms of diffusion processes on Thoma simplex |
| topic | Probability 60J60 |
| url | https://arxiv.org/abs/2408.03553 |