Size-biased diffusion limits and the inclusion process
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910475807096832 |
|---|---|
| author | Chleboun, Paul Gabriel, Simon Grosskinsky, Stefan |
| author_facet | Chleboun, Paul Gabriel, Simon Grosskinsky, Stefan |
| contents | We consider the inclusion process on the complete graph with vanishing diffusivity, which leads to condensation of particles in the thermodynamic limit. Describing particle configurations in terms of size-biased and appropriately scaled empirical measures of mass distribution, we establish convergence in law of the inclusion process to a measure-valued Markov process on the space of probability measures. In the case where the diffusivity vanishes like the inverse of the system size, the derived scaling limit is equivalent to the well known Poisson-Dirichlet diffusion, offering an alternative viewpoint on these well-established dynamics. Moreover, our approach covers all scaling regimes of the system parameters and yields a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. We also discuss in detail connections to known results on related Fleming-Viot processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_09722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Size-biased diffusion limits and the inclusion process Chleboun, Paul Gabriel, Simon Grosskinsky, Stefan Probability Statistical Mechanics Mathematical Physics 60K35, 82C22, 82C26 We consider the inclusion process on the complete graph with vanishing diffusivity, which leads to condensation of particles in the thermodynamic limit. Describing particle configurations in terms of size-biased and appropriately scaled empirical measures of mass distribution, we establish convergence in law of the inclusion process to a measure-valued Markov process on the space of probability measures. In the case where the diffusivity vanishes like the inverse of the system size, the derived scaling limit is equivalent to the well known Poisson-Dirichlet diffusion, offering an alternative viewpoint on these well-established dynamics. Moreover, our approach covers all scaling regimes of the system parameters and yields a natural extension of the Poisson-Dirichlet diffusion to infinite mutation rate. We also discuss in detail connections to known results on related Fleming-Viot processes. |
| title | Size-biased diffusion limits and the inclusion process |
| topic | Probability Statistical Mechanics Mathematical Physics 60K35, 82C22, 82C26 |
| url | https://arxiv.org/abs/2304.09722 |