# SMILE

## Kingman's coalescent with erosion

### 2020

Consider the Markov process taking values in the partitions of $\mathbb{N}$ such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate $d$. This is a special case of exchangeable fragmentation-coalescence process, called Kingman's coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of hierarchically independent diffusions. Moreover, we introduce a new process called Kingman's coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate at rate $d$. By coupling Kingman's coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman's coalescent with erosion to $\{1, \dots, n\}$ converges to the total progeny of a critical binary branching process.

# Bibtex

@article{FRLS2019KingmanErosion,
author = {Foutel-Rodier, F{\'e}lix and Lambert, Amaury and Schertzer, Emmanuel},
title = {Kingman's coalescent with erosion},
journal = {Electronic Journal of Probability},
year = {2020},
pages = {33 pp.},
volume = {25},
url = {https://doi.org/10.1214/20-EJP450},
doi = {10.1214/20-EJP450},
}