# Kingman's coalescent with erosion

## TL;DR

This paper studies a variant of Kingman's coalescent process incorporating erosion and immigration, providing new constructions of its stationary distribution, and linking block sizes to critical branching processes.

## Contribution

It introduces a novel construction of the stationary distribution using flows of bridges and hierarchically independent diffusions, and defines a new coalescent with immigration.

## Key findings

- Stationary distribution constructed via flow of bridges.
- Block size distribution converges to a critical branching process.
- Introduces Kingman's coalescent with immigration.

## Abstract

Consider the Markov process taking values in the partitions of 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,...,n} converges to the total progeny of a critical binary branching process.