SMILE

Stochastic Models for the Inference of Life Evolution

Kingman's coalescent with erosion

Foutel-Rodier, F., Lambert, A., Schertzer, E.

arXiv:1907.05845

2019

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 = {arXiv:1907.05845},
eprint = {arXiv:1907.05845},
year = {2019},
url = {https://arxiv.org/abs/1907.05845}
}

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