Multi-type branching diffusions with small mutation rates
Abstract
Approximate solutions are found for Feller-like neutral multi-type branching diffusions (X(t))t ∈ ℝ≥ 0 in the limit of small mutation rates. The method employed involves solving approximations to the Laplace transformed forward Kolmogorov equation by integrating along characteristics. To leading order in the scale θ of the overall mutation rate the super-critical diffusion is found to collapse onto a line density aligned with the stationary left eigenvector of the rate matrix following a rapid change of behaviour at a critical time tc, which has a weak logarithmic dependence on θ. First order in θ approximations to the density and moments of X(t) are also determined. The first-order approximation corresponds to allowing at most one mutation in the coalescent tree of a sample and is valid for t < tc in the supercritical case, and more generally for the critical and sub-critical cases.
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BibTeX
@misc{burden2026multitype,
title = {{Multi-type branching diffusions with small mutation rates}},
author = {Conrad J. Burden},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05804},
eprint = {2610.05804},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05804},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Multi-type branching diffusions with small mutation rates
AU - Conrad J. Burden
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05804
UR - https://arxiv.org/abs/2610.05804
AB - Approximate solutions are found for Feller-like neutral multi-type branching diffusions (X(t))_t ∈ ℝ_≥ 0 in the limit of small mutation rates. The method employed involves solving approximations to the Laplace transformed forward Kolmogorov equation by integrating along characteristics. To leading order in the scale θ of the overall mutation rate the super-critical diffusion is found to collapse onto a line density aligned with the stationary left eigenvector of the rate matrix following a rapid change of behaviour at a critical time t_c, which has a weak logarithmic dependence on θ. First order in θ approximations to the density and moments of X(t) are also determined. The first-order approximation corresponds to allowing at most one mutation in the coalescent tree of a sample and is valid for t < t_c in the supercritical case, and more generally for the critical and sub-critical cases.
ER -