Scaling limits for Moran processes on metric strategy spaces: an Eulerian derivation of pure replicator and Fleming--Viot measure-valued PDEs
Abstract
We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space 𝒱 (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming--Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over 𝒱.
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BibTeX
@misc{almi2026scaling,
title = {{Scaling limits for Moran processes on metric strategy spaces: an Eulerian derivation of pure replicator and Fleming--Viot measure-valued PDEs}},
author = {Stefano Almi and Riccardo Durastanti and Marco Morandotti and Gianluca Orlando and Francesco Solombrino},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23235},
eprint = {2609.23235},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23235},
}
RIS
TY - UNPB
TI - Scaling limits for Moran processes on metric strategy spaces: an Eulerian derivation of pure replicator and Fleming--Viot measure-valued PDEs
AU - Stefano Almi
AU - Riccardo Durastanti
AU - Marco Morandotti
AU - Gianluca Orlando
AU - Francesco Solombrino
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.23235
UR - https://arxiv.org/abs/2609.23235
AB - We study the large-population limit of a discrete-time Moran process featuring multiple strategies, drawn from a possibly infinite strategy space 𝒱 (a metric space), under both weak and strong selection. In the Eulerian density formulation, the limiting dynamics depend critically on the relative scaling of population size, mutation rate, and selection intensity. Depending on these scalings, the limit behavior is governed either by a purely deterministic replicator-type continuity equation or by a diffusion-enhanced PDE. In the latter case, the diffusion operator recovers the classical Fleming--Viot operator and the Kimura equation as special instances. Methodologically, we derive the limit by reinterpreting the discrete process in Eulerian coordinates, constructing interpolating curves that satisfy an approximate PDE, and establishing convergence via a compactness argument in a suitable topology on the space of probability measures over probabilities over 𝒱.
ER -