The grapheme-valued Wright–Fisher diffusion with mutation

In Athreya et al. (2021), models from population genetics were used to define stochastic dynamics in the space of graphons arising as continuum limits of dense graphs. In the present paper we exhibit an example of a simple neutral population genetics model for which this dynamics is a Markovian diffusion that can be characterized as the solution of a martingale problem. In particular, we consider a Markov chain in the space of finite graphs that resembles a Moran model with resampling and mutation. We encode the finite graphs as graphemes, which can be represented as a triple consisting of a vertex set (or more generally, a topological space), an adjacency matrix, and a sampling (Borel) measure. We equip the space of graphons with convergence of sample subgraph densities and show that the grapheme-valued Markov chain converges to a grapheme-valued diffusion as the number of vertices goes to infinity. We show that the grapheme-valued diffusion has a stationary distribution that is linked to the Griffiths–Engen–McCloskey (GEM) distribution. In a companion paper (Greven et al. 2023), we build up a general theory for obtaining grapheme-valued diffusions via genealogies of models in population genetics.