Vortrag von Frank den Hollander im Rahmen des Workshops
„Geographic spaces for evolution: From lattices to graphs to continuum“
This lecture presents joint work with Siva Athreya (Bangalore) and Adrian Roellin
The aim is to develop a theory of graphon-valued stochastic processes, and to con-
struct and analyze a natural class of such processes arising from population genetics.
We consider finite populations where individuals change type according to Wright-
Fisher resampling. At any time, each pair of individuals is linked by an edge with a
probability that is given by a type-connection matrix, whose entries depend on the
current empirical type distribution of the entire population via a fitness function.
We show that, in the large-population-size limit and with an appropriate scaling of
time, the evolution of the associated adjacency matrix converges to a random process
in the space of graphons, driven by the type-connection matrix and the underlying
Wright-Fisher diffusion on the multi-type simplex. In the limit as the number of types
tends to infinity, the limiting process is driven by the type-connection kernel and the
underlying Fleming-Viot diffusion.