In this joint project with Arno Siri-Jégousse, we introduce a novel research line connecting the fields of mathematical population genetics and self-similar (SS) Markov processes in infinite dimensions. Specifically, we propose a shift in focus from the prevalent paradigm based on the branching property as a tool to analyze the structure of population models, to one based on the self-similarity property, which we also introduce for the first time in the setting of measure-valued (MV) processes. By extending the well-known Lamperti transformation for SS Markov processes to the Banach-valued case, we were able to generalize the celebrated work of Birkner et al. (2005) in population genetics. They describe the frequency process of populations modeled as an MV alpha-stable branching process, in terms of the subfamily of Beta Fleming-Viot processes. We describe the frequency process of populations whose total size evolves as any positive SS Markov process, in terms of general Lambda Fleming-Viot processes. Our results demonstrate the potential power of the SS perspective for the study of population models in which the reproduction dynamics of individuals depend on the total population size, allowing for more complex and realistic models. In parallel, SSMV processes, together with the analytic tools available in population genetics such as duality methods, introduce a promising new template for the development of the theory of SS Markov processes in the infinite-dimensional setting.
