General selection models: Bernstein duality and (minimal) ancestral structures
A talk in the Keine Reihe series by
Fernando Cordero
| Abstract: | Lambda-Wright--Fisher processes provide a flexible framework to describe the type-frequencyevolution of an infinite neutral population. We model interactions via a general polynomial driftvanishing at the boundary. An appropriate decomposition of the drift allows us to approximate thesolution of the associated stochastic differential equation by a sequence of finite population models.The genealogical structure inherent to these models leads in the large population limit to ageneralization of the ancestral selection graph of Krone and Neuhauser. Next, we construct anancestral process that keeps track of the sampling distribution along the ancestral structures and thatsatisfies a duality relation with the type-frequency process. We refer to it as Bernstein coefficientprocess and to the relation as Bernstein duality. As an application, we derive criteria for theaccessibility of the boundary. An intriguing feature in our construction is that multiple ancestralprocesses can be associated to the same forward dynamics. If there is enough time, I will explainhow to characterize the set of optimal ancestral structures.
This talk takes place within the retreat of the CRC 1283. All members of the CRC are invited to attend. Please contact ckoehler@math.uni-bielefeld.de for Zoom-Link, Meeting-ID and Password. |