Boundary behavior of the Λ-Wright--Fisher process with selection

Resumen: Λ-Wright--Fisher processes provide an important modeling framework within mathematical population genetics. We present a variety of parameter-dependent long-term behaviors for a broad class of such processes and explain how to discriminate the different boundary behaviors by explicit criteria. In particular, we describe situations in which both boundary points are asymptotically inaccessible – an apparently new phenomenon in this context. This has interesting biological implications, because it leads to a class of stochastic population models in which selection alone can maintain genetic variation. If at least one boundary point is asymptotically accessible, we derive decay rates for the probability that the boundary is not essentially accessed. To prove this result, we establish and employ Siegmund duality. The dual process can be sandwiched at the boundary in between two transformed Lévy processes. This allows us to relate the boundary behavior of the dual to fluctuation properties of the Lévy processes and it sheds new light on previously established accessibility conditions. This is joint work with Fernando Cordero and Grégoire Véchambre.