Multi-type $Λ$-Wright--Fisher processes with polynomial drift: Bernstein duality, long-term behaviour, and simulation of moments

Source abstract

The $(d+1)$-type $Λ$-Wright--Fisher process with polynomial drift is a Lévy-type process on the $d$-dimensional unit simplex whose generator entails a particular jump and diffusive component, as well as a polynomial drift component. In this work, the process is connected to a (Bernstein) dual process, which admits an interpretation as an urn model with coloured balls. Exploiting this duality, we identify various parameter regimes: one in which the $Λ$-Wright--Fisher process is absorbed almost surely at a vertex of the simplex, and others in which it admits a unique nontrivial ergodic probability measure. The results include verifiable criteria for absorption versus persistence, as well as representations of absorption and stationary measures. Moreover, the duality relation is exploited to devise and implement a simulation-based approach for estimating moments of the $Λ$-Wright--Fisher process.