Dualities for branching-coalescing processes in population genetics

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In mathematical population genetics, two approaches are used to understand the effect of evolutionary forces on a population. On the one hand, one uses stochastic processes describing the allele frequencies forward in time. On the other hand, one considers branching-coalescing processes that arise from genealogical considerations. Ideally, the connection between the two approaches is formalised via a duality relation between two stochastic processes. Together with Prof. Steven Evans, we will develop systematic methods to construct a useful branching-coalescing dual process for a given type-frequency process. In particular, we aim at identifying a class of stochastic processes that can be analysed via ancestral considerations. The systematic construction should considerably widen the scope of duality methods and make them applicable to a wider class of models.Wright-Fisher processes are prominent examples of the classical type-frequency processes. In some special cases, the transition densities of the Wright-Fisher diffusion admits a representation in terms of a branching-coalescing dual process. These representations allow to efficiently simulate the diffusion process; this is important for the inference of mutation and selection parameters from population genetical data. The density representations are derived from a moment duality, which is known only in specific examples. We will construct our dual processes such that they can be used for the representation of the transition densities in a way that is independent of the existence of a moment duality.