Duality for spatially interacting Fleming-Viot processes with mutation and selection

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Consider a system X = ((xξ(t)), ξ∈ Ωₙ)ₜ ≥ ₀ of interacting Fleming-Viot diffusions with mutation and selection which is a strong Markov process with continuous paths and state space (())Ωⁿ, where is the type space, Ωₙ the geographic space is assumed to be a countable group and denotes the probability measures. We establish various duality relations for this process. These dualities are function-valued processes which are driven by a coalescing-branching random walk, that is, an evolving particle system which in addition exhibits certain changes in the function-valued part at jump times driven by mutation. In the case of a finite type space we construct a set-valued dual process, which is a Markov jump process, which is very suitable to prove ergodic theorems which we do here. The set-valued duality contains as special case a duality relation for any finite state Markov chain. In the finitely many types case there is also a further tableau-valued dual which can be used to study the invasion of fitter types after rare mutation. This is carried out in DGsel and DGInvasion.