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Consider a system $X = ((x_\xi(t)), \xi \in \Omega_N)_{t \geq 0}$ ofinteracting Fleming-Viot diffusions with mutation and selection which is astrong Markov process with continuous paths and state space$(\CP(\I))^{\Omega_N}$, where $\I$ is the type space, ${\Omega_N}$ thegeographic space is assumed to be a countable group and $\CP$ denotes theprobability measures.  We establish various duality relations for this process. These dualities arefunction-valued processes which are driven by a coalescing-branching randomwalk, that is, an evolving particle system which in addition exhibits certainchanges in the function-valued part at jump times driven by mutation.  In the case of a finite type space $\I$ we construct a set-valued dualprocess, which is a Markov jump process, which is very suitable to proveergodic theorems which we do here. The set-valued duality contains as specialcase a duality relation for any finite state Markov chain.  In the finitely many types case there is also a further tableau-valued dualwhich can be used to study the invasion of fitter types after rare mutation.This is carried out in \cite{DGsel} and \cite{DGInvasion}.

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