Duality for spatially interacting Fleming-Viot processes with mutation   and selection
  Consider a system $X = ((x_\xi(t)), \xi \in \Omega_N)_{t \geq 0}$ of interacting Fleming-Viot diffusions with mutation and selection which is a strong Markov process with continuous paths and state space $(\CP(\I))^{\Omega_N}$, where $\I$ is the type space, ${\Omega_N}$ the geographic space is assumed to be a countable group and $\CP$ 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 $\I$ 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 \cite{DGsel} and \cite{DGInvasion}. 
Donald A. Dawson and Andreas Greven
math.PR