  Motivated by questions of ergodicity for shift invariant Fleming-Viot
process, we consider the centered Fleming-Viot process $\left(Z_{t}
\right)_{t\geqslant 0}$ defined by $Z_{t} := \tau_{-\left\langle {\rm id},
Y_{t} \right\rangle} \sharp \, Y_{t}$, where $\left(Y_{t}\right)_{t\geqslant
0}$ is the original Fleming-Viot process. Our goal is to characterise the
centered Fleming-Viot process with a martingale problem. To establish the
existence of a solution to this martingale problem, we exploit the original
Fleming-Viot martingale problem and asymptotic expansions. The proof of
uniqueness is based on a weakened version of the duality method, allowing us to
prove uniqueness for initial conditions admitting finite moments. We also
provide counter examples showing that our approach based on the duality method
cannot be expected to give uniqueness for more general initial conditions.
Finally, we establish ergodicity properties with exponential convergence in
total variation for the centered Fleming-Viot process and characterise the
invariant measure.
