  We introduce a class of probability measure-valued diffusions, coined
polynomial, of which the well-known Fleming--Viot process is a particular
example. The defining property of finite dimensional polynomial processes
considered by Cuchiero et al. (2012) and Filipovic and Larsson (2016) is
transferred to this infinite dimensional setting. This leads to a
representation of conditional marginal moments via a finite dimensional linear
PDE, whose spatial dimension corresponds to the degree of the moment. As a
result, the tractability of finite dimensional polynomial processes are
preserved in this setting. We also obtain a representation of the corresponding
extended generators, and prove well-posedness of the associated martingale
problems. In particular, uniqueness is obtained from the duality relationship
with the PDEs mentioned above.
