  We construct a two-parameter family of Feller diffusions on the set of open
subsets of $(0,1)$ that arise as diffusive limits of two-parameter ordered
Chinese Restaurant Process up-down chains. The diffusions we construct are
natural ordered analogues of Petrov's two-parameter extension of Ethier and
Kurtz's infinitely-many-neutral-alleles diffusion model. Recently, there has
been significant interest in ordered analogues of the diffusions Petrov
constructed. Existing methods for constructing such processes have been based
on pathwise methods using marked L\'evy processes and an outstanding conjecture
about these processes is that they are, in fact, the diffusive limit of the
ordered Chinese Restaurant Process up-down chains that we consider here. We
make progress on this conjecture by showing that the diffusive limit of the
ordered Chinese Restaurant Process up-down chains exists. Moreover, our methods
yield a simple, explicit description of the generator of the limiting processes
on a core described in terms of quasisymmetric functions.
