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

Diffusive limits of two-parameter ordered Chinese Restaurant Process up-down chains