Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits

We introduce a three-parameter family of up-down ordered Chinese restaurant processes ${rm PCRP}^{(alpha)}(theta_1,theta_2)$, $alphain(0,1)$, $theta_1,theta_2ge 0$, generalising the two-parameter family of Rogers and Winkel. Our main result establishes self-similar diffusion limits, ${rm SSIP}^{(alpha)}(theta_1,theta_2)$-evolutions generalising existing families of interval partition evolutions. We use the scaling limit approach to extend stationarity results to the full three-parameter family, identifying an extended family of Poisson--Dirichlet interval partitions. Their ranked sequence of interval lengths has Poisson--Dirichlet distribution with parameters $alphain(0,1)$ and $theta:=theta_1+theta_2-alphage-alpha$, including for the first time the usual range of $theta>-alpha$ rather than being restricted to $thetage 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range.