The leftmost column of ordered Chinese restaurant process up-down chains: intertwining and convergence

Recently there has been significant interest in constructing ordered analogues of Petrov’s two-parameter extension Ethier and Kurtz’s infinitely-many-neutral-alleles diffusion model. One method for these processes goes through taking an appropriate diffusive limit Markov chains on integer compositions called Chinese Restaurant Process up-down chains. The resulting are diffusions whose state space is the set open subsets unit interval. In this paper we begin to study nontrivial aspects order structure diffusions. particular, a certain choice parameters, take size first component describe generator limiting process. We then relate leftmost maximal subset open-set valued This challenging because function its discontinuous. Our methods based establishing intertwining relations between study.