Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions

Abstract

constructed (α, θ)-interval partition evolutions for α ∈ (0, 1) and θ ≥ 0, in which the total sums of interval lengths ("total mass") evolve as squared Bessel processes of dimension 2θ, where θ ≥ 0 acts as an immigration parameter. These evolutions have pseudo-stationary distributions related to regenerative Poisson-Dirichlet interval partitions. In this paper we study symmetry properties of (α, θ)-interval partition evolutions. Furthermore, we introduce a three-parameter family SSIP (α) (θ1, θ2) of self-similar interval partition evolutions that have separate left and right immigration parameters θ1 ≥ 0 and θ2 ≥ 0. They also have squared Bessel total mass processes of dimension 2θ, where θ = θ1 + θ2 − α ≥ −α covers emigration as well as immigration. Under the constraint max{θ1, θ2} ≥ α, we prove that an SSIP (α) (θ1, θ2)-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson-Dirichlet distribution with parameters α and θ, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper.

Ranked masses in two-parameter Fleming-Viot diffusions

“…The family K α,θ s , s≥0 is the transition semi-group of an I H -valued diffusion (β s , s≥0) that we call SSIPE(α, θ). This was further extended in [37,Definition 1.3] to a threeparameter family SSIPE (α) (θ 1 , θ 2 ), θ 1 , θ 2 ≥0, so that θ :=θ 1 +θ 2 −α≥−α. The time-change (4.1) ρ(t) = inf s ≥ 0 :…”

“…If β 0 = γ ∈ I H has total mass γ =1, we write (γ t , t≥0)∼PDIPE γ (α, θ), respectively PDIPE (α) γ (θ 1 , θ 2 ). We showed in [17, Theorems 1.3 and 1.6], [37,Theorem 1.4] and [38,Theorem 1.4], that all of these evolutions are interval partition diffusions, and that PDIPE(α, θ) and PDIPE (α) (θ 1 , θ 2 ) have PDIP(α, θ), respectively PDIP (α) (θ 1 , θ 2 ), as their stationary distribution. THEOREM 4.2.…”

“…The first is to calculate relevant parts of the generator of FV(α, θ), which allows us to identify the semi-groups of EKP (α, θ) and ranked FV(α, θ) on the Hilbert space L 2 [α, θ] of functions on ∇ ∞ that are square integrable with respect to the measure PD(α, θ). The second step involves interval partition evolutions [14,15,17,37,38], which can be coupled with FV(α, θ)-processes to have the same ranked masses. We use these couplings to establish sufficient regularity to identify the semi-groups also as operators acting on the space of bounded continuous functions.…”

“…In Section 3, we carry out the first step in the proof of Theorem 1.2. In Section 4, we obtain a version of Theorem 1.2 for the interval partition evolutions of [17,37] and use it to carry out the second step in the proof of Theorem 1.2.…”

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

“…While most SSIP-evolutions have been constructed before [14,15,18,47], in increasing generality, Theorem 1.1 is the first scaling limit result with an SSIP-evolution as its limit. In special cases, this was conjectured in [44].…”