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

Forman et al. (2020+) constructed $(α,θ)$-interval partition evolutions for $α\in(0,1)$ and $θ\ge 0$, in which the total sums of interval lengths ("total mass") evolve as squared Bessel processes of dimension $2θ$, where $θ\ge 0$ 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 ${\rm SSIP}^{(α)}(θ_1,θ_2)$ of self-similar interval partition evolutions that have separate left and right immigration parameters $θ_1\ge 0$ and $θ_2\ge 0$. They also have squared Bessel total mass processes of dimension $2θ$, where $θ=θ_1+θ_2-α\ge-α$ covers emigration as well as immigration. Under the constraint $\max\{θ_1,θ_2\}\geα$, we prove that an ${\rm SSIP}^{(α)}(θ_1,θ_2)$-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.