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

Abstract

Forman et al. (2020+) constructed $(\alpha ,\theta )$-interval partition evolutions for $\alpha \in (0,1)$ and $\theta \ge 0$, in which the total sums of interval lengths ("total mass") evolve as squared Bessel processes of dimension $2\theta$, where $\theta \ge 0$ 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 $(\alpha ,\theta )$-interval partition evolutions. Furthermore, we introduce a three-parameter family ${\mathrm{SSIP}}^{(\alpha )}({\theta }_1,{\theta }_2)$ of self-similar interval partition evolutions that have separate left and right immigration parameters ${\theta }_1\ge 0$ and ${\theta }_2\ge 0$. They also have squared Bessel total mass processes of dimension $2\theta$, where $\theta ={\theta }_1+{\theta }_2-\alpha \ge -\alpha$ covers emigration as well as immigration. Under the constraint $\max \{{\theta }_1,{\theta }_2\}\ge \alpha$, we prove that an ${\mathrm{SSIP}}^{(\alpha )}({\theta }_1,{\theta }_2)$-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson--Dirichlet distribution with parameters $\alpha$ and $\theta$, 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.

arXiv comment22 pages, 3 figures

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