Interval fragmentations with choice: equidistribution and the evolution of tagged fragments

We consider a Markovian evolution on point processes, the $\Psi$--process, on the unit interval in which points are added according to a rule that depends only on the spacings of the existing point configuration. Having chosen a spacing, a new point is added uniformly within it. Building on previous work of the authors and of Junge, we show that the empirical distribution of points in such a process is always equidistributed under mild assumptions on the rule, generalizing work of Junge. A major portion of this article is devoted to the study of a particular growth--fragmentation process, or cell process, which is a type of piecewise--deterministic Markov process (PDMP). This process represents a linearized version of a size--biased sampling from the $\Psi$--process. We show that this PDMP is ergodic and develop the semigroup theory of it, to show that it describes a linearized version of the $\Psi$--process. This PDMP has appeared in other contexts, and in some sense we develop its theory under minimal assumptions.