Branching Interval Partition Diffusions

We introduce and study branching interval partition diffusions in their natural generality.We let interval widths evolve independently according to a general real-valued diffusion subject only to conditions that ensure finite lifetimes of intervals and allow the continuous generation of new intervals.The latter is governed by the Pitman-Yor excursion measure of the real-valued diffusion and an associated spectrally positive Lvy process to order both these excursions and their start times.This generalises previous work by Forman, Pal, Rizzolo and Winkel on the self-similar case and gives rise to a new class of general Markovian homogeneous Crump-Mode-Jagers-type branching processes with characteristics varying diffusively during their lifetimes.