2020/08/06 by Forman, Noah, Rizzolo, Douglas, Shi, Quan +1 · 1 citation
#60G52 #60G55 #60J60 #60J80 #FOS: Mathematics #Primary 60J25 #Probability (math.PR) #Secondary 60G18
paper · doi:10.48550/arxiv.2008.02823
We introduce and study interval partition diffusions with Poisson--Dirichlet(α,θ) stationary distribution for parameters α∈(0,1) and θ≥ 0. This extends previous work on the cases (α,0) and (α,α) and builds on our recent work on measure-valued diffusions. Our methods for dealing with general θ≥ 0 allow us to strengthen previous work on the special cases to include initial interval partitions with dust. In contrast to the measure-valued setting, we can show that this extended process is a Feller process improving on the Hunt property established in that setting. These processes can be viewed as diffusions on the boundary of a branching graph of integer compositions. Indeed, by studying their infinitesimal generator on suitable quasi-symmetric functions, we relate them to diffusions obtained as scaling limits of composition-valued up-down chains.