Year: 2020
Type: preprint
Abstract: We introduce and study interval partition diffusions with Poisson--Dirichlet$(α,θ)$ stationary distribution for parameters $α\in(0,1)$ and $θ\ge 0$. 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 $θ\ge 0$ 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.
Source: arXiv (Cornell University)
Authors: Forman, Noah, Rizzolo, Douglas, Shi, Quan, Winkel, Matthias
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: Growth-fragmentation processes and stochastic geometry, PostDoctoral Research Fellowship, Dynamics in exchangeable partitions and hierarchies, G1945889509, G2271590008, G2506195653, G3111038552, G3180761643, G3685197937, G4782791752, G4803182929, G5666163767, G6517680562, Tree-valued random processes: tree growth, pruning and stationary processes in spaces of continuum trees, G6901256639, Probability Postdoctoral Training Center, G7905952742, G8455129046, G8533620077, Diffusions on Combinatorially Structured State Spaces, G8792570289
OpenAlex ID: W4288015378