Matthias Winkel

Combinatorial up-down chains and their diffusive continuum limits

We study composition-valued continuous-time Markov chains that
appear naturally in the framework of Chinese Restaurant Processes
(CRPs). As time evolves, new customers arrive (up-step) and existing
customers leave (down-step) at suitable rates derived from the ordered
CRP studied in previous work with Jim Pitman. We relate such up-down
CRPs to the splitting trees of Lambert inducing spectrally positive Lévy
processes, which in our setting have jumps marked by integer-valued
paths. We establish limit theorems for the Lévy processes, the
integer-valued paths and finally the composition-valued processes. The
latter give rise to interval-partition-valued diffusions connecting to
work by Forman, Pal, Rizzolo and Winkel on the Aldous diffusion on
continuum trees. This is based on joint work, in part with Dane Rogers,
and in part with Quan Shi.