We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump mechanisms are governed by certain stable laws. The main result gives a lower spectral gap estimate for the generator. As an application, a certain ergodic property is shown for the generalized Fleming-Viot process obtained as the time-changed ratio process.
"Ergodic properties for $\alpha$-CIR models and a class of generalized Fleming-Viot processes." Electron. J. Probab. 19 1 - 25, 2014. https://doi.org/10.1214/EJP.v19-2928