  We construct a pair of related diffusions on a space of interval partitions
of the unit interval $[0,1]$ that are stationary with the Poisson-Dirichlet
laws with parameters (1/2,0) and (1/2,1/2) respectively. These are two
particular cases of a general construction of such processes obtained by
decorating the jumps of a spectrally positive L\'evy process with independent
squared Bessel excursions. The processes of ranked interval lengths of our
partitions are members of a two parameter family of diffusions introduced by
Ethier and Kurtz (1981) and Petrov (2009). The latter diffusions are continuum
limits of up-down Markov chains on Chinese restaurant processes. Our
construction is also a step towards describing a diffusion on the space of real
trees whose existence has been conjectured by Aldous.
