  We introduce and study interval partition diffusions with
Poisson--Dirichlet$(\alpha,\theta)$ stationary distribution for parameters
$\alpha\in(0,1)$ and $\theta\ge 0$. This extends previous work on the cases
$(\alpha,0)$ and $(\alpha,\alpha)$ and builds on our recent work on
measure-valued diffusions. Our methods for dealing with general $\theta\ge 0$
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.
