  We introduce a new class of nonparametric prior distributions on the space of
continuously varying densities, induced by Dirichlet process mixtures which
diffuse in time. These select time-indexed random functions without jumps,
whose sections are continuous or discrete distributions depending on the choice
of kernel. The construction exploits the widely used stick-breaking
representation of the Dirichlet process and induces the time dependence by
replacing the stick-breaking components with one-dimensional Wright-Fisher
diffusions. These features combine appealing properties of the model, inherited
from the Wright-Fisher diffusions and the Dirichlet mixture structure, with
great flexibility and tractability for posterior computation. The construction
can be easily extended to multi-parameter GEM marginal states, which include,
for example, the Pitman--Yor process. A full inferential strategy is detailed
and illustrated on simulated and real data.
