  We consider a stochastic model of gene expression in which transcription
depends on a multistate promoter, including the famous two-state model and
refractory promoters as special cases, and focus on deriving the exact
stationary distribution. Building upon several successful approaches, we
present a more unified viewpoint that enables us to simplify and generalize
existing results. In particular, the original jump process is deeply related to
a multivariate piecewise-deterministic Markov process that may also be of
interest beyond the biological field. In a very particular case of promoter
configuration, this underlying process is shown to have a simple Dirichlet
stationary distribution. In the general case, the corresponding marginal
distributions extend the well-known class of Beta products, involving complex
parameters that directly relate to spectral properties of the promoter
transition matrix. Finally, we illustrate these results with biologically
plausible examples.
