  We study a stochastic model of biosynthesis of proteins in generic bacterial
operons. The stochasticity arises from two different processes, namely from
`bursting' production of either mRNA and/or protein (in the
transcription/translation process) and from standard diffusive fluctuations.
The amount of protein follows the Feller diffusion, while the bursting
introduces random jumps between trajectories of the diffusion process. The
combined effect leads to a process commonly known as a diffusion process with
jumps. We study existence of invariant densities and the long time behavior of
distributions of the corresponding Markov process, proving asymptotic stability
in the evolution of the density.
