  This paper considers adiabatic reduction in both discrete and continuous
models of stochastic gene expression. In gene expression models, the concept of
bursting is a production of several molecules simultaneously and is generally
represented as a compound Poisson process of random size. In a general
two-dimensional birth and death discrete model, we prove that under specific
assumptions and scaling (that are characteristics of the mRNA-protein system)
an adiabatic reduction leads to a one-dimensional discrete-state space model
with bursting production. The burst term appears through the reduction of the
first variable. In a two-dimensional continuous model, we also prove that an
adiabatic reduction can be performed in a stochastic slow/fast system. In this
gene expression model, the production of mRNA (the fast variable) is assumed to
be bursty and the production of protein (the slow variable) is linear as a
function of mRNA. When the dynamics of mRNA is assumed to be faster than the
protein dynamics (due to a mRNA degradation rate larger than for the protein)
we prove that, with the appropriate scaling, the bursting phenomena can be
transmitted to the slow variable. We show that the reduced equation is either a
stochastic differential equation with a jump Markov process or a deterministic
ordinary differential equation depending on the scaling that is appropriate.
  These results are significant because adiabatic reduction techniques seem to
have not been applied to a stochastic differential system containing a jump
Markov process. Last but not least, for our particular system, the adiabatic
reduction allows us to understand what are the necessary conditions for the
bursting production-like of protein to occur.
