  Stochastic gene regulatory networks with bursting dynamics can be modeled
mesocopically as a generalized density-dependent Markov chain (GDDMC) or
macroscopically as a piecewise-deterministic Markov process (PDMP). Here we
prove a limit theorem showing that each family of GDDMCs will converge to a
PDMP as the system size tends to infinity. Moreover, under a simple dissipative
condition, we prove the existence and uniqueness of the stationary distribution
and the exponential ergodicity for the PDMP limit via the coupling method.
Further extensions and applications to single-cell stochastic gene expression
kinetics and bursty stochastic gene regulatory networks are also discussed and
the convergence of the stationary distribution of the GDDMC model to that of
the PDMP model is also proved.
