  We consider multiscale stochastic spatial gene networks involving chemical
reactions and diffusions. The model is Markovian and the transitions are driven
by Poisson random clocks. We consider a case where there are two different
spatial scales: a microscopic one with fast dynamic and a macroscopic one with
slow dynamic. At the microscopic level, the species are abundant and for the
large population limit a partial differential equation (PDE) is obtained. On
the contrary at the macroscopic level, the species are not abundant and their
dynamic remains governed by jump processes. It results that the PDE governing
the fast dynamic contains coefficients which randomly change. The global weak
limit is an infinite dimensional continuous piecewise deterministic Markov
process (PDMP). Also, we prove convergence in the supremum norm.
