A Piecewise Deterministic Limit for a Multiscale Stochastic Spatial Gene Network

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