A Piecewise Deterministic Limit for a Multiscale Stochastic Spatial Gene Network

TL;DR

This paper derives a piecewise deterministic limit for multiscale stochastic spatial gene networks, combining PDEs and jump processes to model different abundance scales and proving convergence of the stochastic model to a PDMP.

Contribution

It introduces a novel multiscale framework linking PDEs and jump processes in stochastic spatial gene networks, establishing convergence to a PDMP.

Findings

Global weak limit is an infinite dimensional PDMP.

Convergence in the supremum norm is proven.

The model captures different abundance scales in gene networks.

Abstract

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.