Uniform convergence of solutions to stochastic hybrid models of gene regulatory networks

Abstract: In a paper by Kurasov, L\"uck, Mugnolo and Wolf, a hybrid gene regulatory network was proposed to model gene expression dynamics by using a stochastic system of coupled partial differential equations. This approach approximates protein counts in cells by modelling them as distributions. In a follow-up paper, the existence and strong convergence of the solutions to equilibrium was proven. In this paper, we make use of a recent convergence theorem for stochastic, irreducible semigroups that contain partial integral operators. In particular, we improve upon their results by showing that the convergence rate is independent of the initial distribution of proteins, therefore proving that the solutions converge not only strongly but even uniformly to equilibrium.