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In a recent paper by Kurasov, L\"uck, Mugnolo and Wolf, a hybrid generegulatory network was proposed to model gene expression dynamics by using astochastic system of coupled partial differential equations. This approachapproximates protein counts in cells by modelling them as distributions. In afollow-up paper, the existence and strong convergence of the solutions toequilibrium was proven. In this paper, we make use of a recent convergencetheorem for stochastic, irreducible semigroups that contain partial integraloperators. In particular, we improve upon their results by showing that theconvergence rate is independent of the initial distribution of proteins,therefore proving that the solutions converge not only strongly but evenuniformly to equilibrium.

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