Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene

We extend the so-called lower-bound technique for equicontinuous families of Markov operators by introducing the new concept of uniform equicontinuity on balls. Combined with a semi-concentrating condition, it yields a new abstract mathematical result on existence and uniqueness of invariant measures for Markov operators. It allows us to show the tightness of the set of invariant measures for some classes of Markov operators. This, in turn, gives a useful tool for proving a continuous dependence on given parameters for semi–concentrating Markov semigroups. In the second part we formulate an abstract modelling framework that defines a piecewise deterministic Markov process whose transition operator at the times of intervention yields a semi-concentrating Markov operator that is uniformly equicontinuous on balls. We show that this framework applies to a detailed stochastic model for an autoregulated gene in a bacterium that takes random transcription delay into account.