We study the operators associated to the stochastic gene regulatory network. We are aimed at proving their stability and continuous dependence upon random deviation. To that end, 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.