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Let S be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in C0(S) with related Feller processes {X A (t), t ≥ 0} and {X B (t), t ≥ 0} and let α and β be two non-negative continuous functions on S with α + β = 1. Assume that the closure C of C 0 = αA + βB with D(C0)=D(A)∩D(B) generates a Feller semigroup {T C (t), t ≥ 0} in C0(S) . It is natural to think of a related Feller process {X C (t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point p∈S , with probability α(p) the process behaves like {X A (t), t ≥ 0} and with probability β(p) it behaves like {X B (t), t ≥ 0}. We provide an approximation of {T C (t), t ≥ 0} via a sequence of semigroups acting in C0(S)×C0(S) that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki et al. [17].
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