On a semigroup generated by a convex combination of two Feller generators

Let be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in 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 with α + β = 1. Assume that the closure C of C ₀ = αA + βB with generates a Feller semigroup {T _C (t), t ≥ 0} in . 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 , 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 that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki et al. [17].