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

Abstract

Let \({\mathcal{S}}\) be a locally compact Hausdorff space. Let A and B be two generators of Feller semigroups in \({C_0(\mathcal{S})}\) with related Feller processes {XA(t), t ≥ 0} and {XB(t), t ≥ 0} and let α and β be two non-negative continuous functions on \({\mathcal{S}}\) with α + β = 1. Assume that the closure C of C0 = αA + βB with \({\mathcal{D}(C_0) = \mathcal{D}(A) \cap \mathcal{D}(B)}\) generates a Feller semigroup {TC(t), t ≥ 0} in \({C_0(\mathcal{S})}\) . It is natural to think of a related Feller process {XC(t), t ≥ 0} as that evolving according to the following heuristic rules. Conditional on being at a point \({p \in \mathcal{S}}\) , with probability α(p) the process behaves like {XA(t), t ≥ 0} and with probability β(p) it behaves like {XB(t), t ≥ 0}. We provide an approximation of {TC(t), t ≥ 0} via a sequence of semigroups acting in \({C_0(\mathcal{S}) \times C_0(\mathcal{S})}\) that supports this interpretation. This work is motivated by the recent model of stochastic gene expression due to Lipniacki et al. [17].