Let $W\left(u\right)\left(x\right)=1/2x^a{(1-x)}^b{u}^{\text{'}\text{'}}\left(x\right)$ with a,b ≥ 2. We consider the C₀-semigroups generated by this operator on the spaces of continuous functions, respectively square integrable functions. The connection between these semigroups together with suitable approximation processes is studied. Also, some qualitative and quantitative properties are derived.

We study the compactness of Feller semigroups generated by second order elliptic partial differential operators with unbounded coefficients in spaces of continuous functions in ${ℝ}^N$.