Markov semigroups and groups of operators

We consider different realizations of the operators $ L_{\theta, a} \ u(x) \ := \ x^{2 a} u''(x) \ + \ (a x^{2 a - 1} + \theta x^a) u'(x)$, $\theta\in\R$, $a\in\R$, acting on suitable spaces of real valued continuous functions. The main results deal with the existence of Feller semigroups generated by $ L_{\theta, a}$ and the representation $L_{\theta, a}=G^2_a+\theta G_a$, where $G_a u=x^a u'$, $0\le a\le 1$, generates a (not necessarily strongly continuous) group. Explicit formulas of the generated semigroups are also deduced.