Markov Semigroups and Groups of Operators

We consider different realizations of the operators Lθ,a u(x) := xu(x) + (ax + θx)u(x), θ ∈ R, a ∈ R, acting on suitable spaces of real valued continuous functions. The main results deal with the existence of Feller semigroups generated by Lθ,a and the representation Lθ,a = G 2 a + θGa, where Gau = xu, 0 ≤ a ≤ 1, generates a (not necessarily strongly continuous) group. Explicit formulas of the generated semigroups are also deduced.