Some classes of non-analytic Markov semigroups

We deal with Markov semigroups T_t corresponding to second order elliptic operators Au = Δu+ Du F, where F is an unbounded locally Lipschitz vector field on R^N. We obtain new conditions on F under which T_t is not analytic in C_b(R^N). In particular, we prove that the one-dimensional operator Au = u'' − x^3 u' with domain {u ∈ C^2(R): u, u'' − x^3 u' \in C_b(R)}, is not sectorial in C_b(R). Under suitable hypotheses on the growth of F, we introduce a new class of non-analytic Markov semigroups in Lp(R^N,μ), where μ is an invariant measure for T_t