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 . We obtain new conditions on F under which T_t is not analytic in . In particular, we prove that the one-dimensional operator Au=u″−x³u′, with domain , , is not sectorial in . Under suitable hypotheses on the growth of F, we introduce a class of non-analytic Markov semigroups in , where μ is an invariant measure for T_t.