Some classes of non-analytic Markov semigroups

We deal with Markov semigroups Tt corresponding to second order elliptic operators Au = Δu+ Du,F , where F is an unbounded locally Lipschitz vector field on RN. We obtain new conditions on F under which Tt is not analytic in Cb(RN). In particular, we prove that the one-dimensional operator Au = u − x3u , with domain {u ∈ C2(R): u, u − x3u ∈ Cb(R)}, is not sectorial in Cb(R). Under suitable hypotheses on the growth of F, we introduce a class of non-analytic Markov semigroups in Lp(RN,μ), where μ is an invariant measure for Tt .  2004 Elsevier Inc. All rights reserved