Elliptic operators with unbounded drift coefficients and Neumann boundary condition

We study the realization A_N of the operator A = 1/2 \Delta - < DU , D\cdot > in L^2(Omega, mu) with Neumann boundary condition, where Omega is a possibly unbounded convex open set in R^N, U is a convex unbounded function, DU(x) is the element with minimal norm in the subdifferential of U at x, and mu(dx) = c exp(-2U(x))dx is a probability measure, infinitesimally invariant for A. We show that A_N is a dissipative self-adjoint operator in L^2(Omega, mu). Log-Sobolev and Poincare' inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by A_N.