Elliptic operators with unbounded drift coefficients and Neumann boundary condition

We study the realization AN of the operator A=1/2(delta)-(left angle bracket)DU, D.(right -pointing angle bracket) in L^2 ((omega),(mu)) with Neumann boundary condition, where (omega) is a possibly unbounded convex open set in ,RN, 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 AN 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 AN.