Elliptic operators with unbounded drift coefficients and Neumann boundary condition

AbstractWe study the realization AN of the operator A=12Δ−〈DU,D·〉 in L2(Ω,μ) with Neumann boundary condition, where Ω 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 μ(dx)=cexp(−2U(x))dx is a probability measure, infinitesimally invariant for A. We show that AN is a dissipative self-adjoint operator in L2(Ω,μ). Log-Sobolev and Poincaré inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by AN