Elliptic operators with unbounded drift coefficients and Neumann boundary condition

Abstract

We study the realization A N of the operator A= 1 2 Δ−〈DU,D·〉 in L 2(Ω,μ) with Neumann boundary condition, where Ω 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 μ(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(Ω,μ) . Log-Sobolev and Poincaré inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by A N .