Ultraboundedness for parabolic equations in convex domains without boundary conditions

We consider the operator Au=12Δu−〈DU,Du〉, where U is a convex real function defined in a convex open set O⊂RN and lim|x|→∞U(x)=limx→∂OU(x)=+∞. We prove that the associated Markov semigroup is ultrabounded with respect to the Gibbs measure e−2U(x)dx.