Ultraboundedness for parabolic equations in convex domains without boundary conditions

We consider the operator A u = 1 2 Δ u − 〈 D U , D u 〉 , where U is a convex real function defined in a convex open set O ⊂ R N and lim | x | → ∞ U ( x ) = lim x → ∂ O U ( x ) = + ∞ . We prove that the associated Markov semigroup is ultrabounded with respect to the Gibbs measure e − 2 U ( x ) d x .