We study the realization $A$ of the operator $\mathcal{A} =\frac{1}{2} \triangle - (DU, D\cdot)$ in $L^{2}(\Omega, \mu)$, where $\Omega$ is a possibly unbounded convex open set in $\mathbb{R}^{N}$, $U$ is a convex unbounded function such that $\lim_{x \rightarrow \partial \Omega, \, x \in \Omega} U(x) = + \infty$ and $\lim_{|x| \rightarrow + \infty, \, x \in \Omega} U(x) = + \infty$, $DU(x)$ is the element with minimal norm in the subdifferential of $U$ at $x$, and $\mu(dx) = c \exp (-2 U(x)) dx$ is a probability measure, infinitesimally invariant for $\mathcal{A}$. We show that $A$, with domain $D(A) = \{u \in H^{2}(\Omega,\mu): (DU, Du) \in L^{2}(\Omega,\mu)\}$ is a dissipative self-adjoint operator in $L^{2}(\Omega,\mu)$. Note that the functions in the domain of $A$ do not satisfy any particular boundary condition. Log-Sobolev and Poincaré inequalities allow then to study smoothing properties and asymptotic behavior of the semigroup generated by $A$.