Dirichlet boundary conditions for elliptic operators with unbounded drift

We study the realisation $A$ of the operator $\mathcal{A} = \Delta - \langle D\Phi, D\cdot \rangle$ in $L^2(\Omega, \mu)$ with Dirichlet boundary condition, where $\Omega$ is a possibly unbounded open set in $\mathbb{R} ^N$, $\Phi$ is a semi-convex function and the measure $d\mu(x) = \exp(-\Phi(x))\,dx$ lets $\mathcal{A}$ be formally self-adjoint. The main result is that $A:D(A)= \{u\in H^2(\Omega, \mu): \langle D\Phi , Du \rangle \in L^2(\Omega, \mu), \,u=0$ at $\partial \Omega\}$ is a dissipative self-adjoint operator in $L^2(\Omega, \mu)$.