On a class of self-adjoint elliptic operators in \(L^2\) spaces with respect to invariant measures (Q870095)

The authors deal with a differential operator \(A\) of the form \[ (Au)(x)=\tfrac12 \Delta u(x)- \langle DU(x),DU(x)\rangle,\;x\in\Omega, \] where \(\Omega\subset\mathbb R^d\) is a (possibly unbounded) convex open set, and \(U: \Omega\to\mathbb R\) is a convex function, such that \(\lim_{|x|\to\infty} U(x)=+\infty\). The symbol \(D\) denotes the gradient. In the present study the authors show that a realization of \(A\) in \(L^2(\Omega,\mu)\), with suitable domain \(D(A)\) and \(\mu(dx)= \exp(-2U(x))\,dx\), is self-adjoint and dissipative. Moreover, several properties of \(A\) and of the measure \(\mu\) are discussed.