Summary: We prove uniqueness of “invariant measures”, i.e., solutions to the equation \(L^*\mu = 0\) where \(L=\Delta+B\cdot \nabla\) on \(\mathbb{R}^n\) with \(B\) satisfying some mild integrability conditions and \(\mu\) being a probability measure on \(\mathbb{R}^n\). This solves an open problem posed by S. R. S. Varadhan in 1980. The same conditions are shown to imply that the closure of \(L\) on \(L^1(\mu)\) generates a strongly continuous semigroup having \(\mu\) as its unique invariant measure. The question whether an extension of \(L\) generates a strongly continuous semigroup on \(L^1(\mu)\) and whether such an extension is unique is addressed separately and answered positively under even weaker local integrability conditions on \(B\). The special case when \(B\) is a gradient of a function (i.e., the “symmetric case”) in particular is studied and conditions are identified ensuring that \(L^*\mu=0\) implies that \(L\) is symmetric on \(L^2(\mu)\) or \(L^*\mu = 0\) has a unique solution. We also prove infinite-dimensional analogues of the latter two results and a new elliptic regularity theorem for invariant measures in infinite dimensions.