Summary: We give necessary and sufficient conditions for the existence of a weak solution of a Kolmogorov equation perturbed by an inverse-square potential. More precisely, using a weighted Hardy’s inequality with respect to an invariant measure \(\mu\), we show the existence of the semigroup solution of the parabolic problem corresponding to a generalized Ornstein-Uhlenbeck operator perturbed by an inverse-square potential in \(L^2(\mathbb{R}^2,\mu)\). In the case of the classical Ornstein-Uhlenbeck operator we obtain nonexistence of positive exponentially bounded solutions of the parabolic problem if the coefficient of the inverse-square function is too large.