Weighted Hardy's inequalities and Kolmogorov-type operators

We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{\varphi^2} {|x|^2}d\mu\leq\int_{\mathbb{R}^N}|\nabla \varphi |^2 d\mu+C\int_{\mathbb{R}^N} \varphi^2d\mu,\quad \varphi\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,\mu}, \] with respect to a probability measure $d\mu$. Moreover, the optimality of the constant $c_{0,\mu}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(\Delta u+\frac{\nabla \mu}{\mu}\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $d\mu$ allow the drift term to be of type $\frac{\nabla \mu}{\mu}= -|x|^{m-2}x$ with $m> 0$.