  We construct a strong Feller process associated with $-\Delta + \sigma \cdot
\nabla$, with drift $\sigma$ in a wide class of measures (weakly form-bounded
measures, e.g. combining weak $L^d$ and Kato class measure singularities), by
exploiting a quantitative dependence of the smoothness of the domain of an
operator realization of $-\Delta + \sigma \cdot \nabla$ generating a
holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$, $p>d-1$, on the value of the
form-bound of $\sigma$. Our method admits extension to other types of
perturbations of $-\Delta$ or $(-\Delta)^{\frac{\alpha}{2}}$, e.g. to yield new
$L^p$-regularity results for Schr\"odinger operators with form-bounded measure
potentials.
