Regularity of solutions to stationary Fokker-Planck-Kolmogorov equations

We discuss regularity of solutions to double divergence form equations of the form
$$ \partial_{x_i}\partial_{x_j}(a^{ij}\mu) - \partial_{x_i}(b^i\mu)=0 $$
with respect to measures on $\mathbb{R}^d$, where $(a^{ij})$ is the diffusion matrix and $b=(b^i)$ is the drift coefficient. The equation is understood in the sense of distributions, so the coefficients can be rather irregular. The key problems concern the existence of solution densities and their regularity properties, and also the existence and uniqueness of probability solutions. In particular, we discuss some recent results obtained jointly with Röckner and Shaposhnikov on Zvonkin's transform of the drift coefficient, which enables one to improve the drift.