  We investigate stability of invariant measures of diffusion processes with
respect to $L^p$ distances on the coefficients, under an assumption of
log-concavity. The method is a variant of a technique introduced by Crippa and
De Lellis to study transport equations. As an application, we prove a partial
extension of an inequality of Ledoux, Nourdin and Peccati relating transport
distances and Stein discrepancies to a non-Gaussian setting via the moment map
construction of Stein kernels.
