This paper is concerned with stochastic systems whose state is a diffusion process governed by an Ito stochastic differential equation (SDE). In the framework of a nominal white-noise model, the SDE is driven by a standard Wiener process. For a scenario of statistical uncertainty, where the driving noise acquires a state-dependent drift and thus deviates from its idealised model, we consider the perturbation of the invariant probability density function (PDF) as a steady-state solution of the Fokker-Planck-Kolmogorov equation. We discuss an upper bound on a logarithmic Dirichlet form for the ratio of the invariant PDF to its nominal counterpart in terms of the Kullback-Leibler relative entropy rate of the actual noise distribution with respect the Wiener measure. This bound is shown to be achievable, provided the PDF ratio is preserved by the nominal steady-state probability flux. The logarithmic Dirichlet form bound is used in order to obtain an upper bound on the relative entropy of the perturbed invariant PDF in terms of quadratic-exponential moments of the noise drift in the uniform ellipticity case. These results are illustrated for perturbations of Gaussian invariant measures in linear stochastic systems involving linear noise drifts.