  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.
