Entropy bounds for invariant measure perturbations in stochastic systems with uncertain noise

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 white-noise model, we consider the resulting perturbation of the invariant probability density function (PDF) for the system 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 perturbed invariant PDF to its nominal counterpart which the system would have in the case of the standard Wiener process at the input. This bound is obtained in terms of the Kullback–Leibler relative entropy rate of the actual noise distribution with respect to the Wiener measure. We show that the bound is 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 under the conditions of uniform ellipticity on the diffusion matrix and strong concavity for the logarithm of the nominal invariant PDF. These results are illustrated for perturbations of Gaussian invariant measures in linear stochastic systems involving linear noise drifts.