Perturbation theory for the Fokker–Planck operator in chaos

•A perturbation expansion for the Fokker–Planck operator of a noisy chaotic map is derived.•Long-time observables of a deterministically perturbed system are approximated by means of the natural measure of the unperturbed one.•Perturbation theory is successfully tested on the 2D Lozi attractor. The stationary distribution of a fully chaotic system typically exhibits a fractal structure, which dramatically changes if the dynamical equations are even slightly modified. Perturbative techniques are not expected to work in this situation. In contrast, the presence of additive noise smooths out the stationary distribution, and perturbation theory becomes applicable. We show that a perturbation expansion for the Fokker–Planck evolution operator yields surprisingly accurate estimates of long-time averages in an otherwise unlikely scenario.