  We study the stochastic stability in the zero-noise limit from a quantitative
point of view.
  We consider smooth expanding maps of the circle, perturbed by additive noise.
We show that in this case the zero-noise limit has a quadratic speed of
convergence, as conjectured by Lin, in 2005, after numerical experiments (see
arXiv:math/0406201 ). This is obtained by providing an explicit formula for the
first and second term in the Taylor's expansion of the response of the
stationary measure to the small noise perturbation. These terms depend on
important features of the dynamics and of the noise which is perturbing it, as
its average and variance.
  We also consider the zero-noise limit from a quantitative point of view for
piecewise expanding maps showing estimates for the speed of convergence in this
case.
