  We present some analytic, non-perturbative results for the invariant density
rho(x) for noisy one-dimensional maps at fully developed chaos. Under periodic
boundary conditions, the Fourier expansion method is used to show precisely how
noise makes rho(x) absolutely continuous and smoothens it out. Simple solvable
models are used to illustrate the explicit dependence of rho(x) on the
amplitude eta of the noise distribution, all the way from the case of zero
noise (eta > 0) to the completely noise-dominated limit (eta=1).
