  We study the problem of estimating the coefficients of a diffusion (X_t,t\geq
0); the estimation is based on discrete data X_{n\Delta},n=0,1,...,N. The
sampling frequency \Delta^{-1} is constant, and asymptotics are taken as the
number N of observations tends to infinity. We prove that the problem of
estimating both the diffusion coefficient (the volatility) and the drift in a
nonparametric setting is ill-posed: the minimax rates of convergence for
Sobolev constraints and squared-error loss coincide with that of a,
respectively, first- and second-order linear inverse problem. To ensure
ergodicity and limit technical difficulties we restrict ourselves to scalar
diffusions living on a compact interval with reflecting boundary conditions.
Our approach is based on the spectral analysis of the associated Markov
semigroup. A rate-optimal estimation of the coefficients is obtained via the
nonparametric estimation of an eigenvalue-eigenfunction pair of the transition
operator of the discrete time Markov chain (X_{n\Delta},n=0,1,...,N) in a
suitable Sobolev norm, together with an estimation of its invariant density.
