  Let $(X_t)_{t \ge 0}$ be solution of a one-dimensional stochastic
differential equation. Our aim is to study the convergence rate for the
estimation of the invariant density in intermediate regime, assuming that a
discrete observation of the process $(X_t)_{t \in [0, T]}$ is available, when
$T$ tends to $\infty$. We find the convergence rates associated to the kernel
density estimator we proposed and a condition on the discretization step
$\Delta_n$ which plays the role of threshold between the intermediate regime
and the continuous case. In intermediate regime the convergence rate is $n^{-
\frac{2 \beta}{2 \beta + 1}}$, where $\beta$ is the smoothness of the invariant
density. After that, we complement the upper bounds previously found with a
lower bound over the set of all the possible estimator, which provides the same
convergence rate: it means it is not possible to propose a different estimator
which achieves better convergence rates. This is obtained by the two hypotheses
method; the most challenging part consists in bounding the Hellinger distance
between the laws of the two models. The key point is a Malliavin representation
for a score function, which allows us to bound the Hellinger distance through a
quantity depending on the Malliavin weight.
