  We aim at estimating in a non-parametric way the density $\pi$ of the
stationary distribution of a $d$-dimensional stochastic differential equation
$(X_t)_{t \in [0, T]}$, for $d \ge 2$, from the discrete observations of a
finite sample $X_{t_0}$, ... , $X_{t_n}$ with $0= t_0 < t_1 < ... < t_n =:
T_n$. We propose a kernel density estimator and we study its convergence rates
for the pointwise estimation of the invariant density under anisotropic
H\"older smoothness constraints. First of all, we find some conditions on the
discretization step that ensures it is possible to recover the same rates as if
the continuous trajectory of the process was available. Such rates are optimal
and new in the context of density estimator. Then we deal with the case where
such a condition on the discretization step is not satisfied, which we refer to
as intermediate regime. In this new regime we identify the convergence rate for
the estimation of the invariant density over anisotropic H\"older classes,
which is the same convergence rate as for the estimation of a probability
density belonging to an anisotropic H\"older class, associated to $n$ iid
random variables $X_1, ..., X_n$. After that we focus on the asynchronous case,
in which each component can be observed at different time points. Even if the
asynchronicity of the observations complexifies the computation of the variance
of the estimator, we are able to find conditions ensuring that this variance is
comparable to the one of the continuous case. We also exhibit that the non
synchronicity of the data introduces additional bias terms in the study of the
estimator.
