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We aim at estimating in a non-parametric way the density $\pi$ of thestationary distribution of a $d$-dimensional stochastic differential equation$(X_t)_{t \in [0, T]}$, for $d \ge 2$, from the discrete observations of afinite 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 ratesfor the pointwise estimation of the invariant density under anisotropicH\"older smoothness constraints. First of all, we find some conditions on thediscretization step that ensures it is possible to recover the same rates as ifthe continuous trajectory of the process was available. Such rates are optimaland new in the context of density estimator. Then we deal with the case wheresuch a condition on the discretization step is not satisfied, which we refer toas intermediate regime. In this new regime we identify the convergence rate forthe estimation of the invariant density over anisotropic H\"older classes,which is the same convergence rate as for the estimation of a probabilitydensity belonging to an anisotropic H\"older class, associated to $n$ iidrandom 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 theasynchronicity of the observations complexifies the computation of the varianceof the estimator, we are able to find conditions ensuring that this variance iscomparable to the one of the continuous case. We also exhibit that the nonsynchronicity of the data introduces additional bias terms in the study of theestimator.

Estimation of the invariant density for discretely observed diffusion processes: impact of the sampling and of the asynchronicity