Minimax rate of estimation for invariant densities associated to continuous stochastic differential equations over anisotropic Holder classes

We study the problem of the nonparametric estimation for the density $\pi$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$. From the continuous observation of the sampling path on $[0, T]$, we study the estimation of $\pi(x)$ as $T$ goes to infinity. For $d\ge2$, we characterize the minimax rate for the $\mathbf{L}^2$-risk in pointwise estimation over a class of anisotropic H\"older functions $\pi$ with regularity $\beta = (\beta_1, ... , \beta_d)$. For $d \ge 3$, our finding is that, having ordered the smoothness such that $\beta_1 \le ... \le \beta_d$, the minimax rate depends on whether $\beta_2 < \beta_3$ or $\beta_2 = \beta_3$. In the first case, this rate is $(\frac{\log T}{T})^\gamma$, and in the second case, it is $(\frac{1}{T})^\gamma$, where $\gamma$ is an explicit exponent dependent on the dimension and $\bar{\beta}_3$, the harmonic mean of smoothness over the $d$ directions after excluding $\beta_1$ and $\beta_2$, the smallest ones. We also demonstrate that kernel-based estimators achieve the optimal minimax rate. Furthermore, we propose an adaptive procedure for both $L^2$ integrated and pointwise risk. In the two-dimensional case, we show that kernel density estimators achieve the rate $\frac{\log T}{T}$, which is optimal in the minimax sense. Finally we illustrate the validity of our theoretical findings by proposing numerical results.