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 rate of estimation of $\pi(x)$ as $T$ goes to infinity. One finding is that, for $d \ge 3$, the rate of estimation depends on the smoothness $\beta = (\beta_1, ... , \beta_d)$ of $\pi$. In particular, having ordered the smoothness such that $\beta_1 \le ... \le \beta_d$, it depends on the fact that $\beta_2 < \beta_3$ or $\beta_2 = \beta_3$. We show that kernel density estimators achieve the rate $(\frac{\log T}{T})^\gamma$ in the first case and $(\frac{1}{T})^\gamma$ in the second, for an explicit exponent $\gamma$ depending on the dimension and on $\bar{\beta}_3$, the harmonic mean of the smoothness over the $d$ directions after having removed $\beta_1$ and $\beta_2$, the smallest ones. Moreover, we obtain a minimax lower bound on the $\mathbf{L}^2$-risk for the pointwise estimation with the same rates $(\frac{\log T}{T})^\gamma$ or $(\frac{1}{T})^\gamma$, depending on the value of $\beta_2$ and $\beta_3$.