We study the problem of the nonparametric estimation for the density $\pi$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{t \in [0, T]}$ with possibly unbounded drift. 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 $\beta_3$.

翻译：我们研究$d$维随机微分方程$(X_t)_{t \in [0, T]}$（可能具有无界漂移项）的平稳分布密度$\pi$的非参数估计问题。基于$[0, T]$上的连续采样路径观测，我们研究当$T$趋于无穷时$\pi(x)$的估计速率。一个发现是，对于$d \ge 3$，估计速率取决于$\pi$的光滑性$\beta = (\beta_1, ... , \beta_d)$。特别地，在将光滑性排序为$\beta_1 \le ... \le \beta_d$后，该速率取决于$\beta_2 < \beta_3$还是$\beta_2 = \beta_3$。我们证明，核密度估计器在第一种情形下达到速率$(\frac{\log T}{T})^\gamma$，在第二种情形下达到速率$(\frac{1}{T})^\gamma$，其中显式指数$\gamma$依赖于维数以及$\bar{\beta}_3$（即移除最小光滑性$\beta_1$和$\beta_2$后，$d$个方向上光滑性的调和均值）。此外，我们获得了点态估计$\mathbf{L}^2$风险的极小极大下界，其速率同样为$(\frac{\log T}{T})^\gamma$或$(\frac{1}{T})^\gamma$，具体取决于$\beta_2$与$\beta_3$的取值。