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.

翻译：我们研究了d维随机微分方程$(X_t)_{t \in [0, T]}$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，我们研究了当$T$趋于无穷时$\pi(x)$的估计。对于$d \ge 2$，我们在各向异性Hölder函数类$\pi$（正则性$\beta = (\beta_1, ... , \beta_d)$）上刻画了点估计$\mathbf{L}^2$风险的极小极大速率。对于$d \ge 3$，我们的发现是：在设定光滑度排序$\beta_1 \le ... \le \beta_d$后，极小极大速率取决于$\beta_2 < \beta_3$或$\beta_2 = \beta_3$两种情形。第一种情形下该速率为$(\frac{\log T}{T})^\gamma$，第二种情形下为$(\frac{1}{T})^\gamma$，其中$\gamma是$显式指数，依赖于维度和排除最小两个光滑度$\beta_1,\beta_2$后$d$个方向光滑度的调和均值$\bar{\beta}_3$。我们同时证明了基于核的估计器可达到最优极小极大速率。此外，我们针对$L^2$积分风险和点态风险提出了自适应过程。在二维情形中，我们证明核密度估计器达到速率$\frac{\log T}{T}$，该速率在极小极大意义下是最优的。最后我们通过数值结果验证了理论发现的有效性。