We introduce a new approach for estimating the invariant density of a multidimensional diffusion when dealing with high-frequency observations blurred by independent noises. We consider the intermediate regime, where observations occur at discrete time instances $k\Delta_n$ for $k=0,\dots,n$, under the conditions $\Delta_n\to 0$ and $n\Delta_n\to\infty$. Our methodology involves the construction of a kernel density estimator that uses a pre-averaging technique to proficiently remove noise from the data while preserving the analytical characteristics of the underlying signal and its asymptotic properties. The rate of convergence of our estimator depends on both the anisotropic regularity of the density and the intensity of the noise. We establish conditions on the intensity of the noise that ensure the recovery of convergence rates similar to those achievable without any noise. Furthermore, we prove a Bernstein concentration inequality for our estimator, from which we derive an adaptive procedure for the kernel bandwidth selection.

翻译：我们提出了一种新方法，用于在高频观测被独立噪声模糊的情况下估计多维扩散的不变密度。我们考虑中间机制，即观测发生在离散时间点 $k\Delta_n$（$k=0,\dots,n$），条件为 $\Delta_n\to 0$ 且 $n\Delta_n\to\infty$。我们的方法涉及构建一个核密度估计器，该估计器采用预平均技术有效去除数据中的噪声，同时保留底层信号的分析特性及其渐近性质。该估计器的收敛速度取决于密度的各向异性正则性和噪声强度。我们建立了噪声强度的条件，以确保恢复与无噪声情况下相似的收敛速度。此外，我们证明了估计器的伯恩斯坦浓度不等式，并据此推导出核带宽选择的自适应程序。