We consider the problem of density estimation in the context of multiscale Langevin diffusion processes, where a single-scale homogenized surrogate model can be derived. In particular, our aim is to learn the density of the invariant measure of the homogenized dynamics from a continuous-time trajectory generated by the full multiscale system. We propose a spectral method based on a truncated Fourier expansion with Hermite functions as orthonormal basis. The Fourier coefficients are computed directly from the data owing to the ergodic theorem. We prove that the resulting density estimator is robust and converges to the invariant density of the homogenized model as the scale separation parameter vanishes, provided the time horizon and the number of Fourier modes are suitably chosen in relation to the multiscale parameter. The accuracy and reliability of this methodology is further demonstrated through a series of numerical experiments.

翻译：我们考虑多尺度Langevin扩散过程中的密度估计问题，其中可推导出单尺度同质化代理模型。具体而言，我们的目标是从完整多尺度系统生成的连续时间轨迹中学习同质化动力学不变测度的密度。我们提出一种基于截断傅里叶展开的谱方法，以Hermite函数作为正交基。借助遍历定理，傅里叶系数可直接从数据计算得出。我们证明：当尺度分离参数趋近于零时，只要时间范围和傅里叶模式数量相对于多尺度参数进行适当选择，所得密度估计量具有鲁棒性，并收敛于同质化模型的不变密度。通过一系列数值实验进一步验证了该方法的准确性和可靠性。