We estimate nonparametrically the spatially varying diffusivity of a stochastic heat equation from observations perturbed by additional noise. To that end, we employ a two-step localization procedure, more precisely, we combine local state estimates into a locally linear regression approach. Our analysis relies on quantitative Trotter--Kato type approximation results for the heat semigroup that are of independent interest. The presence of observational noise leads to non-standard scaling behaviour of the model. Numerical simulations illustrate the results.

翻译：本文针对受附加噪声干扰的观测数据，对随机热方程的空间变化扩散率进行非参数估计。为此，我们采用两步局部化方法，具体而言，将局部状态估计与局部线性回归方法相结合。本研究的理论分析基于对热半群的定量Trotter-Kato型逼近结果，该结果本身具有独立的学术价值。观测噪声的存在导致模型呈现出非标准的尺度特性。数值模拟结果验证了本文所提方法的有效性。