The coefficients in a second order parabolic linear stochastic partial differential equation (SPDE) are estimated from multiple spatially localised measurements. Assuming that the spatial resolution tends to zero and the number of measurements is non-decreasing, the rate of convergence for each coefficient depends on its differential order and is faster for higher order coefficients. Based on an explicit analysis of the reproducing kernel Hilbert space of a general stochastic evolution equation, a Gaussian lower bound scheme is introduced. As a result, minimax optimality of the rates as well as sufficient and necessary conditions for consistent estimation are established.

翻译：本文基于多重空间局部化测量数据，对二阶抛物型线性随机偏微分方程（SPDE）的系数进行估计。假设空间分辨率趋于零且测量次数非递减，各系数的收敛速率取决于其微分阶数，高阶系数具有更快的收敛速率。通过对一般随机发展方程再生核希尔伯特空间的显式分析，引入了高斯下界估计方案。由此建立了速率的最小最大最优性，以及一致估计的充分必要条件。