We investigate pointwise estimation of the function-valued velocity field of a second-order linear SPDE. Based on multiple spatially localised measurements, we construct a weighted augmented MLE and study its convergence properties as the spatial resolution of the observations tends to zero and the number of measurements increases. By imposing H\"older smoothness conditions, we recover the pointwise convergence rate known to be minimax-optimal in the linear regression framework. The optimality of the rate in the current setting is verified by adapting the lower bound ansatz based on the RKHS of local measurements to the nonparametric situation.

翻译：我们研究二阶线性随机偏微分方程中函数值速度场的逐点估计问题。基于多个空间局部测量，我们构造了加权增广极大似然估计量，并研究其在观测空间分辨率趋于零且测量数量增加时的收敛性质。通过施加赫尔德光滑性条件，我们恢复了在线性回归框架下已知的极小极大最优逐点收敛速率。通过将基于局部测量再生核希尔伯特空间的下界方法推广至非参数情形，验证了当前设定下该速率的最优性。