The non-parametric estimation of a non-linear reaction term in a semi-linear parabolic stochastic partial differential equation (SPDE) is discussed. The estimation error can be bounded in terms of the diffusivity and the noise level. The estimator is easily computable and consistent under general assumptions due to the asymptotic spatial ergodicity of the SPDE as both the diffusivity and the noise level tend to zero. If the SPDE is driven by space-time white noise, a central limit theorem for the estimation error and minimax-optimality of the convergence rate are obtained. The analysis of the estimation error requires the control of spatial averages of non-linear transformations of the SPDE, and combines the Clark-Ocone formula from Malliavin calculus with the Markovianity of the SPDE. In contrast to previous results on the convergence of spatial averages, the obtained variance bound is uniform in the Lipschitz-constant of the transformation. Additionally, new upper and lower Gaussian bounds for the marginal (Lebesgue-) densities of the SPDE are required and derived.

翻译：讨论了半线性抛物型随机偏微分方程（SPDE）中非线性反应项的非参数估计问题。估计误差可表示为扩散率与噪声水平的函数。基于SPDE在扩散率和噪声水平均趋近于零时的渐近空间遍历性，该估计量易于计算且在一般假设下具有相合性。当SPDE由时空白噪声驱动时，得到了估计误差的中心极限定理及收敛速度的极小极大最优性。误差分析需要控制SPDE非线性变换的空间平均值，并融合了Malliavin分析中的Clark-Ocone公式与SPDE的马尔可夫性。与以往空间平均值收敛性结果不同，本文获得的方差界对变换的Lipschitz常数具有一致性。此外，还推导并使用了SPDE边缘（勒贝格）密度的新高斯上下界。