Nonparametric estimation for semilinear SPDEs, namely stochastic reaction-diffusion equations in one space dimension, is studied. We consider observations of the solution field on a discrete grid in time and space with infill asymptotics in both coordinates. Firstly, we derive a nonparametric estimator for the reaction function of the underlying equation. The estimate is chosen from a finite-dimensional function space based on a least squares criterion. Oracle inequalities provide conditions for the estimator to achieve the usual nonparametric rate of convergence. Adaptivity is provided via model selection. Secondly, we show that the asymptotic properties of realized quadratic variation based estimators for the diffusivity and volatility carry over from linear SPDEs. In particular, we obtain a rate-optimal joint estimator of the two parameters. The result relies on our precise analysis of the H\"older regularity of the solution process and its nonlinear component, which may be of its own interest. Both steps of the calibration can be carried out simultaneously without prior knowledge of the parameters.

翻译：本文研究了半线性随机偏微分方程（即一维空间中的随机反应-扩散方程）的非参数估计问题。我们考虑在时间与空间离散网格上观测解场，并采用两个坐标方向上的渐近稠密填充方法。首先，我们构建了底层方程反应函数的非参数估计量。该估计量基于最小二乘准则从有限维函数空间中选择。Oracle不等式提供了该估计量实现通常非参数收敛率的条件，并通过模型选择实现自适应性。其次，我们证明了基于实现二次变分的扩散率与波动率估计量的渐近性质可从线性随机偏微分方程推广而来。特别地，我们得到了这两个参数的速率最优联合估计量。这一结果依赖于对解过程及其非线性分量的Hölder正则性的精确分析，该分析本身可能具有独立研究价值。校准的两个步骤可在无需参数先验知识的情况下同步完成。