Non-parametric estimation of the reaction term in semi-linear SPDEs with spatial ergodicity

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.