In this talk, we will revisit the problem of nonparametric estimation in a second-order linear stochastic partial differential equation (SPDE) with additive noise. In the first part, we study point estimators of a space-dependent diffusivity and of a space-dependent noise level. For discrete observations in time and space of the corresponding Gaussian random field, we construct moment estimators and obtain central limit theorems. The proofs rely on semigroup theory and adapt the ideas from [1] for local measurements, which average the solution over small neighbourhoods in space. In the second part, we will discuss some ideas on obtaining a nonparametric LAN (local asymptotic normality) expansion of the likelihood process, relative to the diffusivity function in the setting of [1]. This opens up the road to efficient estimation. The proof reveals interesting aspects of partially observed space-time random fields. It relies on stochastic filtering techniques, on the reproducing kernel Hilbert space of the observation process, which was recently computed in [2], as well as on the solution of certain Wiener–Hopf integral equations.

This is partly based on joint work with F. Hildebrandt and M. Trabs.

[1] R. Altmeyer and M. Reiß. Nonparametric estimation for linear SPDEs from local measurements. Annals of Applied Probability, 2021.

[2] R. Altmeyer, A. Tiepner and M. Wahl. Optimal parameter estimation for linear SPDEs from multiple measurements. arXiv preprint arXiv:2211.02496, 2022.