Stochastic partial differential equations (SPDEs) form a convenient class of mathematical models for random spatio-temporal dynamics. Of particular interest are stochastic reaction-diffusion equations, or more generally, semi-linear SPDEs. In this talk, I will first give a brief introduction to semi-linear SPDEs and the statistical questions related to them. Subsequently, an estimator for the reaction intensity will be deduced. Consistent inference is achieved by studying a small diffusivity level, which is realistic in applications. The main result is a central limit theorem for the estimation error of a parametric estimator, from which confidence intervals can be constructed. Statistical efficiency is demonstrated by establishing local asymptotic normality. Local observations allow for non-parametric estimation of a reaction intensity varying in time and space. The statistical analysis requires advanced tools from stochastic analysis like Malliavin calculus for SPDEs, the infinite-dimensional Gaussian Poincaré inequality and regularity results for SPDEs in $L^p$-interpolation spaces.
This is joint work with Markus Reiß.