  We consider the estimation of a non-linear reaction term in the stochastic
heat or more generally in a semi-linear stochastic partial differential
equation (SPDE). Consistent inference is achieved by studying a small
diffusivity level, which is realistic in applications. Our 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. The estimation method
is extended to local observations in time and space, which allows for
non-parametric estimation of a reaction intensity varying in time and space.
Furthermore, discrete observations in time and space can be handled. The
statistical analysis requires advanced tools from stochastic analysis like
Malliavin calculus for SPDEs, the infinite-dimensional Gaussian Poincar\'e
inequality and regularity results for SPDEs in $L^p$-interpolation spaces.
