  The coefficient function of the leading differential operator is estimated
from observations of a linear stochastic partial differential equation (SPDE).
The estimation is based on continuous time observations which are localised in
space. For the asymptotic regime with fixed time horizon and with the spatial
resolution of the observations tending to zero, we provide rate-optimal
estimators and establish scaling limits of the deterministic PDE and of the
SPDE on growing domains. The estimators are robust to lower order perturbations
of the underlying differential operator and achieve the parametric rate even in
the nonparametric setup with a spatially varying coefficient. A numerical
example illustrates the main results.
