  For the stochastic heat equation with multiplicative noise we consider the
problem of estimating the diffusivity parameter in front of the Laplace
operator. Based on local observations in space, we first study an estimator
that was derived for additive noise. A stable central limit theorem shows that
this estimator is consistent and asymptotically mixed normal. By taking into
account the quadratic variation, we propose two new estimators. Their limiting
distributions exhibit a smaller (conditional) variance and the last estimator
also works for vanishing noise levels. The proofs are based on local
approximation results to overcome the intricate nonlinearities and on a stable
central limit theorem for stochastic integrals with respect to cylindrical
Brownian motion. Simulation results illustrate the theoretical findings.
