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.

翻译：针对带有乘性噪声的随机热方程，我们考虑拉普拉斯算子前扩散系数参数的估计问题。基于空间局部观测，我们首先研究了为加性噪声情形设计的估计量。稳定的中心极限定理表明该估计量具有相合性与渐近混合正态性。通过引入二次变分，我们提出两种新型估计量。其极限分布具有更小的（条件）方差，且最后一种估计量在噪声强度趋近于零时仍然有效。证明过程基于局部逼近结果以克服复杂的非线性项，并利用柱形布朗运动随机积分的稳定中心极限定理。数值模拟结果验证了理论发现。