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：针对带乘法噪声的随机热方程，研究了拉普拉斯算子前扩散系数的估计问题。基于空间局部观测，首先分析了针对加性噪声设计的估计量。稳定的中心极限定理表明该估计量具有相合性和渐近混合正态性。通过引入二次变差，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趋零时仍有效。证明过程依托于局部近似结果以克服复杂的非线性结构，并利用关于柱布朗运动的随机积分稳定中心极限定理。数值模拟结果验证了理论发现。