The aim of this paper is to study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient for fractional stochastic heat equation driven by an additive space-time noise. We consider the traditional for stochastic partial differential equations statistical experiment when the measurements are performed in the spectral domain, and in contrast to the existing literature, we study the asymptotic properties of the maximum likelihood (type) estimators (MLE) when both, the number of Fourier modes and the time go to infinity. In the first part of the paper we consider the usual setup of continuous time observations of the Fourier coefficients of the solutions, and show that the MLE is consistent, asymptotically normal and optimal in the mean-square sense. In the second part of the paper we investigate the natural time discretization of the MLE, by assuming that the first N Fourier modes are measured at M time grid points, uniformly spaced over the time interval [0, T]. We provide a rigorous asymptotic analysis of the proposed estimators when \(N\rightarrow \infty \) and/or \(T,M\rightarrow \infty \). We establish sufficient conditions on the growth rates of N, M and T, that guarantee consistency and asymptotic normality of these estimators.

本文的目的是研究由附加时空噪声驱动的分数随机热方程漂移系数的最大似然估计器（MLE）的渐近性质。当在频谱域中进行测量时，我们考虑传统的随机偏微分方程统计实验，并且与现有文献相反，我们研究最大似然（类型）估计量（MLE）的渐近性质，当两者傅立叶模态和时间到无穷大。在本文的第一部分中，我们考虑了对溶液的傅立叶系数进行连续时间观测的常规方法，并证明了MLE在均方意义上是一致的，渐近正态的和最优的。在时间间隔[0，T ]上均匀间隔的M个时间网格点 上测量N个傅立叶模式。当\（N \ rightarrow \ infty \）和/或\（T，M \ rightarrow \ infty \）时，我们对拟议的估计量提供了严格的渐近分析。我们对N，  M和T的增长率建立了充分的条件，以保证这些估计量的一致性和渐近正态性。