Parameter estimation for fractional stochastic heat equations : Berry-Ess\'een bounds in CLTs

Abstract

The aim of this work is to estimate the drift coefficient of a fractional heat equation driven by an additive space-time noise using the Maximum likelihood estimator (MLE). In the first part of the paper, the first $N$ Fourier modes of the solution are observed continuously over a finite time interval $[0, T ]$. The explicit upper bounds for the Wasserstein distance for the central limit theorem of the MLE is provided when $N \rightarrow \infty$ and/or $T \rightarrow \infty$. While in the second part of the paper, the $N$ Fourier modes are observed at uniform time grid : $t_i = i \frac{T}{M}$, $i=0,..,M,$ where $M$ is the number of time grid points. The consistency and asymptotic normality are studied when $T,M,N \rightarrow + \infty$ in addition to the rate of convergence in law in the CLT.