Cramér-type moderate deviations of drift estimation in the stochastic heat equation

For the stochastic fractional heat equation driven by additive noise, we study the asymptotic properties of the maximum likelihood estimator (MLE) of the drift coefficient. Using parameter-dependent change of measure methods along with asymptotic analysis techniques, we establish (self-normalized) Cramér-type moderate deviations, (non-)uniform Berry-Esseen bounds, moderate deviations and precise large deviations. The study covers three asymptotic regimes: large time asymptotics, increasing number of Fourier modes, large time asymptotics and increasing number of Fourier modes. Our main results reveal how the asymptotic behaviors depend on the time horizon, the number of Fourier modes, the spatial dimension, and the fractional order.