  We study the simple hypothesis testing problem for the drift coefficient for
stochastic fractional heat equation driven by additive noise. We introduce the
notion of asymptotically the most powerful test, and find explicit forms of
such tests in two asymptotic regimes: large time asymptotics, and increasing
number of Fourier modes. The proposed statistics are derived based on Maximum
Likelihood Ratio. Additionally, we obtain a series of important technical
results of independent interest: we find the cumulant generating function of
the log-likelihood ratio; obtain sharp large deviation type results for
$T\to\infty$ and $N\to\infty$.
