Hypothesis testing for stochastic PDEs driven by additive noise

We study the simple hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive space-time white noise colored in space. We assume that the first $N$ Fourier modes of the solution are observed continuously over time interval $[0,T]$. We introduce the notion of asymptotically the most powerful test, and find explicit forms of such test in two asymptotic regimes: large time asymptotics $T\to\infty$, and increasing number of Fourier modes $N\to\infty$. The proposed statistics are derived based on Maximum Likelihood Ratio. Over the course of proving the main results, we obtain a series of technical results that are also of independent interest. In particular, we find the cumulant generating function of the log-likelihood ratio, we obtain some sharp large deviation type results for both $T\to\infty$ and $N\to\infty$, and find some useful asymptotics for the power of the likelihood ratio type tests.