We study hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive space-time... Show moreWe study hypothesis testing problem for the drift/viscosity coefficient for stochastic fractional heat equation driven by additive space-time white noise colored in space. Since it is the first attempt to deal with hypothesis testing in SPDEs, we assume that the first N Fourier modes of the solution are observed continuously over time interval [0, T], similar methodology could be developed later for discrete sampling. The highlight of this article lies in the notion of “asymptotically the most powerful test” we introduce, which is a brand new idea for hypothesis testing not only in stochastic PDEs but in general stochastic processes. This conception provides a definite criterion how we compare the convergence rates of errors of two tests and how we maximize this convergence rate in a given rejection class when T or N is near infinity. And also we will give some equally important results for controlling the errors with finite T and N. We will build up asymptotic rejection class and find explicit forms of “the most powerful test” in two asymptotic regimes: large time asymptotics T →∞, and increasing number of Fourier modes N → ∞. The proposed statistics are derived based on Maximum Likelihood Ratio. We first consider a simple hypothesis testing, for which we exploit the key technic, by which we continue considering for more general issues. Over the course of proving the main results, we obtain a series of technical results on the asymptotic behaviors of the probabilities related to likelihood ratio, which are also, in some sense, of high value for study in probability theory. In particular, we find the cumulant generating function of the log-likelihood ratio, we obtain some sharp large deviation type results for both T → ∞ and N → ∞, and develop some useful strategies in probability convergence for studying asymptotic properties of the power of the likelihood ratio type tests. M.S. in Applied Mathematics, December 2013 Show less
