A note on error estimation for hypothesis testing problems for some linear SPDEs

The aim of the present paper is to estimate and control the Type I and Type II errors of a simple hypothesis testing problem of the drift/viscosity coefficient for stochastic fractional heat equation driven by additive noise. Assuming that one path of the first $N$ Fourier modes of the solution is observed continuously over a finite time interval $[0,T]$, we propose a new class of rejection regions and provide computable thresholds for $T$, and $N$, that guarantee that the statistical errors are smaller than a given upper bound. The considered tests are of likelihood ratio type. The main ideas, and the proofs, are based on sharp large deviation bounds. Finally, we illustrate the theoretical results by numerical simulations.Comment: Forthcoming in Stochastic Partial Differential Equations: Analysis and Computation

Asymptotic Criticality of the Navier-Stokes Regularity Problem

The problem of global-in-time regularity for the 3D Navier-Stokes equations, i.e., the question of whether a smooth flow can exhibit spontaneous formation of singularities, is a fundamental open problem in mathematical physics. Due to the super-criticality of the equations, the problem has been super-critical in the sense that there has been a fixed `scaling gap' between any regularity criterion and the corresponding a priori bound (regardless of the functional setup utilized). The purpose of this work is to present a mathematical framework--based on a suitably defined `scale of sparseness' of the super-level sets of the positive and negative parts of the components of the higher-order spatial derivatives of the velocity field--in which the scaling gap between the regularity class and the corresponding a priori bound (in the vicinity of a possible singular time) vanishes as the order of the derivative goes to infinity. This reveals asymptotically critical nature of the Navier-Stokes regularity problem.Comment: 53 pp; additional concepts and techniques have been developed so that some key dynamical signatures of the derivatives at different levels can be precisely quantified, leading to a more transparent formulation of the main theorem, and in preparation for applying the mechanism to other super-critical fluid model

HYPOTHESIS TESTING FOR STOCHASTIC PDES DRIVEN BY ADDITIVE NOISE

We 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 201

Time-Global Regularity of the Navier-Stokes System with Hyper-Dissipation--Turbulent Scenario

The question of whether the hyper-dissipative Navier-Stokes (NS) system can exhibit spontaneous formation of singularities in the super-critical regime--the hyper-dissipation being generated by a fractional power of the Laplacian confined to interval $\bigl(1, \frac{5}{4}\bigr)$--has been a major open problem in the mathematical fluid dynamics since the foundational work of J.L. Lions in 1960s. In this work, a mathematical evidence of the criticality of the Laplacian is presented. While the framework for the proof is based on the `scale of sparseness' of the super-level sets of the positive and negative parts of the components of the higher-order derivatives of the velocity or vorticity fields recently introduced by the authors, a major novelty in the current work is the classification of the hyper-dissipative flows near a potential spatiotemporal singularity in two main categories, that is, `homogeneous' (the flows exhibiting a near-steady behavior, possibly after a coordinate transformation) and `non-homogenous' (a generic case consistent with the formation and decay of turbulence). The main result states that in the non-homogeneous case, any power of the Laplacian greater than 1 yields a contradiction, preventing a blow-up.Comment: 49p