Matching upper and lower moment bounds for a large class of stochastic PDEs driven by general space-time Gaussian noises

In this paper, we obtain matching upper and lower moment bounds for the solution to stochastic partial differential equation driven by a general Gaussian noise, giving a complete answer to the open problem of the matching lower moment bounds for the stochastic wave equations driven by a general Gaussian noise. Two new conditions are introduced for the Green's function of the equation to assure this intermittency property: small ball nondegeneracy and bounded Hardy-Littlewood-Sobolev total mass, which are satisfied by a large class of stochastic PDEs, including stochastic heat equations, stochastic wave equations, stochastic heat equations with fractional Laplacians, and stochastic partial differential equations with fractional derivatives both in time and in space. The main technique to obtain the lower moment bounds is to develop a Feynman diagram formula for the moments of the solution, to find the manageable main terms, and to carefully analyse these terms of sophisticated multiple integrals by exploring the above two properties.