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In this paper, we study intermittency properties for various stochastic PDEswith varieties of space time Gaussian noises via matching upper and lowermoment bounds of the solution. Due to the absence of the powerful Feynman Kacformula, the lower moment bounds have been missing for many interestingequations except for the stochastic heat equation. This work introduces andexplores the Feynman diagram formula for the moments of the solution and thesmall ball nondegeneracy for the Green's function to obtain the lower boundsfor all moments which match the upper moment bounds. Our upper and lowermoments are valid for various interesting equations, including stochastic heatequations, stochastic wave equations, stochastic heat equations with fractionalLaplcians, and stochastic diffusions which are both fractional in time and inspace.

Intermittency properties for a large class of stochastic PDEs driven by fractional space-time noises