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