Comparison Principles for parabolic stochastic partial differential equations

We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations. We use this fact to build moment compar- ison principles for stochastic heat equations with smooth spatially homogeneous noises (SHE(1)), and then use them to approximate the solution of stochastic heat equations with spatially homogeneous noise with Riesz kernels (SHE(2)), and obtain moment comparison principles for SHE(2) as well.