Comparison principle for stochastic heat equations

We consider the comparison theorems for the 1-dimensional stochastic heat equations with white noise and the d-dimensional stochastic equations with colored noise that is spatially correlated. The equations take the forms, 6u6t=Du+a u+bu W&d2;t,x 0<t<infinity, x∈R where W˙(t, x) is a white noise and 6u6t=Du+a u+bu F&d2;t,x 0<t<infinity, x∈Rd where F˙(t, x) is a d + 1 parameter colored noise. The main approach to develop the comparison principle is to transform the SPDEs into SDEs after discretizing the continuous time equations on small grids.