| Abstract: |
| We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^d$
\[
\left(\frac{\partial }{\partial t} -\frac{1}{2}\Delta \right) u(t,x) = \rho(u(t,x))
\:\dot{M}(t,x),
\]
where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $\rho$ is a Lipschitz continuous function that vanishes at zero. These results are obtained for rough initial data and under Dalang`s condition, namely,
$\int_{\mathbb{R}^d}(1+|\xi|^2)^{-1}\hat{f}(\text{d} \xi) |
|