Comparison principle for stochastic heat equation on Rd.

We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on Rd (..t.12.)u(t,x)=.(u(t,x))M.(t,x), for measure-valued initial data, where M. is a spatially homogeneous Gaussian noise that is white in time and . is Lipschitz continuous. These results are obtained under the condition that .Rd(1+|.|2)..1f^(d.)<. for some ..(0,1], where f^ is the spectral measure of the noise. The weak comparison principle and nonnegativity of solutions to the same equation are obtained under Dalang.s condition, that is, .=0. As some intermediate results, we obtain handy upper bounds for Lp(.)-moments of u(t,x) for all p.2, and also prove that u is a.s. Hölder continuous with order ... in space and ./2.. in time for any small .>0.