In this talk, I will present a recent work on the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^d$

$$\left(\frac{\partial }{\partial t} -\frac{1}{2}\Delta \right) u(t,x)= \rho(u(t,x))\:\dot{M}(t,x),$$

for measure-valued initial data, where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and $\rho$ is Lipschitz continuous. These results are obtained under the condition that $\int_{\mathbb{R}^d}(1+|\xi|^2)^{\alpha-1}\hat{f}(\text{d}\xi)<\infty$ for some $\alpha\in(0,1]$, where $\hat{f}$ 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, i.e., $\alpha=0$. As some intermediate results, we obtain handy upper bounds for $L^p(\Omega)$-moments of $u(t,x)$ for all $p\ge 2$, and also prove that $u$ is a.s. H\"older continuous with order $\alpha-\epsilon$ in space and $\alpha/2-\epsilon$ in time for any small $\epsilon>0$.

This talk is based on the joint work with Jingyu Huang (AOP '19, Vol. 47, No. 2, 989--1035).