Stochastic heat equation (SHE) with multiplicative noise is an important model, that has deep relations with many models in statistical physics. When the diffusion coefficient is linear (i.e., sigma(u)=u), this model is also called the parabolic Anderson model, the solution of which traditionally gives rise to the Hopf-Cole solution to the Kardar-Parisi-Zhang (KPZ) equation. Obtaining various fine properties of its solution will certainly deepen our understanding of these important models.

In this talk, I will highlight several interesting properties of SHE and then focus on the probability densities of the solution. This second part is based on a recent joint work with Y. Hu and D. Nualart where we have established a necessary and sufficient condition for the existence and regularity of the probability density of the solution to SHE with measure-valued initial conditions. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders.