**Abstract:** | The stochastic heat equation (SHE) is a canonical model that is related to many models in mathematical physics, mathematical biology, particle systems, etc. It usually takes the following form: ? ?t ?1/2? u(t, x) = ρ(u(t, x)) W˙ (t, x), u(0, o) = ?, t > 0, x ∈ R d, where ? is the initial data, M˙is a spatially homogeneous Gaussian noise that is white in time and ρ is a Lipschitz continuous function. In this talk, we will first make a short introduction to SHE. Then we will concentrate on the study of a particular set of properties of this equation - the comparison principles, which include both sample-path comparison principles and stochastic comparison principles. These results are obtained for general initial data and under Dalang's condition. For the sample-path comparison, one can compare solutions pathwisely with respect to different but comparable initial conditions, while for the stochastic comparison, one can compare certain functionals of the solutions either with respect to different diffusion coefficients ρ or different correlation functions of the noise. This talk is based on some joint works with Jingyu Huang and Kunwoo Kim. |