**Propagation of high moments for parabolic Anderson model**

The parabolic Anderson model is the heat equation perturbed by a multiplicative noise. In case of Gaussian noise with non-trivial constant initial datum, the \(n\)-th moment of the solution grows exponentially fast in long term over the whole spatial domain. If the initial datum is localized, the moment grows exponentially only inside a space-time cone. Outside of the cone, the moment decays exponentially in long term. We will discuss how to specify these cones. The talk is based on a joint work with Jingyu Huang and David Nualart (available on arXiv:1509.00897).