Applied Mathematics Colloquium: Dr. Le Chen

Title: Stochastic heat equation: intermittency and densities

Stochastic heat equation (SHE) with multiplicative noise is an important model which 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.