Regularity and strict positivity of densities for the nonlinear stochastic heat equation
Seminar | November 29 | 3:10-4 p.m. | 1011 Evans Hall
Le Chen, University of Nevada, Las Vegas
In this talk, I will present some recent progress in understanding the existence, regularity and strict positivity of the (joint-) density of the solution to a semilinear stochastic heat equation. The talk will consists two parts. In the first part, I will show that under a mild cone condition for the diffusion coefficient, one can establish the smooth joint density at multiple points. 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. In the second half of the talk, we will show that the (joint) density is strictly positive in the interior of the support of the law. The contributions in both parts lie in that one can allow both singular diffusion coefficient, such as the Parabolic Anderson Model, and the rough initial data, such as the Dirac delta measure.
The talk is based on the joint-work with Yaozhong Hu and David Nualart (https://arxiv.org/abs/1611.03909).