Stochastic heat equation with rough dependence in space

2018-08-02 16:00-17:00 Room 1418, Sciences Building No. 1

Abstract: First I will present some results on the stochastic heat equaton driven by various Gaussian noises. Then I will talk about the recent result on nonlinear partial differential equation \[ \frac{\partial }{\partial t} u=\frac12 \Delta u+\sigma(u) \dot W\,, \] where $\Delta$ is the Laplacian and $\dot W$ is a Gaussian noise which is white in time and fractional white in space with Hurst parameter $H\in (1/4, 1/2)$ and $\sigma(u)$ is a nonlinear function of $u$. When $\dot W$ is white both in time and space and when $\sigma(u)=u$, this equation is called parabolic Anderson model. It is related to the so-called Anderson localization and is also related to the KPZ equation by a Cole-Hopf transformation. The noise we studied is more singular. We shall find a proper space so that the solution exists uniquely on this space. The results are a joint work with ingyu Huang, Khoa Le, David Nualart and Samy Tindel.