Stochastic heat equation with general nonlinear spatial rough Gaussian noise 

In this talk, we consider the following one dimensional (in space variable) nonlinear  stochastic heat equation driven by the Gaussian noise which is white in time and fractional in space:

$$\frac{\partial u(t,x)}{\partial t}=\frac{{\partial }^{2}u(t,x)}{\partial x^2}+\sigma (u(t,x))\dot{W}(t,x),$$

where $W(t,x) $ is a centered Gaussian process with covariance given by

$$\mathbf{E}[W(t,x)W(s,y)]=\frac{1}{2}(|x{|}^{2H}+|y{|}^{2H}-|x-y{|}^{2H})(s\wedge t).$$

Here the Hurst parameter $H$ is between 1/4 and 1/2 and $\dot{W}(t,x)=\frac{\partial^2 W}{\partial t\partial x}$.   We remove the technical condition $\sigma(0)=0$ previously assumed. The idea is to introduce a weight for the solution.

When $\sigma(t,u)$ is a constant the solution is a Gaussian random field and we obtain the bound of the solution $\sup_{0\le t\le T, |x|\le L} |u(t,x)|$ when $T$ and $L$ goes to infinity.  This is a joint work with Xiong Wang.