Necessary and sufficient conditions to solve parabolic Anderson model with rough noise

Shuhui Liu, Yaozhong Hu, Xiong Wang

Published 2022-06-06Version 1

We obtain necessary and sufficient condition for the existence of $n$-th chaos of the solution to the parabolic Anderson model $\frac{\partial}{\partial t}u(t,x)=\frac{1}{2}\Delta u(t,x)+u(t,x)\dot{W}(t,x)$, where $\dot{W}(t,x)$ is a fractional Brownian field with temporal Hurst parameter $H_0\ge 1/2$ and spatial parameters $H$ $ =(H_1, \cdots, H_d)$ $ \in (0, 1)^d$. When $d=1$, we extend the condition on the parameters under which the chaos expansion of the solution is convergent in the mean square sense, which is both sufficient and necessary under some circumstances.