In search of necessary and sufficient conditions to solve parabolic Anderson model with rough noise

Abstract

This paper attempts to obtain necessary and sufficient conditions to solve the parabolic Anderson model with fractional Gaussian noises: $\frac{\partial }{\partial t}u(t,x)=\frac{1}{2}\Delta u(t,x)+u(t,x)\dot{W}(t,x)$, where $W(t,x)$ is the fractional Brownian field with temporal Hurst parameter $H_0\in [1/2,1)$ and spatial Hurst parameters $H$ $=(H_1,\cdots ,H_d)$ $\in (0,1{)}^d$, and $\dot{W}(t,x)=\frac{{\partial }^{d+1}}{\partial t\partial x_1\cdots \partial x_d}W(t,x)$. When $d=1$ and when $(H_0,H)\in (\frac{1}{2},1)\times (\frac{1}{20},\frac{1}{2})$ we show that the condition $2H_0+H>5/2$ is necessary and sufficient to ensure the existence of a unique solution for the parabolic Anderson Model. When $d\ge 2$, we find the necessary and sufficient condition on the Hurst parameters so that each chaos of the solution candidate is square integrable.

50 pages, 4 figures