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

TL;DR

This paper establishes necessary and sufficient conditions involving fractional Brownian noise parameters for the existence and uniqueness of solutions to the parabolic Anderson model, extending understanding to rough noise settings.

Contribution

It derives precise conditions on Hurst parameters that guarantee solution existence and uniqueness for the model with fractional Gaussian noise.

Findings

Condition $2H_0+H>5/2$ is necessary and sufficient for $d=1$.

Criteria for chaos square integrability in higher dimensions.

Extension of solution theory to rough fractional noises.

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 $(H_0,H)\in(\frac 12,1)\times(\frac 1{20},\frac 12)$ 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.