In search of necessary and sufficient conditions to solve the parabolic Anderson model with fractional Gaussian noises
Shuhui Liu, Yaozhong Hu, Xiong Wang · Electronic Journal of Probability · 2024
This paper attempts to obtain necessary and sufficient conditions to solve the parabolic Anderson model with fractional Gaussian noises: ∂ ∂tu(t,x)=1 2Δu(t,x)+u(t,x)W˙(t,x), where W(t,x) is the fractional Brownian field with temporal Hurst parameter H0∈[1∕2,1) and spatial Hurst parameters H =(H1,⋯,Hd) ∈(0,1)d, and W˙(t,x)=∂d+1∂t∂x 1⋯∂xdW(t,x). When d=1 and when (H0,H)∈(1 2,1)×(1 20,1 2) we show that the condition 2H0+H>5∕2 is necessary and sufficient to ensure the existence of a unique solution for the parabolic Anderson Model. When d≥2, we find the necessary and sufficient condition on the Hurst parameters so that each chaos of the solution candidate is square integrable.