In search of necessary and sufficient conditions to solve the parabolic Anderson model with fractional Gaussian noises

Abstract

This paper attempts to obtain necessary and sufficient conditions to solve the parabolic ∂ Anderson model with fractional Gaussian noises: [Formula Presented] 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 Ẇ (t, x) = [Formula Presented]. When d = 1 and when (H0, H) ∈ [Formula Presented] 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.