Parabolic Anderson model with a fractional Gaussian noise that is rough in time

This paper concerns the parabolic Anderson equation ∂u ∂t = 1 2 ∆u+ u ∂d+1WH ∂t∂x1 · · · ∂xd generated by a (d + 1)-dimensional fractional noise with the Hurst parameter H = (H0, H1, · · · , Hd) with special interest in the setting that some of H0, · · · , Hd are less than half. In the recent work [9], the case of the spatial roughness has been investigated. To put the last piece of the puzzle in place, this work investigates the case when H0 < 1/2 with the concern on solvability, Feynman-Kac’s moment formula and intermittency of the system. Key-words: parabolic Anderson equation, Dalang’s condition, fractional, rough and critical Gaussian noises, Feynman-Kac’s representation, Brownian motion, moment asymptotics AMS subject classification (2010): 60F10, 60H15, 60H40, 60J65, 81U10. ∗Research partially supported by the Simons Foundation #585506. 1