Discrete stochastic heat equation driven by fractional noise : Feynman-Kac representation and asymptotic behavior

We consider the parabolic Anderson model on $\mathbb{Z}^d$ driven by fractional noise. We prove that it has a mild solution given by Feynman-Kac representation which coincides with the partition function of a directed polymer in a fractional Brownian environment. Our argument works in a unified way for every Hurst parameter in $(0,1)$. We also study the asymptotic time evolution of this solution. We show that for $H\leq1/2$, almost surely, it converges asymptotically to $e^{\lambda t}$ for some deterministic strictly positive constant $\lambda$'. Our argument is robust for every jump rate and non-pathological spatial covariance structures.\\ For $H>1/2$ on one hand, we demonstrate that the solution grows asymptotically no faster than $e^{k t\sqrt{\log t}}$, for some positive deterministic constant $k$'. On the other hand, the asymptotic growth is lower-bounded by $e^{c t}$ for some positive deterministic constant `$c$'. \ Invoking Malliavin calculus seems inevitable for our results.