Moment Intermittency in the PAM with Asymptotically Singular Noise

Abstract: Let $\xi$ be a singular Gaussian noise on $\mathbb R^d$ that is either white, fractional, or with the Riesz covariance kernel; in particular, there exists a scaling parameter $\omega>0$ such that $c^{\omega/2}\xi(c\cdot)$ is equal in distribution to $\xi$ for all $c>0$. Let $(\xi_\varepsilon){\varepsilon>0}$ be a sequence of smooth mollifications such that $\xi\varepsilon\to\xi$ as $\varepsilon\to0$. We study the asymptotics of the moments of the parabolic Anderson model (PAM) with noise $\xi_\varepsilon$ as $\varepsilon\to0$, both for large (i.e., $t\to\infty$) and fixed times $t$. This approach makes it possible to study the moments of the PAM with regular and singular noises in a unified fashion, as well as interpolate between the two settings. As corollaries of our main results, we obtain the following: $\textbf{(1)}$ When $\xi$ is subcritical (i.e., $0<\omega<2$), our results extend the known large-time moment and tail asymptotics for the Stratonovich PAM with noise $\xi$. Our method of proof clarifies the role of the maximizers of the variational problems (known as Hartree ground states) that appear in these moment asymptotics in describing the geometry of intermittency. We take this opportunity to prove the existence and study the properties of the Hartree ground state with a fractional kernel, which we believe is of independent interest. $\textbf{(2)}$ When $\xi$ is critical or supercritical (i.e., $\omega=2$ or $\omega>2$), our results provide a new interpretation of the moment blowup phenomenon observed in the Stratonovich PAM with noise $\xi$. That is, we uncover that the latter is related to an intermittency effect that occurs in the PAM with noise $\xi_\varepsilon$ as $\varepsilon\to0$ for $\textit{fixed finite times}$ $t>0$.