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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 ascaling parameter $\omega>0$ such that $c^{\omega/2}\xi(c\cdot)$ is equal indistribution to $\xi$ for all $c>0$. Let $(\xi_\varepsilon)_{\varepsilon>0}$ bea sequence of smooth mollifications such that $\xi_\varepsilon\to\xi$ as$\varepsilon\to0$. We study the asymptotics of the moments of the parabolicAnderson model (PAM) with noise $\xi_\varepsilon$ as $\varepsilon\to0$, bothfor large (i.e., $t\to\infty$) and fixed times $t$. This approach makes itpossible to study the moments of the PAM with regular and singular noises in aunified fashion, as well as interpolate between the two settings. Ascorollaries of our main results, we obtain the following:  $\textbf{(1)}$ When $\xi$ is subcritical (i.e., $0<\omega<2$), our resultsextend the known large-time moment and tail asymptotics for the StratonovichPAM with noise $\xi$. Our method of proof clarifies the role of the maximizersof the variational problems (known as Hartree ground states) that appear inthese moment asymptotics in describing the geometry of intermittency. We takethis opportunity to prove the existence and study the properties of the Hartreeground state with a fractional kernel, which we believe is of independentinterest.  $\textbf{(2)}$ When $\xi$ is critical or supercritical (i.e., $\omega=2$ or$\omega>2$), our results provide a new interpretation of the moment blowupphenomenon observed in the Stratonovich PAM with noise $\xi$. That is, weuncover that the latter is related to an intermittency effect that occurs inthe PAM with noise $\xi_\varepsilon$ as $\varepsilon\to0$ for $\textit{fixedfinite times}$ $t>0$.

Moment Intermittency in the PAM with Asymptotically Singular Noise