Intermittency conjecture for the parabolic Anderson model with moving catalysts

Fix $n\in\mathbb{N}$. Let $d$ be the spatial dimension, let $\kappa\geq 0$ be the diffusion constant of the catalyst, and let $\kappa_p^{(n)}$ denote the critical values governing $p$-intermittency. The system is called fully intermittent when it is $p$-intermittent for every $p\geq 1$. Intermittency conjecture. For $1\leq d\leq 2$, the system is fully intermittent. For $d\geq 3$, the intermittency vanishes as $\kappa$ increases, with three regimes: for $\kappa\in[0,\kappa_2^{(n)})$, the system is fully intermittent; for $\kappa\in[\kappa_2^{(n)},nG_d(0))$, there exists $p=p(\kappa)\geq 3$ such that the system is $q$-intermittent for every $q\geq p$; and for $\kappa\in[nG_d(0),\infty)$, the system is not $p$-intermittent for any $p\geq 2$. The first regime is proved in a small neighborhood of $0$, and full intermittency is proved for $n=1,2$ when $1\leq d\leq 2$. This conjecture describes the expected transition from full intermittency to its disappearance as catalyst diffusion increases; the stated partial results leave the intermediate and higher-dimensional regimes open.