Conjecture on the critical moment blowup threshold for the parabolic Anderson model

Let $\xi$ be a critical noise, meaning that its singularity parameter is $\omega=2$, and let $t_0(p)\in(0,\infty)$ be a moment blowup threshold such that

$$\langle u(t,x)^p\rangle<\infty\quad\text{if }t<t_0(p),\qquad \langle u(t,x)^p\rangle=\infty\quad\text{if }t>t_0(p).$$

Let $\kappa$ be the model parameter and let $\mathcal G$ be the best constant

$$\mathcal G:=\inf\bigg\{C>0:\iint_{(\mathbb R^d)^2}f(x)^2\gamma(x-y)f(y)^2\,\mathrm d x\,\mathrm d y\leq C\int_{\mathbb R^d}|\nabla f(x)|_2^2\,\mathrm d x\quad\forall f\in H^1(\mathbb R^d)\text{ such that }\|f\|_2=1\bigg\}.$$

Critical moment blowup threshold conjecture. For all $p>0$,

$$t_0(p)=\frac{2\kappa}{p\mathcal G}.$$

This conjecture identifies the finite-time moment blowup threshold with the variational constant governing the critical asymptotics. A corresponding threshold is known or conjectured in related critical PAM settings, but the equality for all $p>0$ in the Stratonovich setting remains open.