22 problems

Let $u(t,x)$ solve the one-dimensional parabolic Anderson model with space-time white noise $xi$ and narrow wedge initial condition: … For fixed $t>0$, Huang–Lê's spatial asymptoti…

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…

Successive-merging conjecture. In dimension $d=3$, the curves in Fig. 3 never merge, whereas for $d\geq 4$ the curves merge successively.

High-dimensional non-intermittency conjecture. There is no intermittency for sufficiently large diffusion constant $\kappa$ in dimensions $d\ge 4$.

Let $b_2=2/g(0,0)$, where $g(0,0)$ is the Green's function at the origin of a simple random walk on $\mathbb{Z}^d$, and let $b_\ast$ be the critical parameter for survival of the p…

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

Let $u$ denote the parabolic Anderson model with two-dimensional white noise, and let its moment at time $t$ be understood in the sense considered by Gu and Wu. Gu and Wu's moment…

Let $(Z_j(t))_{t\geq 0}$ solve the parabolic Anderson model on $\mathbb{Z}^d$ with inverse temperature $\beta>0$, and let $\operatorname{Corr}$ denote the correlation coefficient.…

Infinite-moment conjecture. For $d=3$, or for $d=2$ when $t$ is sufficiently large, one has

Let $cxi(z)=cxi(-z)$ with probability $p=p(|z|)$, where $p(n)cto1$ as $nctocinfty$, and fix a Pareto parameter $calpha>2$. Complete localisation is the property that the parabolic…

In the parabolic Anderson model with an i.i.d. random potential, complete localisation means that the solution is asymptotically concentrated at a single spatial site, while double…

Sharp lower-bound constant conjecture. The constant in the left-hand side should be $1$, so the lower bound should have the same leading constant as the upper bound. This conjectur…

Let ${\mathcal K}=\{\mathcal K(x,y):x,y\in\mathbb Z^d,\,x\sim y\}$ be a field of random conductances on neighbouring sites, and let $\lambda_0({\mathcal K})$ denote the associated…

Let $u(s,\cdot)$ be the solution considered in the parabolic Anderson model, and call a connected high-potential region an island. As $s\to\infty$, consider the spatial concentrati…

Edwards–van der Hofstad–Holmes conjecture. If $\xi$ is ergodic in space and time, then

Let $\lambda(\kappa)$ denote the Lyapunov exponent of the continuum parabolic Anderson model with diffusivity $\kappa>0$, and suppose that the previously established lower bound ……

Let $\lambda_p^{u_0}(\kappa)$ denote the quenched Lyapunov exponent of order $p\in\mathbb{N}$ for mobility parameter $\kappa$, and let $\lambda_0^{u_0}(\kappa)$ be the correspondin…

Let $u(t,z)$ be the solution of the parabolic Anderson model with independent Weibull potential $\xi(z)$ of parameter $0<\gamma<2$, and let $U(t)=\sum_{z\in\mathbb{Z}^d}u(t,z)$. Co…

Let $u(t,z)$ be the solution of the parabolic Anderson model with a heavy-tailed random potential, and let $U(t)=\sum_{z\in\mathbb Z^d}u(t,z)$ denote its total mass. One-point loca…

Fix $n\geq 1$. For each $p$, let $\kappa_p^{(n)}$ denote the critical diffusion constant associated with $p$-intermittency. Distinct critical values conjecture. For all fixed…

All-diffusivity intermittency conjecture. In dimension $d=3$, the parabolic Anderson model is intermittent for every $\kappa>0$.

Let $\xi$ be the random potential satisfying the essential-supremum and bounded independent-identically-distributed assumptions, let $\kappa>0$ be the diffusion parameter, and let…