35 problems

Exact Hausdorff-measure conjecture. There exist positive finite constants $C_1$ and $C_2$ such that

Non-explosion conjecture. In this regime, solutions cannot explode in finite time.

Compact support conjecture. The same compact-support and unbounded-union-of-supports results should hold for solutions to $$ .

Let $H\in(\frac12,1)$ be the Hurst parameter of the fractional noise, let $\alpha$ be the exponent associated with the renewal/pinning model, and consider the stochastic heat equat…

Critical stochastic heat flow conjecture. In $d=2$, at this scaling, the limiting object should be the 2d critical stochastic heat flow $Z_{s,t}^{\theta}(\mathrm d x,\mathrm d y)$.

Assume $\beta_N$ as in the paper's critical scaling. The critical-point scaling-limit conjecture. At the critical point $\hat\beta=\hat\beta_c$, the properly rescaled partition fun…

Critical KPZ scaling-limit conjecture. There exists a non-random function $b(\varepsilon)>0$ with $b(\varepsilon)\to\infty$ as $\varepsilon\to0$ such that, for every…

Let $M_1:=M^{\theta}_{[0,1]}$, and let $M'$ be the time-$[0,1]$ marginal of $M^{\theta}_{[0,2]}$. These are random measures arising from polymer measures on different time interval…

Let $M^{\theta}_{[s,t]}$ be the polymer measure on paths over the spacetime interval $[s,t]$ in $1+2$-dimensional Euclidean space. More precisely, paths sampled from…

Let $Z^{\beta_N}_N(0)$ denote the point-to-plane partition function of the directed polymer at the critical intermediate-disorder scale, and let $N$ tend to infinity. Critical one-…

Let the $1d$ stochastic heat equation have space-time white noise, and consider its spatial average over boxes whose growth rate depends on time. Stable-limit conjecture. Non-Gauss…

Consider directed polymers in an independent and identically distributed random environment, with partition function formed from Gibbs weights involving an inverse-temperature para…

Let $K_n(f)$ be the spatially averaged fluctuation observable, written as a sum of identically distributed terms $K_n(x)$, and suppose these terms possess all moments. Gaussian sca…

Let $f\in\mathcal C_c(\mathbb{R}^d)$, let $S_n(f)$ be the centered, spatially averaged directed-polymer partition function defined by … and let $p^*(\beta)$ denote the relevant mom…

Let $d\geq 3$, and let $(\beta_{L^N})_{N\geq 2}$ denote the critical coupling constants associated with the $N$-th moments of the stochastic heat equation. Strict-ordering conjectu…

Let the two-dimensional critical stochastic heat flow be the flow of random measures on $\mathbb R^2$ constructed as the scaling limit of intermediate-disorder directed-polymer par…

Extension conjecture. The results of Tao should extend to the broader regime

Let $Z^{hs}(x,s\mid y,t)$ be the half-space polymer partition function with Dirichlet boundary conditions. For a real parameter $v<0$, let $Z^{\rm stat}_2(0,y;v,-v)$ denote the sta…

Let $Z^{\rm stat}_{\ell}(\vec y;\vec a)$ denote the stationary polymer partition-function process constructed from Brownian motions with drift vector $\vec a$. Let denote the stoch…

Let $\ell$ be a positive integer, let $a_1\leqslant\dots\leqslant a_{\ell}$, and set $\vec a=(a_1,\dots,a_{\ell})$. For starting points $x_i=a_i t$ at time $-t$, let…

Let $\mathcal Z^{{\omega},a}_{\beta}(t,x)$ be the solution of the stochastic heat equation with delta initial condition, and define the random probability measure … For $\beta=0$,…

Consider the stochastic heat equation on $\mathbb{R}^d$ driven by Lévy noise that is white in both time and space. Strong-intermittency conjecture. The solution should exhibit stro…

Let $Z:=Z_{x,s}^{\infty}$ be the limiting partition function and let $\Phi=-\ln Z$. Assume that the probability distribution of the disorder variable $\xi$ has compact support. Tai…

Joint-sampling extension conjecture. Similar to the results of Cialenco, Delgado-Vences, and Kim and of Bibinger and Trabs, the theorems on these estimators should remain true when…

Let $Z(t)=Z(0,t)$ denote the solution of the half-space stochastic heat equation at the boundary, let $k$ be a positive integer, and let $A,B$ be parameters satisfying $A>k-1$ and…