24 problems

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 $\mathcal{P}$ be the operator associated with the one-parameter family of analytic Markov maps, acting on the Hilbert space $\mathcal{H}$, and let $r\in[0,1]$ denote the parame…

Nonlinear saturation conjecture. The resulting phase coherence will lead to a nonlinear amplitude saturation which will stop the wave predicted by the theory, and this will in turn…

Consider a branching random walk in random environment on $\mathbb Z^d$: particles move as continuous-time simple random walks and branch at position $x$ at rate $\xi_x$, where…

Let $U\subset\mathbb{R}^d$ be a bounded domain satisfying the hypotheses of either Theorem or Theorem, and let $u(t,x)$ be the corresponding solution to the stochastic heat equatio…

Mixing and correlation-decay conjecture. The random systems from Theorem 1.1b are mixing. Moreover, if $\Sigma_R=\{1\}$, then the rate of the annealed decay of correlations is

No absolutely continuous stationary measure conjecture. If $\rho=1$, then $F$ admits no absolutely continuous stationary probability measure.

Sharpness conjecture. The exponent $3/2$ in the upper bound

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…

Multiscale-limit-theorem conjecture. Multiscale phenomena are in the background of many peculiar limit theorems, including those established in the cited works.

Endpoint fourth-moment conjecture. The conjecture is

Let the underlying state space be the lattice $\mathbb{Z}^d$. Consider the weakly self-avoiding random walk in a random potential, whose paths are penalized for spending too much t…

Let $F$ be a p.c.f. fractal, and let $d_H$ denote its Hausdorff dimension with respect to the resistance metric. For the Lyapunov exponent … where defined, write $\Theta(\cdot)$ fo…

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…

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…

Let $p_t(x,y)$ be the transition probability at time $t$ for the rate-one random walk with kernel $p(,)$, and let $\blambda_p(kappa)$ denote the associated Lyapunov exponent. Suppo…

Let $S_N(x,t)$ be a time-dependent Steinhaus series and let $\mathcal{E}_N(x,t)$ denote its exponential-square observable. Suppose that, for every $x\in\mathbb{R}^d$, $S_N(x,t)$ te…

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

Self-similar time-series conjecture. At least qualitatively, the similarity of observed burst size and duration distributions in solar wind and magnetospheric quantities to those f…

Voter-model Lyapunov-exponent conjecture. On $[0,\infty)$, $\kappa\mapsto\lambda_p(\kappa)$ is strictly decreasing and convex, with

SEP merging conjecture. The same behavior occurs for SEP as in the two intermittency conjectures for the independent simple random walk catalyst: in $d=3$ the curves never merge, w…

Threshold conjecture. In $d\geq4$, there exists a strictly increasing sequence $0<\kappa_2<\kappa_3<\ldots$ such that, for $p=2,3,\ldots$, the system is $p$-intermittent if and onl…