15 problems

Let $F(s)$ be the Laplace transform expression … For $1 \leq \gamma < 2$, the parameters satisfy $s,K>0$, and $F(s)$ is associated with the first-passage-time problem. Complete mon…

Let $M$ be a Markov process independent of an almost surely increasing $alpha$-stable process $\sigma=\{\sigma(t),\ t\in\mathbb{R}_+\}$ with inverse process $\psi$, and define the…

Diffusive-limit conjecture. As $\eta$ goes to zero, the processes $({\mathbf{X}}^\eta_t({\mathbf{x}}))_{t\ge 0}$ converge in law to the $2$-dimensional process…

Extended states conjecture. If

Let $d\in\{2,3\}$. Let $u\in C^0([0,1];C^\alpha(\mathbb T^d))\cap C^\alpha([0,1];C^0(\mathbb T^d))$ be a weak solution of the incompressible Euler equations, for some Hölder expone…

Geometric-scale separation problem. Prove anomalous diffusion for a variant of the construction with geometrically separated active scales.

Let $(W_n,G_n)$ be the barycentric excluded-volume random walk described in the paper, with $W_n\in\mathbb R^2$ its position, and let $\|\cdot\|$ denote Euclidean norm. Let…

Let $v$ be the stationary solution of the free-particle generalized Langevin equation … Here $K$ is the memory kernel, and the mean squared displacement is…

Extreme blocking conjecture. The model's qualitative blocking behaviour is similar to, but more extreme than, that of the present article.

Crossing-time scaling conjecture. The time scale on which $X^{\beta,0}$ is able to cross such gaps is $n^{1+1/\rho}$.

Let $R(t)$, for $t\geq 0$, be a given reaction-rate matrix that does not depend on spatial position, and let $\mathbf{p}$ denote the vector of state densities. Let…

Let $X_n$ be the position of simple random walk after $n$ steps on the uniform infinite planar triangulation, and suppose that its displacement exponent $\beta$ exists in the sense…

Let $\gamma\in(\frac12,1)$, let $v:X\to\mathbb{R}^d$ and $h:X\to G$ be the observable and cocycle from the system, and let $\operatorname{Fix} h(0)=\{w\in\mathbb{R}^d:h(0)w=w\}$. W…

A bead-spring network is modeled by particles connected through springs, and the Rouse model is the standard bead-spring model whose qualitative behavior is under consideration. Ro…

Let $K_r$ be the Markov process on ${\mathbb T}$ whose densities obey the master equation generated by $A$, and let $V:{\mathbb T}\to{\mathbb R}$ be a dispersion relation. Define ……