17 problems

Localized Hölder contraction estimate. For each $f\in C^{0,\beta}(\mathbb{R}^d)$,

Let $d\geq 3$ and let $\phi^{(\xi)}$ be the corrector. For $\varepsilon>0$, define the random distribution … for $f\in C_c^\infty(\mathbb{R}^d)$. Gaussian free field scaling-limit…

Let the transparent mass FPUT lattice be approximated by the corresponding KdV solution on the long time scale considered in the paper, and let the absolute approximation error be…

Let $\mathbb{P}$ satisfy the condition $\mathsf{CFS}(\beta,\Psi)$, where $\beta\in(\frac12,1)$, and let ${\mathbf{b}}_r(x)$ and $\bar{\mathbf{a}}$ denote the corresponding rescaled…

Let $\mathcal P^\circ=\{x_n^\circ\}_n$ be a Poisson process, and let $\mathcal P=\{x_n\}_n$ be one of the Matérn I, II, or III hardcore processes obtained from it by the thinning r…

Let $\mathcal P=\{x_n\}_n$ be a Poisson process in $\mathbb R^d$, with associated Voronoi tessellation $\mathcal V=\{V_n\}_n$. Write $r_n$ for the diameter of $V_n$, and define … F…

In stochastic homogenization, consider a divergence-form linear elliptic equation with random coefficients and its ensemble-averaged solution. The conjecture concerns describing th…

Let $\boldsymbol{\mathrm{P}}\subset\Omega$ be a corresponding prototype with the strong extension property, and let $\mathcal{V}_{\mathrm{pot}}^{p}(\boldsymbol{\mathrm{P}})$ denote…

Periodization approximation conjecture. For all $1\le n\le2d$,

Bourgain–Spencer conjecture. Under suitable mixing conditions on the random coefficient field $\boldsymbol a$, such an operator $B_0$ exists and its Fourier symbol $\hat B_0$ is of…

Stochastic homogenization conjecture. The Hamiltonian $H_{\ell}(p,x,\omega)$ is stochastically homogenizable.

Let $u_\omega$ be the solution from Definition, under the hypotheses of Theorem (i) but without its limitation on the dimension $d$, and let $L_{u_\omega,\eta}(t)$ denote the trans…

Let $(\Omega,\mathcal{F},\mathbb{P})$ be the probability space underlying the stationary random potential $V(x,\omega)$, and let $\varphi:[0,\infty)\to\mathbb{R}$ be continuous and…

Fix $x\in\mathbb{R}^d$, $\omega\in\Omega$, and $n\geq 0$. Let $P_{y,\omega}$ denote the law of the diffusion started at $y$ in environment $\omega$, and let $X^*_{L_n^2}$ be its ma…

Variance-scaling conjecture. The variance of the time constant should behave like

General comparison with measurable ingredients conjecture. Under these assumptions, the conclusion of the comparison proposition holds with $M^+_{CS}$ in place of $M^+_A$.

Let $d$ be the spatial dimension and let $\mathfrak{d}$ denote the spectral measure associated with the homogenization problem. A pair $( u,q)$ is a spectral exponent pair for…