8 problems

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

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…

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…

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

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