4 problems

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

Let $Y^\lambda$ be the biased diffusion, let $R^\omega_\lambda(t)$ be defined by … and define its exponentially weighted time average by … For the corrector amplitude … where…

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…

the exponential maximal-excursion tail control asserts