3 problems

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 $f\in C^{0,\beta}(\mathbb{R}^d)$,

the exponential maximal-excursion tail control asserts