Exponential maximal-excursion estimate for diffusions in random environments

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 maximal excursion up to time $L_n^2$. Exponential maximal-excursion estimate. For each $v\geq D_n$ and every $y$ satisfying

$$|y-x|\leq 30\sqrt{d}L_n,$$

we have

$$P_{y,\omega}(X^*_{L_n^2}\geq v)\leq \exp\left(-\frac{v}{D_n}\right).$$

This estimate provides the tail control needed to handle localization errors in stochastic homogenization. As with the Hölder control, the source says that it is not valid for every $(x,\omega,n)$; according to the cited result, it is available for large $n$ with high probability on a large portion of space.