The exponential maximal-excursion tail control for random-environment diffusions

Fix $x\in\mathbb{R}^d$, $\omega\in\Omega$ and $n\geq 0$. For each $v\geq D_n$ and every $y$ satisfying

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

the exponential maximal-excursion tail control asserts

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

The paper notes that this estimate is not valid in general for every $(x,\omega,n)$, while related controls are available for large $n$, with high probability, on a large portion of space. The supplied text does not establish a resolution status for the displayed universal formulation.