The region $Y$ introduces a delay time $\tau$, the average time between entry and exit. A simple shape-independent expression for $\tau$ results if the dynamics in $Y$ is ergodic:
$$\tau_{2D}=\frac{\pi A}{Wv},$$
with $A$ the area of the two-dimensional region $Y$ and $W$ the width of the tunnel. (The velocity of the particle is $v$.) Similarly, for a three-dimensional region $Y$ (volume $V$, cross-sectional area $S$ of the tunnel):
$$\tau_{3D}=\frac{4V}{Sv},$$
again shape-independent. Roughly speaking, ergodic dynamics results if the step length $l$ of the Brownian walk is less than the linear dimension of $Y$ but greater than the width of the tunnel, $W\ll l\ll \sqrt{A}$.