About estimating escape time of gradient Langevin dynamics
Observe that the following event contains your event $E$: either (1) for some $i$, $\|\xi_{i,2}\| &gt; C_{\zeta}$ (in which case $x_{i+1}$ and $x_{i+1}'$ are no longer the same), or (2) for some $i$, $x_i'$ leaves the ball. It might be easier to see containment (in the opposite direction) of the complement events: if for all $i$ both $\|\xi_{i,2}\| \le C_{\xi}$ and also $x_i'$ is inside the ball, this surely means that $x_i=x_i'$ for all $i$ and hence also $x_i$ is always inside the ball. 

The union bound should now be clear.