  The mean first exit (passage) time characterizes the average time of a
stochastic process never leaving a fixed region in the state space, while the
escape probability describes the likelihood of a transition from one region to
another for a stochastic system driven by discontinuous (with jumps) L\'evy
motion. This paper discusses the two deterministic quantities, mean first exit
time and escape probability, for the anomalous processes having the tempered
L\'{e}vy stable waiting times with the tempering index $\lambda>0$ and the
stability index $0<\alpha \le 1$; as for the distribution of jump lengths or
the type of the noises driving the system, two cases are considered, i.e.,
Gaussian white noise and non-Gaussian (tempered) $\beta$-stable ($0<\beta<2$)
L\'{e}vy noise. Firstly, we derive the nonlocal elliptic partial differential
equations (PDEs) governing the mean first exit time and escape probability.
Based on the derived PDEs, it is observed that the mean first exit time depends
strongly on the domain size and the values of $\alpha$, $\beta$ and $\lambda$;
when $\lambda$ is close to zero, the mean first exit time tends to $\infty$. In
particular, we also find an interesting result that the escape probability of a
particle with (tempered) power-law jumping length distribution has no relation
with the distribution of waiting times for the model considered in this paper.
For the solutions of the derived PDEs, the boundary layer phenomena are
observed, which inspires the motivation for developing the boundary layer
theory for nonlocal PDEs.
