Mean first-passage time of a random walker under Galilean transformation

We consider a continuous-time random walk model with finite-mean waiting-times and we study the mean first-passage time (MFPT) as estimated by an observer in a reference frame $\mathcal{S}$, that is co-moving with a target, and by an observer in a reference frame $\mathcal{S}'$, that is in uniform motion with respect to the target through a Galilean transformation. We found that the simple picture emerging in $\mathcal{S}$, where the mean first-passage time depends on the whole jump distribution but only on the mean value of the waiting-times, does indeed not hold in $\mathcal{S}'$ where the estimation depends on the whole jump distribution and also on the whole distribution of the waiting-times. We derive the class of jump-size distributions such that the dependence of the MFPT on the mean waiting-time only is conserved also in $\mathcal{S}'$. However, if the MFPT is finite, the dependence on the specific waiting-time distribution disappears in $\mathcal{S}'$ when the initial position is sufficiently far-away from the target. While the MFPT emerges to be Galilean invariant with both two-sided and one-sided jump distributions with finite moments, the MFPT is not a Galilean invariant for one-sided jump distribution with power-law tails (one-sided L\'evy distributions).