First-passage and first-arrival problems in continuous-time random walks: Beyond the diffusion approximation

Some exact solutions of the first-passage and first-arrival problems for the continuous-time random-walk model are obtained. On the basis of these exact solutions, the following has been revealed. First, for some jump-length distributions with a finite variance, the approximate solutions obtained in the diffusion approximation can differ significantly from the exact solutions. Second, for some waiting time distributions with a finite mean, the times of first passage and the times of first arrival can significantly depend on the ensemble under consideration. In particular, the mean first-passage time corresponding to the stationary ensemble can be significantly greater than the mean first-passage time corresponding to the nonaged ensemble. Third, for any continuous distribution of jump lengths, the probability of first arrival is zero for a point-like target. This last result is contrary to existing opinion, but it is consistent with the fact that a single point has a probability measure equal to zero in the probability space defined by a continuous distribution of jump lengths.