One- and two-boundary problems for jump processes that slowly converge to Gaussian behavior.

Abstract

We consider jump processes with finite variance which, from the point of view of experimental studies, apparently do not converge to the expected Gaussian behavior. A mathematically tractable example of such processes are those with a distribution of jump lengths in the form of a weighted sum of exponential distributions. For such processes on a half-line and on a finite interval, we calculate the probability densities of arrival at a given time at a given point on the line. The found probability densities allow us to calculate many quantities important for applications: coordinate probability density functions, first passage times, leapover-length distributions, splitting probabilities, conditional exit times, and others. The following results are obtained: for a given variance of the jump-length distribution, the ratio of the mean leapover length to the mean jump length can be arbitrarily large; for a given variance of the jump-length distribution and a given interval length, the transmission probability can be arbitrarily small; for a given variance of the jump-length distribution, a given interval length, and a given mean waiting time, the mean exit time can be arbitrarily large. These results are radically different from those previously obtained using the continuous limit for processes rapidly converging to Gaussian behavior.