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

Area of Science:

• Probability Theory
• Stochastic Processes
• Mathematical Physics

Background:

• Jump processes with finite variance often deviate from expected Gaussian behavior in experimental settings.
• Processes with jump lengths modeled as weighted sums of exponential distributions offer mathematical tractability.

Purpose of the Study:

• To calculate probability densities for jump processes on a half-line and finite interval.
• To analyze key application-relevant quantities such as first passage times and transmission probabilities.
• To investigate behaviors distinct from those predicted by the continuous limit.

Main Methods:

• Mathematical analysis of jump processes with weighted exponential jump-length distributions.
• Calculation of probability densities for arrival times and positions.
• Derivation of formulas for coordinate probability densities, first passage times, and other metrics.

Main Results:

• The ratio of mean leapover length to mean jump length can be arbitrarily large for a fixed variance.
• Transmission probability can be arbitrarily small for a fixed variance and interval length.
• Mean exit time can be arbitrarily large for fixed variance, interval length, and mean waiting time.

Conclusions:

• Jump processes with finite variance exhibit unique properties not captured by Gaussian approximations.
• The findings challenge conventional understanding derived from continuous limits, particularly for non-Gaussian behaviors.
• This work provides a framework for analyzing complex stochastic systems with potential applications in various scientific fields.