Escape from the potential well: competition between long jumps and long waiting times

TL;DR

This paper investigates how the interplay of long waiting times and long jumps affects escape dynamics from a potential well, revealing universal power-law decay behaviors in survival probabilities.

Contribution

It introduces a comprehensive analysis of the competition between long waiting times and jumps, highlighting the universal asymptotic properties of survival probabilities in such systems.

Findings

Survival probability decays as a power-law unaffected by jump length distribution exponent.

Long waiting times with diverging mean dominate the asymptotic behavior.

Universal behavior observed at both small and large times.

Abstract

Within a concept of the fractional diffusion equation and subordination, the paper examines the influence of a competition between long waiting times and long jumps on the escape from the potential well. Applying analytical arguments and numerical methods, we demonstrate that the presence of long waiting times distributed according to a power-law distribution with a diverging mean leads to very general asymptotic properties of the survival probability. The observed survival probability asymptotically decays like a power-law whose form is not affected by the value of the exponent characterizing the power-law jump length distribution. It is demonstrated that this behavior is typical of and generic for systems exhibiting long waiting times. We also show that the survival probability has a universal character not only asymptotically but also at small times. Finally, it is indicated which properties of the first passage time density are sensitive to the exact value of the exponent characterizing the jump length distribution.