Diffusion equation and rare fluctuations of the biased aging continuous-time random walk model

TL;DR

This paper investigates the fractional advection-diffusion equation and rare fluctuations in the biased aging continuous-time random walk model, highlighting the role of rare events and establishing connections with large deviations, supported by simulations.

Contribution

It introduces a space-based fractional operator for ACTRW with finite mean waiting times and explores the impact of rare events on positional distribution and large deviations.

Findings

Rare events dominate the tail of the positional distribution.

A strong relationship between renewals and large deviations is established.

Theoretical results are validated through simulations.

Abstract

We explore the fractional advection-diffusion equation and rare events associated with the ACTRW model. When waiting times have a finite mean but infinite variance, and the displacements follow a narrow distribution, the fractional operator is defined in terms of space rather than time. The far tail of the positional distribution is governed by rare events, which exhibit a different scaling compared to typical fluctuations. Additionally, we establish a strong relationship between the number of renewals and the positional distribution in the context of large deviations. Throughout the manuscript, the theoretical results are validated through simulations.