Large Deviation in Continuous Time Random Walks

TL;DR

This paper analyzes large deviation properties of continuous-time random walks, deriving a general rate expression and exploring specific examples to reveal fundamental behaviors of such stochastic processes.

Contribution

It provides a unified framework for large deviations in CTRW, linking step and waiting time distributions, with explicit formulas for Gaussian steps and various waiting time distributions.

Findings

Derived a general large deviation rate expression for CTRW.

Reduced the expression to Legendre transforms for Gaussian steps.

Explored examples revealing properties of large deviations in different CTRW models.

Abstract

We discuss large deviation properties of continuous-time random walks (CTRW) and present a general expression for the large deviation rate in CTRW in terms of the corresponding rates for the distributions of steps' lengths and waiting times. In the case of Gaussian distribution of steps' lengths the general expression reduces to a sequence of two Legendre transformations applied to the cumulant generating function of waiting times. The discussion of several examples (Bernoulli and Gaussian random walks with exponentially distributed waiting times, Gaussian random walks with one-sided L\'evy and Pareto-distributed waiting times) reveals interesting general properties of such large deviations.