On the Gap and Time Interval between the First Two Maxima of Long Continuous Time Random Walks

TL;DR

This paper analyzes the joint distribution of the gap and time interval between the first two maxima of a one-dimensional continuous time random walk with heavy-tailed waiting times and jump distributions, revealing complex behaviors depending on parameters.

Contribution

It provides the first exact analytical characterization of the joint PDF of the first two maxima gap and time interval in CTRWs with heavy-tailed waiting times, extending previous discrete-time results.

Findings

Joint PDF reaches a stationary limit as T→∞

Rich behavior depending on parameters γ and μ

Results confirmed by numerical simulations

Abstract

We consider a one-dimensional continuous time random walk (CTRW) on a fixed time interval $T$ where at each time step the walker waits a random time $\tau$, before performing a jump drawn from a symmetric continuous probability distribution function (PDF) $f(\eta)$, of L\'evy index $0 < \mu \leq 2$. Our study includes the case where the waiting time PDF $\Psi(\tau)$ has a power law tail, $\Psi(\tau) \propto \tau^{-1 - \gamma}$, with $0< \gamma < 1$, such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if $\gamma < \mu/2$ (and super-diffusive if $\gamma > \mu/2$). We investigate the joint PDF of the gap $g$ between the first two highest positions of the CTRW and the time $t$ separating these two maxima. We show that this PDF reaches a stationary limiting joint distribution $p(g,t)$ in the limit of long CTRW, $T \to \infty$. Our exact analytical results show a very rich behavior of this joint PDF in the $(\gamma, \mu)$ plane, which we study in great detail. Our main results are verified by numerical simulations. This work provides a non trivial extension to CTRWs of the recent study in the discrete time setting by Majumdar et al. (J. Stat. Mech. P09013, 2014).