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

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 $τ$, before performing a jump drawn from a symmetric continuous probability distribution function (PDF) $f(η)$, of Lévy index $0 < μ\leq 2$. Our study includes the case where the waiting time PDF $Ψ(τ)$ has a power law tail, $Ψ(τ) \propto τ^{-1 - γ}$, with $0< γ< 1$, such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if $γ< μ/2$ (and super-diffusive if $γ> μ/2$). 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 $(γ, μ)$ 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).

36 pages, 10 figures. arXiv admin note: text overlap with arXiv:1405.1222