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 $\tau$, before performing a jump drawn from a symmetric continuous probability distribution function (PDF) $f(\eta )$, of Lévy index $0<\mu \le 2$. 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).

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

References in corpus (5)

• On distributions of functionals of anomalous diffusion paths
• On the Gap and Time Interval between the First Two Maxima of Long Random Walks
• Survival probability of an immobile target surrounded by mobile traps
• Exact Statistics of the Gap and Time Interval Between the First Two Maxima of Random Walks
• Asymmetric Lévy flights in the presence of absorbing boundaries

Cited by in corpus (2)

• Extreme value statistics of correlated random variables: a pedagogical review
• First Gap Statistics of Long Random Walks with Bounded Jumps