On the gap and time interval between the first two maxima of long continuous time random walks

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) , of Lévy index 0<μ≤ 2. . Our study includes the case where the waiting time PDF has a power law tail, , with , such that the average time between two consecutive jumps is infinite. The random motion is sub-diffusive if (and super-diffusive if ). 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, . Our exact analytical results show a very rich behavior of this joint PDF in the 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 (2014 J. Stat. Mech. P09013).