Correlated continuous-time random walks—scaling limits and Langevin picture

Abstract

In this paper we analyze correlated continuous-time random walks introduced recently by Tejedor and Metzler (2010 J. Phys. A: Math. Theor. 43 082002). We obtain the Langevin equations associated with this process and the corresponding scaling limits of their solutions. We prove that the limit processes are self-similar and display anomalous dynamics. Moreover, we extend the model to include external forces. Our results are confirmed by Monte Carlo simulations.

Flow Intermittency, Dispersion, and Correlated Continuous Time Random Walks in Porous Media

“…Hence, while in many applications the assumption of uncorrelated temporal increments is convenient and allows for simple developments, for transport in divergence-free velocity fields it is not valid. Furthermore, correlations between consecutive velocities, thus, temporal increments, are expected to impact the scaling properties of dispersion in CTRW frameworks [29][30][31][32][33][34][35].…”

Asymptotic behaviour of random walks with correlated temporal structure

“…The introduced model generalizes those studied in Chechkin et al [12] and Tejedor & Metzler [13]. Recent progress in the field of CTRWs with correlated temporal and spatial structure can be found in Meerschaert et al [14] and Magdziarz et al [15,16]. The Langevin description of some classes of anomalous diffusions has been recently studied in Magdziarz [17], Magdziarz et al [18] and Teuerle et al [19].…”

Non-Ergodicity and Ageing in Anomalous Diffusion

“…109 show a pronounced amplitude scatter. 107 A graphical comparison of the correlated CTRW with the regular, renewal CTRW is shown in Fig.…”

“…107,108 This model features a stretched exponential mode relaxation P (k, t) exp(−ct 1/2 ) in the limit γ = 2, while for for 0 < γ < 2 a power-law form P (k, t) t −γ is obtained. 109 There also exist alternative models to correlated jump processes, see the discussion in Refs. [110,111] and the citations therein.…”