Continuous Time Random Walks (CTRWs): Simulation of continuous trajectories

Continuous time random walks have been developed as a straightforward generalisation of classical random walk processes. Some 10 years ago, Fogedby introduced a continuous representation of these processes by means of a set of Langevin equations [H. C. Fogedby, Phys. Rev. E 50 (1994)]. The present work is devoted to a detailed discussion of Fogedby’s model and presents its application for the robust numerical generation of sample paths of continuous time random walk processes.

💡 Research Summary

The paper provides a comprehensive examination of continuous‑time random walks (CTRWs) through the lens of Fog B. H. C. Fogedby’s continuous representation introduced in 1994. Unlike the traditional discrete CTRW, which is defined by a sequence of jumps separated by random waiting times, Fogedby’s formulation recasts the process as a pair of coupled stochastic differential equations (SDEs) driven by independent noises. The first SDE describes the evolution of the spatial coordinate (x(s)) as a Langevin equation with a deterministic force term (F(x)) and a Gaussian white noise (\eta(s)). The second SDE governs the physical time (t(s)) as an accumulation of a “waiting‑time” noise (\tau(s)), which is taken from a one‑sided Lévy stable distribution characterized by an index (\alpha) (0 < α < 1 for sub‑diffusive behavior). By treating the internal parameter (s) as an operational time, the model automatically embeds the heavy‑tailed waiting‑time statistics that give rise to non‑Markovian memory effects in the observable time domain.

The authors first prove mathematically that the marginal process (x(t)=x(s(t))) generated by these SDEs reproduces the same probability density functions, mean‑square displacement, and propagators as the classic discrete CTRW with the same waiting‑time distribution (\psi(\tau)). This equivalence is established by showing that the random time change (t(s)) is a strictly increasing Lévy subordinator whose inverse (s(t)) has the same distributional properties as the renewal process underlying the discrete CTRW.

A major contribution of the paper is a detailed discussion of the numerical implementation. Two practical challenges are identified: (1) discretizing the internal time (s) with a step (\Delta s) small enough to resolve the Lévy increments while keeping computational cost reasonable, and (2) handling the fact that the accumulated physical time (t(s)) may not increase monotonically if the Lévy increments are allowed to be negative. The authors resolve the second issue by restricting (\tau(s)) to a strictly positive stable law (or by taking absolute values) and by introducing an “inverse‑time tracking” algorithm. In this algorithm, one advances (s) step by step, accumulating (t), until the target physical time (t_{\text{target}}) is reached or exceeded; the final spatial coordinate is then obtained by linear interpolation between the last two points. This procedure yields a continuous trajectory (x(t)) without the need for post‑hoc time‑re‑sampling.

To validate the method, the paper presents extensive simulations of an Ornstein‑Uhlenbeck (OU) process subjected to sub‑diffusive waiting times with (\alpha=0.8). The OU force (F(x)=-\gamma x) provides a restoring drift, while the Gaussian noise drives fluctuations. Results show that the continuous‑time simulation reproduces the analytically known autocorrelation function and mean‑square displacement of the sub‑diffusive OU process, while delivering smoother sample paths than a naïve discrete CTRW implementation. Moreover, the authors demonstrate that the choice of (\Delta s) influences numerical accuracy in a predictable way: smaller (\Delta s) reduces discretization error but increases the number of Lévy draws, which can be mitigated by using optimized stable‑distribution samplers.

The discussion section highlights the advantages of the continuous formulation: (i) it yields trajectories that are directly comparable to experimental time series because the physical time axis is continuous; (ii) it allows straightforward inclusion of arbitrary external potentials, time‑dependent forces, or higher‑dimensional dynamics; (iii) it avoids artefacts associated with the “staircase” nature of discrete renewal events. Limitations are also acknowledged. Generating Lévy stable random numbers remains computationally intensive, especially for very small (\alpha) where the variance of increments diverges. Additionally, the method assumes that the waiting‑time distribution is strictly one‑sided; extensions to bidirectional or tempered stable laws would require further algorithmic development.

Finally, the authors outline future research directions: extending the framework to multi‑dimensional CTRWs, incorporating space‑dependent waiting‑time exponents (\alpha(x)), coupling to fractional Fokker‑Planck equations, and employing machine‑learning techniques to infer model parameters from empirical data. In summary, the paper not only revisits Fogedby’s continuous CTRW model with rigorous theoretical grounding but also delivers a robust, practical algorithm for generating continuous trajectories, thereby offering a valuable tool for physicists, biologists, and quantitative analysts studying anomalous diffusion and related stochastic phenomena.