Continuous-time random walks: Simulation of continuous trajectories

Continuous-time random walks have been developed as a straightforward generalization of classical random-walk processes. Some ten 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, 1657 (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.

Figure 1

Examples for the fully skewed Levy distribution according to Eq. (5) for different characteristic exponents $\alpha$. For $\alpha \to 1$, the PDF converges to $\delta (x-1)$.

Figure 2

Trajectories of the process $t\left(s\right)$ (upper panel) and the associated inverse process $s\left(t\right)$ (lower panel) simulated for $\alpha =0.9$. The finite jumps of the process $t\left(s\right)$ that are characteristic for Levy processes are evident. Due to these jumps the inverse process $s\left(t\right)$, however, a priori is not defined on $[0,\infty [$.

Figure 3

Determination of the intrinsic increment $\mathrm{\Delta }s$. From the transition PDF of Ornstein-Uhlenbeck processes that is available in a closed form, the square root of the means square deviation of the increment $\mathrm{\Delta }x$ as function of the increment $\mathrm{\Delta }s$ can be derived. From inspection of this graph, $\mathrm{\Delta }s=0.0001$ seems to be sufficient for the current purpose.

Figure 4

Sample trajectories of CTRWs with linear repelling force $F\left(x\right)=-x$ for different stability indices $\alpha$. The solid line corresponds to the process $x\left(t\right)$ whereas the dashed line indicates the corresponding $s\left(t\right)$. $\alpha =1$ complies with the ordinary Ornstein-Uhlenbeck process. With decreasing $\alpha$ the process is dominated by waiting events indicated by constants $x$ and $s$.

Figure 5

Increments distributions for $\mathrm{\Delta }x≔x(t+\tau )-x\left(t\right)$ with $\tau =0.001$ for some processes depicted in Fig. 4. For reasons of clearness the individual PDFs are shifted in vertical direction by a constant factor. It is evident that with decreasing stability index $\alpha$ a central peak evolves that corresponds to persistent regions due to waiting events. On the other hand, the distributions broaden indicating a higher probability of the occurrence of extreme increments.

Figure 6

Correlation function of the process depicted in Fig. 4 for $\alpha =0.8$. The solid line marks the analytical solution (15) derived by Baule et al. [15]. The circles indicate the correlation obtained from the analysis of an ensemble of $500000$ trajectories. Since a perfect coincidence is observed, this evaluation is proposed as a benchmark for accuracy of the numerical implementation.

Figure 7

Performance of the generation of skewed $\alpha$-stable random numbers. The solid line exhibits the PDF obtained by numerical integration of Eq. (5) for $\alpha =0.8$. For $x\gg 1$ the PDF shows power-law decay with the exponent $1+0.8$ that is depicted dashed. The points mark the PDF obtained from a sample of ${10}^7$ random numbers that have been generated by means of Eq. (A3) for the same stability index. The analytical PDF evidently is well reproduced by the sample of random numbers.