Statistics of occupation times and connection to local properties of nonhomogeneous random walks

We consider the statistics of occupation times, the number of visits at the origin, and the survival probability for a wide class of stochastic processes, which can be classified as renewal processes. We show that the distribution of these observables can be characterized by a single exponent, that is connected to a local property of the probability density function of the process, viz., the probability of occupying the origin at time $t$, $P\left(t\right)$. We test our results for two different models of lattice random walks with spatially inhomogeneous transition probabilities, one of which of non-Markovian nature, and find good agreement with theory. We also show that the distributions depend only on the occupation probability of the origin by comparing them for the two systems: When $P\left(t\right)$ shows the same long-time behavior, each observable follows indeed the same distribution.

Figure 1

Distribution of the fraction of time spent in the positive axis for the Gillis random walk, in semilogarithmic scale. To enhance readability of the outer values, it has been performed the transformation $x\to arccos(2x-1)$ on the $x$ axis. The (light blue) dots represent the simulation results, the (red) line the theoretical prediction. (a) The case $\epsilon =-0.2$; (b) $\epsilon =0.2$. In both cases the results are obtained simulating ${10}^6$ walks of ${10}^4$ steps.

Figure 2

Distribution of the fraction of time spent in the positive axis for the Gillis random walk, ergodic case, with $\epsilon =0.8$. Data are obtained simulating ${10}^6$ walks of different numbers of steps. As the maximum number of steps grows, the distribution converges to a Dirac delta, centered around $k_n/n=1/2$.

Figure 3

Distribution of the fraction of time spent in the positive axis for the averaged Lévy-Lorentz gas, in semilogarithmic scale. To enhance readability of the outer values, it has been performed the transformation $x\to arccos(2x-1)$ on the $x$ axis. The (light blue) dots represent the simulation results, the (red) line the theoretical prediction. (a) The superdiffusive case, with $\alpha =0.7$. Data are obtained simulating ${10}^6$ walks of ${10}^4$ steps. (b) The subdiffusive version, with $\alpha =0.3$. In this case data are obtained simulating ${10}^7$ walks of ${10}^5$ steps.

Figure 4

Distribution of the random variable $\xi$ representing the rescaled number of steps in which the process occupies the origin. The (light blue) dots represent the simulation results, the (red) line the theoretical prediction. (a) The case $\epsilon =-0.2$; (b) $\epsilon =0.2$. In both cases the results are obtained simulating ${10}^6$ walks of ${10}^4$ steps.

Figure 5

Distribution of the random variable $\xi$ representing the rescaled number of steps in which the process occupies the origin, ergodic case, with $\epsilon =0.8$. Data are obtained simulating ${10}^6$ walks of different numbers of steps. As the maximum number of steps grows, the distribution converges to a Dirac delta, centered around ${\xi }_0=1$.

Figure 6

Distribution of the random variable $\xi$ representing the rescaled number of steps in which the process occupies the origin. The (light blue) dots represent the simulation results, the (red) line the theoretical prediction. (a) The superdiffusive case, with $\alpha =0.7$. The results are obtained simulating ${10}^6$ walks of ${10}^4$ steps. (b) Corresponds to the subdiffusive version, with $\alpha =0.3$. Data are obtained simulating ${10}^7$ walks of ${10}^5$ steps.

Figure 7

Exponents of the asymptotic power-law decay of the persistence and survival probabilities for the Gillis random walk. Data are obtained simulating ${10}^7$ walks of ${10}^5$ steps. The (green) squares represent the persistence probability, while the (light blue) circles refer to the survival probability.

Figure 8

Exponents of the asymptotic power-law decay of the persistence and survival probabilities for the averaged Lévy-Lorentz gas. (a) The superdiffusive version; (b) the subdiffusive one. In both cases data are obtained simulating ${10}^7$ walks of ${10}^5$ steps. The (green) squares represent the persistence probability, while the (light blue) circles refer to the survival probability.

Figure 9

Distributions of the occupation time (a) and the number of returns (b) for the averaged Lévy-Lorentz gas, superdiffusive version (green points), and the Gillis random walk (light blue squares), compared to the theoretic result. The values of the corresponding parameters are $\alpha =0.7$ and $\epsilon =-0.0882$, which in both cases yield $\rho =7/17$. For both systems we considered ${10}^7$ walks evolved for ${10}^4$ steps.

Figure 10

Distributions of the occupation time (a) and the number of returns (b) for the averaged Lévy-Lorentz gas, subdiffusive version (green points), and the Gillis random walk (light blue squares), compared to the theoretic result. The values of the corresponding parameters are $\alpha =0.3$ and $\epsilon =0.1296$, which in both cases yield $\rho =17/27$. For both systems we considered ${10}^7$ walks evolved for ${10}^4$ steps.