Universal behaviors of the multi-time correlation functions of random processes with renewal: the step noise case (the random velocity of a Lévy walk)

TL;DR

The study develops a general, time-domain framework to compute all $n$-time correlation functions for renewal step-noise processes (random velocities in Lévy walks), showing that the correlations are governed by the waiting-time PDF $\psi$ and moments of the jump variable $\xi$. It reveals a universal endpoint-dominance in the large-lag limit: for power-law WT with $\mu>2$, any $n$-time correlation reduces to the two-time correlation evaluated at the earliest and latest times, and, when $p(\xi)$ has fat tails, this convergence is accelerated and largely independent of $\psi$; in the Poisson case, multi-time correlations factorize into products of two-time terms (telegraph-noise limit). The results extend the prior two-time universality to the full hierarchy of multi-time correlations and are supported by comprehensive numerical simulations across dichotomous, Gaussian, flat, and power-law jump distributions. These findings have broad implications for non-Markovian transport and stochastic dynamics under renewal noise, enabling simplified analytic treatments and universal master-equation approaches for complex systems.

Abstract

Stochastic processes with renewal properties are powerful tools for modeling systems where memory effects and long-time correlations play a significant role. In this work, we study a broad class of renewal processes where a variable's value changes according to a prescribed Probability Density Function (PDF), $p(ξ)$, after random waiting times $θ$. This model is relevant across many fields, including classical chaos, nonlinear hydrodynamics, quantum dots, cold atom dynamics, biological motion, foraging, and finance. We derive a general analytical expression for the $n$-time correlation function by averaging over process realizations. Our analysis identifies the conditions for stationarity, aging, and long-range correlations based on the waiting time and jump distributions. Among the many consequences of our analysis, two new key results emerge. First, for Poissonian waiting times, the correlation function quickly approaches that of telegraphic noise. Second, for power-law waiting times with $μ>2$, , \emph{any $n$-time correlation function asymptotically reduces to the two-time correlation evaluated at the earliest and latest time points}. This second result reveals a universal long-time behavior where the system's full statistical structure becomes effectively two-time reducible. Furthermore, if the jump PDF $p(ξ)$ has fat tails, this convergence becomes independent of the waiting time PDF and is significantly accelerated, requiring only modest increases in either the number of realizations or the trajectory lengths. Building upon earlier work that established the universality of the two-point correlation function (i.e., a unique formal expression depending solely on the variance of $ξ$ and on the waiting-time PDF), the present study extends that universality to the full statistical description of a broad class of renewal-type stochastic processes.

Figures (22)

• Figure 1: Schematic representation of a trajectory realization $\mathord{\xi}(t)$ for the noise of Lévy flight-CTRW process (see text for details). In actual cases, the pulse heights are infinite, as the trajectory consists of a sum of shifted Dirac delta functions. Here, for visualization purposes, the pulses are depicted as very thin boxes of equal width, with heights determined by the random values of $\mathord{\xi}$.
• Figure 2: Schematic representation of a trajectory realization $\mathord{\xi}(t)$ for the noise of Lévy walk random velocity (LWRV, see text for details).
• Figure 3: Log-plots of the 6-time correlation function for $t_1 = 100$, in the case of exponential waiting times (i.e., a Poisson process) with $\mathord{\tau} = 1$, for the PDF of Eq. \ref{['dichotomous']} (dichotomous case). This figure is meant as an example of the tests carried out to validate the numerical simulations. Dots represent the results of numerical simulations, while dashed lines correspond to the factorized expression $\exp[-(t_2 - t_1)/\mathord{\tau}] \exp[-(t_4 - t_3)/\mathord{\tau}] \exp[-(t_6 - t_5)/\mathord{\tau}]$, which is exact in this case. Different colors show correlations computed considering different intermediate time values, as indicated in the legend. The agreement between simulations and theory is excellent.
• Figure 4: Log-plots of the 4-time correlation function for $t_1 = 0$, in the case of exponential waiting times with $\mathord{\tau} = 1$, for the PDF of Eq. \ref{['pNormal']} (Normal PDF). Dots represent the results of numerical simulations. Solid lines, which are nearly indistinguishable from the numerical simulations, represent the exact theoretical result obtained by inserting Eq. \ref{['pois']} into Eq. \ref{['corr4_2']}. Dashed lines are the functions $\exp[-(t_2 - t_1)/\mathord{\tau}] \exp[-(t_4 - t_3)/\mathord{\tau}]$ and represent the factorization limit: asymptotically, they tend to the exact theoretical results. The dotted line shows the 2-time correlation function, illustrating the so-called “universal limit result” which, however, fails in Poissonian cases when the PDF lacks heavy tails.
• Figure 5: Log-plots of the 4-time correlation function and $t_1 = 100$, in the case of exponential waiting times with $\mathord{\tau} = 1$, for the PDF of Eq. \ref{['pNormal']} (Normal PDF). Dots represent the results of numerical simulations. Solid lines, which are nearly indistinguishable from the numerical simulations, represent the exact theoretical result obtained by inserting Eq. \ref{['pois']} into Eq. \ref{['corr4_2']}. Dashed lines are the functions $\exp[-(t_2 - t_1)/\mathord{\tau}] \exp[-(t_4 - t_3)/\mathord{\tau}]$ and represent the factorization limit: asymptotically, they tend to the exact theoretical results. The dotted line shows the 2-time correlation function, illustrating the so-called “universal limit result” which, however, fails in Poissonian cases when the PDF lacks heavy tails.

Theorems & Definitions (7)

• Proposition 1
• Proposition 2
• Lemma 1
• Proposition 3
• Proposition 4
• Lemma 2
• Proposition 5