Scaling limits of Lévy walks with random velocities

Abstract: This paper investigates Lévy walks with random velocities, extending classical models beyond constant speed assumptions. We derive scaling limits, demonstrating that diffusion depends on interplay between heavy-tailed duration and velocity distributions. Three distinct scaling regimes are identified, including a critical case with logarithmic corrections, offering a precise framework for modeling anomalous transport in heterogeneous systems.

• The paper derives the scaling limits under three regimes by analyzing the interplay between heavy-tailed waiting times and random velocities.
• It employs stable Lévy processes and subordination techniques to characterize non-Gaussian, non-Markovian transport phenomena.
• The study highlights a critical regime with logarithmic corrections, advancing the understanding of anomalous diffusion in heterogeneous media.

Scaling Limits of Lévy Walks with Random Velocities: A Technical Summary

Introduction and Motivation

The paper "Scaling limits of Lévy walks with random velocities" (2604.23610) addresses Lévy walk (LW) models in the context where the velocity of each displacement is a random variable, as opposed to the traditional assumption of constant velocity. The LW framework is a canonical model for anomalous diffusion, accurately capturing a range of transport phenomena with finite propagation speed and non-Gaussian spreading—especially relevant in heterogeneous physical, biological, and complex systems. The extension to random velocities provides a more physically plausible mechanism for highly variable transport, as occurs in disordered media, heterogeneous materials, cold atomic ensembles, and biological systems.

The analysis focuses on the critical interplay between the heavy-tailed distributions governing the durations of walks (waiting times) and velocities, identifying three principal scaling regimes, including the regime where both distributions have identical asymptotic exponents, which yields nontrivial logarithmic corrections to scaling.

Model Definition and Preliminaries

Consider a sequence of IID waiting times $\{T_i\}$ with heavy-tailed distribution $\mathbb{P}(T_i > t) \sim t^{-\alpha}$ for $\alpha \in (0,1)$ and IID velocities $\{V_i\}$ with $\mathbb{P}(V_i > v) \sim v^{-\beta}$ for $\beta \in (0,1)$. The direction $\{\mathbf{I}_i\}$ is a sequence of unit vectors in $\mathbb{R}^d$. The displacement at each step is $\mathbf{J}_i = V_i \mathbf{I}_i T_i$, representing ballistic motion at random speed $V_i$ and direction $\mathbb{P}(T_i > t) \sim t^{-\alpha}$0 over time $\mathbb{P}(T_i > t) \sim t^{-\alpha}$1.

Three natural versions of the LW process with random velocities are studied:

• Wait-first LW: $\mathbb{P}(T_i > t) \sim t^{-\alpha}$2, where the walk begins with a waiting event.
• Jump-first LW: $\mathbb{P}(T_i > t) \sim t^{-\alpha}$3.
• Continuous LW: $\mathbb{P}(T_i > t) \sim t^{-\alpha}$4, incorporating continuous updating.

The process is described in terms of the accumulated sum of such jumps up to the random number of steps $\mathbb{P}(T_i > t) \sim t^{-\alpha}$5 determined by the cumulative waiting times.

Main Theoretical Results

Scaling Regimes

The asymptotic scaling limits of these processes are fully characterized by the relation between the indices $\mathbb{P}(T_i > t) \sim t^{-\alpha}$6 (waiting times) and $\mathbb{P}(T_i > t) \sim t^{-\alpha}$7 (velocities). The main findings are:

1. $\mathbb{P}(T_i > t) \sim t^{-\alpha}$8: Heavy-tailed waiting times dominate

The walk is controlled by the large waiting times, with the process scaling as $\mathbb{P}(T_i > t) \sim t^{-\alpha}$9. Here, the joint scaling of displacement and time sums leads to limit processes subordinated by the same stable subordinator, reflecting strong spatiotemporal coupling:

• Both position and time increments are governed by the minimum of the heavy-tailed indices.
• The scaling limits are given by stable Lévy processes subordinated by the inverse of a stable subordinator.
• The process is non-Markovian and non-Gaussian, with dependent increments.

2. $\alpha \in (0,1)$0: Heavy-tailed velocities dominate

Transport is now limited by large fluctuations in the velocities, irrespective of the durations, so scaling proceeds as $\alpha \in (0,1)$1:

• Position increments are affected by the extremal velocities, with waiting times playing a secondary role.
• In this regime, jumps and waiting times decouple asymptotically; the limiting processes for position and time are independent.
• The subordination structure persists, but with position governed by a $\alpha \in (0,1)$2-stable process and time by an independent $\alpha \in (0,1)$3-stable process.

3. $\alpha \in (0,1)$4: Critical regime—product of heavy-tailed variables

This case yields novel scaling, with the asymptotics of the product of the two heavy-tailed variables ($\alpha \in (0,1)$5 and $\alpha \in (0,1)$6) being regularly varying with index $\alpha \in (0,1)$7 but acquiring logarithmic corrections. Specifically,

$\alpha \in (0,1)$8

The corresponding walk must be rescaled by $\alpha \in (0,1)$9 for convergence. Here, jumps and waiting times are asymptotically independent, but the scaling limit involves an additional logarithmic factor in its normalization.

Limit Laws and Pathwise Descriptions

In each regime, functional limit theorems are derived for the various forms of the LW (wait-first, jump-first, continuous). The limiting processes are described as Lévy processes (potentially dependent in the first regime) evaluated at stable inverse subordinators, precisely characterizing both the probabilistic structure and the stochastic dynamics of the resulting LWs with random velocity.

The main technical contributions involve:

• Rigorous derivation of the limit processes and their Lévy characteristics.
• Precise treatment of the product of regularly varying random variables, including detailed analysis of the critical logarithmic regime.
• Uniform and Skorokhod topologies are both considered for the functional convergence.

Implications and Discussion

The results have several significant theoretical and practical implications:

• Physical Relevance: This framework provides rigorously justified stochastic models for transport where not only temporal but also kinetic disorder is present. Many real-world systems, especially those exhibiting strong heterogeneity or disorder, cannot be adequately modeled with constant velocity, and thus require the present treatment.
• Anomalous Transport: The three scaling regimes encapsulate the qualitative shifts in transport mechanisms as system heterogeneity (in waiting vs. velocity) changes. The existence of a critical regime with logarithmic corrections demonstrates the nontriviality of scaling analysis when multiple sources of heavy-tailed disorder interact.
• Subordination Structure: The limiting processes confirm that spatiotemporal coupling persists even in the presence of velocity disorder but can become more intricate due to the velocity’s tail dominance or balanced contributions.
• Mathematical Contribution: The detailed asymptotic analysis of products of heavy-tailed random variables in the context of stochastic processes extends classical results about heavy-tailed sums and is relevant for broader applications in random media and kinetic theory.
• Future Directions: This work opens the path to further investigation of statistical properties (e.g., propagators, FPT distributions), associated kinetic equations (such as those involving fractional material derivatives), and multidimensional or interacting variants of random walks with random velocities. Additionally, the impact of spatiotemporal correlations between waiting time and velocity, beyond mere independence, could be explored for more complex physical models.

Conclusion

This paper provides a comprehensive analysis of Lévy walks with random velocities, rigorously characterizing the scaling limits in terms of the joint asymptotics of velocity and waiting time distributions. The identification of three universality classes—including a critical regime with explicit logarithmic scaling corrections—advances the understanding of anomalous transport beyond the constant velocity paradigm. The theoretical results have immediate implications for modeling diffusion in complex, heterogeneous, and disordered systems and lay the groundwork for further studies into the stochastic dynamics of kinetic processes with multiple sources of disorder.