Mathematical models of Lévy Walks have found many recent applications to physical and biological systems, and such applications have generally approached Lévy Walks from the traditional perspective of discrete models for random walks. In 2016, however, Fedotov showed that the statistical properties of a Lévy Walk in one dimension can be fully captured by a system of hyperbolic first-order differential equations involving partial probability densities. These account for the local direction of motion, parametrized with respect to the particle age defined as the time elapsed from the last transition in the current direction of motion. This theoretical approach offers a radical change of paradigm, as it takes the physical time t as the primitive variable, and thus represents statistical properties in physical space-time in a direct way that is more amenable to analytical solutions.
As LML External Fellow Rainer Klages and colleagues note in a recent study, this simple change in perspective has many implications. In particular, it connects the theory of Lévy Walks with classical Poisson-Kac approaches developed to characterise stochastic processes possessing finite propagation velocity, which originated in work by Goldstein and Kac. This useful connection, however, comes at the price of a seemingly increased complexity of the formalism, as it introduces an extra independent variable (the age) to parametrize the partial densities of Lévy Walks. In their paper, Klages and colleagues explore the consequences of this framework for the formulation of statistical theories of Lévy Walk dynamics.
As they demonstrate, the additional level of complexity can actually be removed from the model by defining the system evolution in terms of a single function h(x, t) of spatial and temporal coordinates x and t. Consequently no extra degree of complexity appears, other than the intrinsic convolutional nature of the resulting integral equation, which is the fingerprint of a Lévy Walk process. Their analysis indicates that any coarse model based exclusively on the overall density P(x, t) corresponds to a specific initial preparation of the system involving symmetries and constraints on the initial distribution of ages and velocity. The authors expect their analysis to prove useful in material science and polymer physics for modeling complex nonlinear viscoelastic fluctuations. The study of finite velocity processes may also have implications for developing models of the foraging of biological organisms.
The paper is available at https://iopscience.iop.org/article/10.1088/1751-8121/ab38eb