Realization of Lévy walks as Markovian stochastic processes

Abstract

Based on multivariate Langevin processes we present a realization of Lévy flights as a continuous process. For the simple case of a particle moving under the influence of friction and a velocity-dependent stochastic force we explicitly derive the generalized Langevin equation and the corresponding generalized Fokker-Planck equation describing Lévy flights. Our procedure is similar to the treatment of the Kramers-Fokker-Planck equation in the Smoluchowski limit. The proposed approach may open a way to treat Lévy flights in inhomogeneous media or systems with boundaries in the future.

Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns

“…As a consequence, model agreement with experimental data does not add much to our understanding of spontaneous movements of these cells beyond demonstrating that they can be modelled phenomenologically. However, a slight re-parametrization and re-interpretation of the driving noise lead to the model of Lubashevsky et al (2009) which realizes LW as Markovian stochastic processes (Reynolds 2010). The distinction between CRW and LW therefore appears to be superficial one, in the context of cell mobility modelling.…”

Can spontaneous cell movements be modelled as Lévy walks?

“…This brings forth new biological insight as Lévy walks are advantageous when searching in the absence of external stimuli and without knowledge of the target distribution, as may be the case with cells of the epidermis that form new tissue by locating and then attaching on to one another. The Hänggi-Klimontovich interpretation of the driving noise in the model of Lubashevsky et al (2009) and Cauchy distributions of predicted velocities do, however, appear problematic, even unphysical. Here it is shown that these are perceived rather than actual difficulties.…”

“…Here it is shown that these are perceived rather than actual difficulties. Intermittent stop-start motions of the kind displayed by some cells and protozoan are found to underlie the formulation of the model of Lubashevsky et al (2009) and the velocities of starved Dictyostelium discoideum (a unicellular organism) are found to be Cauchy distributed to a good approximation. It is therefore suggested that the model of Lubashevsky et al (2009) can describe the spontaneous movements of some cells, and that some cells have spontaneous movement patterns that can be approximated by Lévy walks, as first proposed by Schuster…”

Lévy Random Walks as Nonlinear Brownian Motion

“…As seen, the animal motion described within the given approach does undergo a variety of sharp turns. Moreover, appealing to the results of [8,9] it is reasonable to expect that the random variable a as well as the turn angles should obey statistics of truncated Lévy random walks with the cutoff about a s . Assuming quantities such as the values λ and ∆ to depend on the position r of a given organism in space and its velocity v as well as the same collection of variables of other organisms one can couple the animal behavior with the state of environment.…”