Random coefficient autoregressive processes describe Brownian yet non-Gaussian diffusion in heterogeneous systems

Abstract

Many studies on biological and soft matter systems report the joint presence of a linear mean-squared displacement and a non-Gaussian probability density exhibiting, for instance, exponential or stretched-Gaussian tails. This phenomenon is ascribed to the heterogeneity of the medium and is captured by random parameter models such as 'superstatistics' or 'diffusing diffusivity'. Independently, scientists working in the area of time series analysis and statistics have studied a class of discrete-time processes with similar properties, namely, random coefficient autoregressive models. In this work we try to reconcile these two approaches and thus provide a bridge between physical stochastic processes and autoregressive models. We start from the basic Langevin equation of motion with time-varying damping or diffusion coefficients and establish the link to random coefficient autoregressive processes. By exploring that link we gain access to efficient statistical methods which can help to identify data exhibiting Brownian yet non-Gaussian diffusion.

Universal spectral features of different classes of random diffusivity processes

“…Here we consider the common example of squared Ornstein-Uhlenbeck process and related models. We note that diffusing-diffusivity models are intimately related to random-coefficient autoregressive processes [79].…”

“…Then, from (24), we readily get the coefficient of variation, γ = 19/8. The small-A asymptotic behavior of P (A) can be deduced directly from (79). Expanding the exponential function in the integral into the Taylor series in powers of A and expressing the emerging generalized elliptic integrals via their representations in terms of the toroidal functions P n−1/2 (cosh(η)) (see (C3) in Appendix C), we get…”

Cusp of Non-Gaussian Density of Particles for a Diffusing Diffusivity Model

“…Then, the marginal distribution for the displacements follows As we did in Section 2.2.1.1, using the general forms obtained above, i.e., Equation ( 9), Equation (17), Equation (18), Equation (23), and Equation (28), we can analyze P(x, t) in the short and long time limits.…”

“…This feature is becoming a more frequent observation for the spreading of molecules. Phenomenological approaches are diffusing diffusivity models, in which non-Gaussianity is obtained by coupled stochastic differential equations with random diffusion coefficients [ 7 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 ], and path integrals formalism for Brownian motion in the presence of a sink [ 21 ]. More recently, theoretical frameworks describing this behavior emerged from continuous time random walk (CTRW) approaches employing large deviations theory [ 22 , 23 , 24 ] and microscopical models like molecular dynamics of tracer particles in polymer networks [ 25 , 26 ] and interacting particles with fluctuating sizes [ 27 , 28 , 29 ], the so-called Hitchhiker model [ 28 ].…”

Brownian motion and beyond: first-passage, power spectrum, non-Gaussianity, and anomalous diffusion

“…Moreover, we mention a study by Barkai and Burov [91], in which the authors use extreme value statistic arguments to derive a robust exponential shape of the displacement PDF. Finally, in a recent work Ślęzak et al [92] show that random coefficient autoregressive processes of the ARMA type can be used to describe Brownian yet non-Gaussian processes, and thus connect the world of physics of such dynamics with the world of time series analysis.…”