Fractional Brownian motion with Hurst index less then 1/2 and continuous-time random walk heavy tailed waiting times (and the corresponding fractional Fokker-Planck equation) are two different processes that lead to a subdiffusive behavior widespread in complex systems. We propose simple test, based on analysis of so-called p variations, which allows distinguishing between models basis one realization unknown process. apply test data Golding Cox [Phys. Rev. Lett. 96, 098102...

Single particle tracking is an essential tool in the study of complex systems and biophysics it commonly analyzed by time-averaged mean square displacement (MSD) diffusive trajectories. However, past work has shown that MSDs are susceptible to significant errors biases, preventing comparison assessment experimental studies. Here, we attempt extract practical guidelines for estimation anomalous time averaged through simulation multiple scenarios with fractional Brownian motion as a...

A canonical decomposition of H-self-similar Lévy symmetric alpha-stable processes is presented. The resulting components completely described by both deterministic kernels and the corresponding stochastic integral with respect to motion are shown be related dissipative conservative parts dynamics. This result provides analysis tools for study anomalous diffusion phenomena in Langevin equation framework. For example, a simple computer test testing origins self-similarity implemented four real...

We show in this paper that the sample (time average) mean-squared displacement (MSD) of fractional L\'evy $\ensuremath{\alpha}$-stable motion behaves very differently from corresponding ensemble average (second moment). While MSD diverges for $\ensuremath{\alpha}<2$, may exhibit either subdiffusion, normal diffusion, or superdiffusion. Thus, $H$-self-similar stable processes can model a subdiffusive, diffusive superdiffusive dynamics sense MSD. character process is controlled by sign memory...

Stochastic motion on the surface of living cells is critical to promote molecular encounters that are necessary for multiple cellular processes. Often complexity cell membranes leads anomalous diffusion, which under certain conditions it accompanied by non-ergodic dynamics. Here, we unravel two manifestations ergodicity breaking in dynamics membrane proteins somatic hippocampal neurons. Three different tagged molecules studied soma: voltage-gated potassium and sodium channels Kv1.4 Nav1.6...

Abstract Accurately characterizing the anomalous diffusion of a tracer particle has become central issue in biophysics. However, measurement errors raise difficulty characterization single trajectories, which is usually performed through time-averaged mean square displacement (TAMSD). In this paper, we study fractionally integrated moving average (FIMA) process as an appropriate model for data with errors. We compare FIMA and traditional TAMSD estimators exponent. The ability framework to...

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise the anomalous diffusion behavior in great variety of physical systems. The correlation and properties this random motion are fully characterized by its index self-similarity or Hurst exponent. However, recent single-particle tracking experiments biological cells revealed highly complicated phenomena that cannot be attributed class processes....

We propose a generalization of the widely used fractional Brownian motion (FBM), memory-multi-FBM (MMFBM), to describe viscoelastic or persistent anomalous diffusion with time-dependent memory exponent α(t) in changing environment. In MMFBM built-in, long-range is continuously modulated by α(t). derive essential statistical properties such as its response function, mean-squared displacement (MSD), autocovariance and Gaussian distribution. contrast existing forms FBM time-varying exponents...

Abstract Diffusion of nanoparticles in the cytoplasm live cells has frequently been reported to exhibit an anomalous and even heterogeneous character, i.e. particles seem switch gears during their journey. Here we show by means a hidden Markov model that individual trajectories quantum dots living cultured feature dichotomous switching between two distinct mobility states with overall subdiffusive mode motion fractional Brownian (FBM) type. Using extracted features experimental as input for...

We address the problem of recognizing α-stable Lévy distribution with index close to 2 from experimental data. are interested in case when sample size available data is not large, thus power law asymptotics clearly detectable, and shape empirical probability density function a Gaussian. propose testing procedure combining simple visual test based on fourth moment Anderson-Darling Jarque-Bera statistical tests we check efficiency method simulated Furthermore, apply our analysis turbulent...

In this survey paper we present a systematic methodology which demonstrates how to identify the origins of fractional dynamics. We consider three mechanisms lead it, namely Brownian motion, Lévy stable motion and an autoregressive fractionally integrated moving average (ARFIMA) process but concentrate on ARFIMA modelling. The is based statistical tools for identification validation dynamics, in particular parameter estimator, ergodicity test, self-similarity index estimator sample...

Anomalous diffusion in crowded fluids, e.g., cytoplasm of living cells, is a frequent phenomenon. A common tool by which the anomalous single particle can be classified time-averaged mean square displacement (TAMSD). classical mechanism leading to fractional Brownian motion (FBM). validation such process for single-particle tracking data great interest experimentalists. In this paper we propose rigorous statistical test FBM based on TAMSD. To end analyze distribution TAMSD statistic, given...

Abstract The stochastic trajectories of molecules in living cells, as well the dynamics many other complex systems, often exhibit memory their path over long periods time. In addition, these systems can show dynamic heterogeneities due to which motion changes along trajectories. Such effects manifest themselves spatiotemporal correlations. Despite broad occurrence heterogeneous nature, analysis is still quite poorly understood and tools model them are largely missing. We contribute tackling...

The Langevin equation is a common tool to model diffusion at single-particle level. In nonhomogeneous environments, such as aqueous two-phase systems or biological condensates with different coefficients in phases, the solution not unique unless interpretation of stochastic integrals involved selected. We analyze particles and evaluate mean, mean square displacement, distribution particles, well variance time-averaged mean-square displacements. Our analytical results provide method choose...

Brownian motion in one or more dimensions is extensively used as a stochastic process to model natural and engineering signals, well financial data. Most works dealing with multidimensional consider the different independent components. In this article, we investigate of correlated R2, where individual components are not necessarily independent. We explore various statistical properties under consideration, going beyond conventional analysis second moment. Our particular focus lies on...

A time series of soft X-ray emission observed by the Geostationary Operational Environment Satellites from 1974 to 2007 is analyzed. We show that in solar-maximum periods energy distribution solar flares for C, M, and X classes well described a fractional autoregressive integrated moving average model with Pareto noise. The incorporates two effects detected our empirical studies. One effect long-term dependence (long-term memory), another corresponds heavy-tailed distributions. parameters...

In this paper we propose an algorithm to distinguish between light- and heavy-tailed probability laws underlying random datasets. The idea of the algorithm, which is visual easy implement, check whether law belongs domain attraction Gaussian or non-Gaussian stable distribution by examining its rate convergence. method allows discriminate various non-stable distributions. test differentiate distributions, appear same according standard Kolmogorov–Smirnov test. particular, it helps Student's t...

We apply the results of Baryshnikov, Mayo and Taylor (1998) to calculate non-arbitrage prices a zero-coupon coupon CAT bond. First, we derive pricing formulae in compound doubly stochastic Poisson model framework. Next, study $10$-year catas