scholarly journals Time-averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

2021 ◽  
Author(s):  
Wei Wang ◽  
Andrey G. Cherstvy ◽  
Holger Kantz ◽  
Ralf Metzler ◽  
Igor M. Sokolov
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Diffusion Processes ◽  
Stochastic Simulations ◽  
Time Averaging ◽  
Weak Ergodicity ◽  
Experimental Systems ◽  
Long Time ◽  
Breaking Parameter ◽  
Ergodicity Breaking

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does the process of stochastic resetting impact nonergodicity? These are the main questions addressed in this study. Specifically, we examine, both analytically and by stochastic simulations, the implications of resetting on the MSD-and TAMSD-based spreading dynamics of fractional Brownian motion (FBM) with a long-time memory, of heterogeneous diffusion processes (HDPs) with a power-law-like space-dependent diffusivity D(x) = D0 |x| γ, and of their “combined” process of HDP-FBM. We find, i.a., that the resetting dynamics of originally ergodic FBM for superdiffusive choices of the Hurst exponent develops distinct disparities in the scaling behavior and magnitudes of the MSDs and mean TAMSDs, indicating so-called weak ergodicity breaking (WEB). For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD, and additionally observe a new trimodal form of the probability density function (PDF) of particle’ displacements. For all three reset processes (FBM, HDPs, and HDP-FBM) we compute analytically and verify by stochastic computer simulations the short-time (normal and anomalous) MSD and TAMSD asymptotes (making conclusions about WEB) as well as the long-time MSD and TAMSD plateaus, reminiscent of those for “confined” processes. We show that certain characteristics of the reset processes studied are functionally similar, despite the very different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity breaking parameter EB as a function of the resetting rate r. For all the reset processes studied, we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB ∼ (1/r)-decay at large r values. Together with the emerging MSD-versus-TAMSD disparity, this pronounced r-dependence of the EB parameter can be an experimentally testable prediction. We conclude via discussing some implications of our results to experimental systems featuring resetting dynamics.

scholarly journals Inertia triggers nonergodicity of fractional Brownian motion

2021 ◽  
Author(s):  
Andrey Cherstvy ◽  
Wei Wang ◽  
Ralf Metzler ◽  
Igor Sokolov
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Particle Tracking ◽  
Single Particle ◽  
Single Particle Tracking ◽  
Langevin Equations ◽  
Ergodic Properties ◽  
Massive Particles ◽  
Breaking Parameter ◽  
Ergodicity Breaking

How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also concurrently dramatically alter the values of the ergodicity breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped ot massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of extensive stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying e.g. the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM.

scholarly journals Time averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

2021 ◽  
Vol 104 (2) ◽  
Author(s):  
Wei Wang ◽  
Andrey G. Cherstvy ◽  
Holger Kantz ◽  
Ralf Metzler ◽  
Igor M. Sokolov
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Diffusion Processes ◽  
Time Averaging

scholarly journals Modeling Anomalous Diffusion by a Subordinated Integrated Brownian Motion

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Long Shi ◽  
Aiguo Xiao
Keyword(s):  
Brownian Motion ◽  
Continuous Time ◽  
Waiting Times ◽  
Mean Square Displacement ◽  
Mean Square ◽  
Weak Ergodicity ◽  
Normal Diffusion ◽  
Weak Ergodicity Breaking ◽  
The Mean ◽  
Ergodicity Breaking

We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times are correlated. In a continuum limit, the process can be defined by an integrated Brownian motion subordinated by an inverse α-stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when 0<α<1/3, normal diffusion when α=1/3, and superdiffusion when 1/3<α<1. The time-averaged mean square displacement is also employed to show weak ergodicity breaking occurring in the proposed process. An extension to the fractional case is also considered.

FRACTIONAL BROWNIAN MOTION WITH STOCHASTIC VARIANCE: MODELING ABSOLUTE RETURNS IN STOCK MARKETS

2008 ◽  
Vol 19 (08) ◽  
pp. 1221-1242 ◽  
Author(s):  
H. E. ROMAN ◽  
M. PORTO
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Markets ◽  
Distribution Functions ◽  
Conditional Heteroskedasticity ◽  
Arch Models ◽  
Variance Modeling ◽  
Probability Distribution Functions ◽  
Second Moments ◽  
Long Time

We discuss a model for simulating a long-time memory in time series characterized in addition by a stochastic variance. The model is based on a combination of fractional Brownian motion (FBM) concepts, for dealing with the long-time memory, with an autoregressive scheme with conditional heteroskedasticity (ARCH), responsible for the stochastic variance of the series, and is denoted as FBMARCH. Unlike well-known fractionally integrated autoregressive models, FBMARCH admits finite second moments. The resulting probability distribution functions have power-law tails with exponents similar to ARCH models. This idea is applied to the description of long-time autocorrelations of absolute returns ubiquitously observed in stock markets.

scholarly journals Weak ergodicity breaking and ageing in anomalous diffusion

2015 ◽  
Vol 36 ◽  
pp. 1560007 ◽  
Author(s):  
Ralf Metzler
Keyword(s):  
Anomalous Diffusion ◽  
Large Scale ◽  
Diffusion Processes ◽  
Tracer Particle ◽  
Single Particle Tracking ◽  
Weak Ergodicity ◽  
Ergodic Processes ◽  
Mean Squared Displacement ◽  
Weak Ergodicity Breaking ◽  
Ergodicity Breaking

Modern single particle tracking techniques and many large scale simulations produce time series r(t) of the position of a tracer particle. Standardly these are evaluated in terms of the time averaged mean squared displacement. For ergodic processes such as Brownian motion, one can interpret the results of such an analysis in terms of the known theories for the corresponding ensemble averaged mean squared displacement, if only the measurement time is sufficiently long. In anomalous diffusion processes, that are widely observed over many orders of magnitude, the equivalence between (long) time and ensemble averages may be broken (weak ergodicity breaking). In such cases the time averages may no longer be interpreted in terms of ensemble theories. Here we collect some recent results on weakly non-ergodic systems with respect to the time averaged mean squared displacement and the inherent irreproducibility of individual measurements. We also address the phenomenon of ageing, the dependence of physical observables on the time span between initial preparation of the system and the start of the measurement.

scholarly journals Survival Exponents for Some Gaussian Processes

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
G. Molchan
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Power Law ◽  
Gaussian Processes ◽  
Similar Process ◽  
Fixed Level ◽  
Long Time ◽  
Self Similar

The problem is a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during long time. The exponent in such asymptotics is estimated for some Gaussian processes, including the fractional Brownian motion (FBM) in , and the integrated FBM in , .

STOCHASTIC POROUS MEDIA EQUATION DRIVEN BY FRACTIONAL BROWNIAN MOTION

2013 ◽  
Vol 13 (04) ◽  
pp. 1350010 ◽  
Author(s):  
JAN BÁRTEK ◽  
MARÍA J. GARRIDO-ATIENZA ◽  
BOHDAN MASLOWSKI
Keyword(s):  
Porous Media ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives ◽  
Multiplicative Noise ◽  
Stochastic Equation ◽  
Random Dynamical System ◽  
Porous Media Equation ◽  
Media Equation ◽  
Long Time

The present work deals with stochastic porous media equation with multiplicative noise, driven by fractional Brownian motion B(H) with Hurst index H > 1/2. The stochastic integral with integrator B(H) is defined pathwise following the theory developed by Zähle [24], based on the so-called fractional derivatives. It is shown that there is a one-to-one correspondence between solutions to the stochastic equation and solutions to its deterministic counterpart. By means of this correspondence and exploiting properties of the deterministic porous media equation, the existence, uniqueness, regularity and long-time properties of the solution is established. We also prove that the solution forms a random dynamical system in an appropriate function space.

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