Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Wang, Wei; Cherstvy, Andrey G.; Liu, Xianbin; Metzler, Ralf

doi:10.1103/physreve.102.012146

Cited by 71 publications

(76 citation statements)

References 145 publications

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“…(32) in Ref. 150 ). For all these processes at each elementary time-step of dt = 10 −2 or dt = 10 −3 in simulations a constant rate of resetting is set r, so that resetting probability to the initial position…”

Section: Some Applications Of Fbm and Hdpsmentioning

confidence: 93%

“…The primary focus of the current study is on the effects of resetting onto the TAMSD characteristics for stochastic processes of nearly ergodic (see Refs. [132][133][134][135] ) fractional BM (FBM) 42,[136][137][138][139][140][141][142][143][144] and of nonergodic heterogeneous diffusion processes (HDPs) 126,143,[146][147][148][149][150] as well as for a combination of FBM with varying Hurst exponent H 151,152 and HDPs with varying exponent of the power-law-like diffusivity γ 150 . Note that a "hybrid" process of SBM-HDP was also introduced 126 and recently applied to the experimental data 153 .…”

Section: Outline Of the Papermentioning

confidence: 99%

“…We employ the same simulation scheme as in the recent study of a numerical and analytical investigation of the "compound" process of FBM-HDPs introduced recently in Ref. 150 . Shortly, the overdamped Langevin equation…”

Section: Main Equations For Fbm Hdps and Hdp-fbmmentioning

confidence: 99%

“…We use below the exact theoretical results and asymptotic relations of Ref. 150 for FBM, HDPs, and HDP-FBM processes (without re-deriving them here) and employ the middle-point, physically motivated, Stratonovichconvention-based simulation scheme for Eqs. (13), (15), and (19) (see the details of the Itô-Stratonovich conversion employed and Eq.…”

Section: Some Applications Of Fbm and Hdpsmentioning

confidence: 99%

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

Wang

Cherstvy

Kantz

et al. 2021

Preprint

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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.

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Section: Some Applications Of Fbm and Hdpsmentioning

confidence: 93%

Section: Outline Of the Papermentioning

confidence: 99%

Section: Main Equations For Fbm Hdps and Hdp-fbmmentioning

confidence: 99%

Section: Some Applications Of Fbm and Hdpsmentioning

confidence: 99%

See 2 more Smart Citations

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

Wang

Cherstvy

Kantz

et al. 2021

Preprint

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“…Building an explanatory model for the volatility in terms of gGBM would bring novel insights regarding the theoretical and empirical characteristics of the asset prices. We also leave for future analysis the problem of gGBM with stochastic volatility, which can be treated in the framework of the Fokker-Planck equation for gGBM with time varying volatility σ(t), in analogy of the diffusing-diffusivity models for heterogeneous media [56,[70][71][72][73].…”

Section: Discussionmentioning

confidence: 99%

Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

Stojkoski

Sandev

Basnarkov

et al. 2020

Entropy

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Classical option pricing schemes assume that the value of a financial asset follows a geometric Brownian motion (GBM). However, a growing body of studies suggest that a simple GBM trajectory is not an adequate representation for asset dynamics, due to irregularities found when comparing its properties with empirical distributions. As a solution, we investigate a generalisation of GBM where the introduction of a memory kernel critically determines the behaviour of the stochastic process. We find the general expressions for the moments, log-moments, and the expectation of the periodic log returns, and then obtain the corresponding probability density functions using the subordination approach. Particularly, we consider subdiffusive GBM (sGBM), tempered sGBM, a mix of GBM and sGBM, and a mix of sGBMs. We utilise the resulting generalised GBM (gGBM) in order to examine the empirical performance of a selected group of kernels in the pricing of European call options. Our results indicate that the performance of a kernel ultimately depends on the maturity of the option and its moneyness.

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Great Barrier Reef degradation, sea surface temperatures, and atmospheric CO2 levels collectively exhibit a stochastic process with memory

et al. 2021

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Quantifying ecological memory could be done at several levels from the rate of physiological changes in an ecosystem all the way down to responses at the genetic level. One way of unlocking the information encoded in a collective environmental memory is to examine the recorded time-series data generated by different components of an ecosystem. In this paper, we probe into the case of the Great Barrier Reef (GBR) which is threatened by elevated sea surface temperatures (SST) and ocean acidification attributed to rising atmospheric CO2 levels. Specifically, we investigate the interrelated dynamics between the degradation of the GBR, SST, and rising atmospheric CO2 levels, by considering three datasets: (a) the mean percentage hard coral cover of the GBR from the archives of the Australian Institute of Marine Science; (b) SST close to the GBR from the National Oceanic and Atmospheric Administration; and (c) the Keeling curve for atmospheric CO2 levels measured by the Mauna Loa Observatory. We show that fluctuating observables in these datasets have the same memory behavior described by a non-Markovian stochastic process. All three datasets show a good match between empirical and analytical mean square deviation. An explicit analytical form for the corresponding probability density function is obtained which obeys a modified diffusion equation with a time dependent diffusion coefficient. This study provides a new perspective on the similarities of and interaction between the GBR's declining hard coral cover, SST, and rising atmospheric CO2 levels by putting all three systems into one unified framework indexed by a memory parameter and a characteristic frequency . The short-time dynamics of CO2 levels and SST fall in the superdiffusive regime, while the GBR exhibits hyperballistic fluctuation in percent coral cover with the highest values for and .

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Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise

Cited by 71 publications

References 145 publications

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

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

Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

Great Barrier Reef degradation, sea surface temperatures, and atmospheric CO2 levels collectively exhibit a stochastic process with memory

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