Generalized Ornstein-Uhlenbeck processes

Bezuglyy, Vlad; Mehlig, B.; Wilkinson, M.; Nakamura, Katsuhiro; Arvedson, E.

doi:10.1063/1.2206878

Cited by 31 publications

(64 citation statements)

References 19 publications

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“…Also, if the potential is time dependent, the energies of the particles will not be constant. The existence of such "anomalous diffusion" has been demonstrated for classical dynamics with spatially and temporally fluctuating potentials [5][6][7][8][9][10]. These works claim universal behavior in the sense that for generic random potentials the diffusion coefficient exhibits a universal power-law dependence on instantaneous velocity v, such that D(v) ∼ |v| −3 as |v| → ∞.…”

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confidence: 98%

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Universality classes of transport in time-dependent random potentials

Krivolapov

Fishman

2012

Phys. Rev. E

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The growth of the average kinetic energy of classical particles is studied for potentials that are random both in space and time. Such potentials are relevant for recent experiments in optics and in atom optics. It is found that for small velocities uniform acceleration takes place, and at a later stage fluctuations of the potential are encountered, resulting in a regime of anomalous diffusion. This regime was studied in the framework of the Fokker-Planck approximation. The diffusion coefficient in velocity was expressed in terms of the average power spectral density, which is the Fourier transform of the potential correlation function. This enabled to establish a scaling form for the Fokker-Planck equation and to compute the large and small velocity limits of the diffusion coefficient. A classification of the random potentials into universality classes, characterized by the form of the diffusion coefficient in the limit of large and small velocity, was performed. It was shown that one-dimensional systems exhibit a large variety of universality classes, contrary to systems in higher dimensions, where only one universality class is possible. The relation to Chirikov resonances, which are central in the theory of chaos, was demonstrated. The general theory was applied and numerically tested for specific physically relevant examples.

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confidence: 98%

“…In the literature, this behavior is considered as universal [5,7,8,10] for any dimension given that the correlation function of the potential is sufficiently differentiable. For dimensions two and higher, as shown above, this is indeed the case, however for one-dimensional systems other behaviors are possible.…”

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confidence: 99%

Universality classes of transport in time-dependent random potentials

Krivolapov

Fishman

2012

Phys. Rev. E

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“…While originally proposed to describe stationary Gauss-Markov process [2][3][4][5], a number of generalizations of the OrnsteinUhlenbeck (OU) process have been suggested under less restrictive assumptions [6][7][8][9][10][11][12]. Indeed, non-Gaussian processes of the OU-type are becoming popular given their flexibility in modelling financial data [7].…”

Section: Introductionmentioning

confidence: 98%

On the time-homogeneous Ornstein–Uhlenbeck process in the foreign exchange rates

et al. 2015

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Since Gaussianity and stationarity assumptions cannot be fulfilled by financial data, the time-homogeneous Ornstein-Uhlenbeck (THOU) process was introduced as a candidate model to describe time series of financial returns [1]. It is an Ornstein-Uhlenbeck (OU) process in which these assumptions are replaced by linearity and time-homogeneity. We employ the OU and THOU processes to analyze daily foreign exchange rates against the US dollar. We confirm that the OU process does not fit the data, while in most cases the first four cumulants patterns from data can be described by the THOU process. However, there are some exceptions in which the data do not follow linearity or time-homogeneity assumptions.

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“…This scaling is responsible for the interesting effects discussed in the following. The addition of a linear force to the logarithmic potential breaks this scaling and leads to a very different behavior [30][31][32][33]. ) and their time average (thick red) for the Brownian particle moving in an asymptotically logarithmic potential U (x) = (U0/2) ln(1 + x 2 ) (top panel, inset).…”

Section: Fokker-planck Equation For the Logarithmic Potentialmentioning

confidence: 99%

“…However, neither the super-aging behavior nor the long-time limit Eq. (46) can be obtained from the equilibrium distribution: both require the infinite covariant density (31). Figure 3 shows the normalized correlation function (48) for different values of t 0 .…”

Section: Correlation Functionmentioning

confidence: 99%

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

et al. 2012

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We consider an overdamped Brownian particle moving in a confining asymptotically logarithmic potential, which supports a normalized Boltzmann equilibrium density. We derive analytical expressions for the two-time correlation function and the fluctuations of the time-averaged position of the particle for large but finite times. We characterize the occurrence of aging and nonergodic behavior as a function of the depth of the potential, and we support our predictions with extensive Langevin simulations. While the Boltzmann measure is used to obtain stationary correlation functions, we show how the non-normalizable infinite covariant density is related to the super-aging behavior.

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Generalized Ornstein-Uhlenbeck processes

Cited by 31 publications

References 19 publications

Universality classes of transport in time-dependent random potentials

Universality classes of transport in time-dependent random potentials

On the time-homogeneous Ornstein–Uhlenbeck process in the foreign exchange rates

Superaging correlation function and ergodicity breaking for Brownian motion in logarithmic potentials

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