Maximum of a Fractional Brownian Motion: Probabilities of Small Values

Molchan, G.

doi:10.1007/s002200050669

Cited by 139 publications

(187 citation statements)

References 8 publications

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“…This conjecture shows excellent agreement with both anomalous diffusive systems on all time scales [63]. The Molchan long-time prediction [52] is given to guide the eye. The fBm FPTD consists of two different absorption points (∆x = 50, ∆x = 100, 6 × 10 4 simulations) [64] and displays good agreement with SFD results.…”

Section: Sfd Data Versus Approximationsmentioning

confidence: 76%

First passage times for a tracer particle in single file diffusion and fractional Brownian motion

Sanders

Ambjörnsson

2012

The Journal of Chemical Physics

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We investigate the full functional form of the first passage time density (FPTD) of a tracer particle in a single-file diffusion (SFD) system whose population is: (i) homogeneous, i.e. all particles having the same diffusion constant and (ii) heterogeneous, with diffusion constants drawn from a heavytailed power-law distribution. In parallel, the full FPTD for fractional Brownian motion [fBmdefined by the Hurst parameter, H ∈ (0, 1)] is studied, of interest here as fBm and SFD systems belong to the same universality class. When H < 1/3, which includes homogeneous SFD (H = 1/4), and heterogeneous SFD (H < 1/4), the WFA fails to agree with any temporal scale of the simulations and Molchan's long-time result.SFD systems are compared to their fBm counter parts; and in the homogeneous system both scaled FPTDs agree on all temporal scales including also, the result by Molchan, thus affirming that SFD and fBm dynamics belong to the same universality class. In the heterogeneous case SFD and fBm results for heterogeneity-averaged FPTDs agree in the asymptotic time limit. The non-averaged heterogeneous SFD systems display a lack of self-averaging. An exponential with a power-law argument, multiplied by a power-law pre-factor is shown to describe well the FPTD for all times for homogeneous SFD and sub-diffusive fBm systems.

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Section: Sfd Data Versus Approximationsmentioning

confidence: 76%

First passage times for a tracer particle in single file diffusion and fractional Brownian motion

Sanders

Ambjörnsson

2012

The Journal of Chemical Physics

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“…Nonetheless, the absence of inertial subrange intermittency makes incremental changes in Eulerian and Lagrangian (tracer-particle) velocities resolutely Gaussian, in contrast with laboratory results which exhibit highly non-Gaussian statistics [20]. The same shortcoming can be expected for fluid velocities along the trajectories of heavy particles, and this may impact on model predictions for Lévy exponents [25,26].…”

Section: Introductionmentioning

confidence: 95%

“…u 2 ( t) , will be proportional to t 2/3 over an associated range of scales [24]. It follows from u 2 ( t) ∝ t 2/3 that the distribution of time-intervals between consecutive bouts of strong velocity fluctuations will have a power-law tail [25,26]. The characteristic power-law exponent is dependent on the underling dynamics.…”

Section: Introductionmentioning

confidence: 99%

Lévy flight movement patterns in marine predators may derive from turbulence cues

Reynolds

2014

Proc. R. Soc. A.

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The Lévy-flight foraging hypothesis states that because Lévy flights can optimize search efficiencies, natural selection should have led to adaptations for Lévy flight foraging. Some of the strongest evidence for this hypothesis has come from telemetry data for sharks, bony fish, sea turtles and penguins. Here, I show that the programming for these Lévy movement patterns does not need to be very sophisticated or clever on the predator's part, as these movement patterns would arise naturally if the predators change their direction of travel only after encountering patches of relatively strong turbulence (a seemingly natural response to buffeting). This is established with the aid of kinematic simulations of three-dimensional turbulence. Lévy flights movement patterns are predicted to arise in all but the most quiescent of oceanic waters.

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“…Statistical properties of these currents for the Sinai problem (H = 1/2) are known [36,37,38,39,40,41]. Using the results of [42], and noting that with V (x) = 0, J + (L) and J − (L) have equal moments, we have for arbitrary H and to leading order in…”

mentioning

confidence: 99%

Diffusion in periodic, correlated random forcing landscapes

Dean¹,

Gupta²,

Oshanin³

et al. 2014

J. Phys. A: Math. Theor.

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Abstract. We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically-extended (with period L) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent H ∈ (0, 1). While the periodicity ensures that the ultimate long-time behavior is diffusive, the generalised Sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times. , and the negative ones as D′ and a ′ being numerical constants and β the inverse temperature. These results demonstrate that D L is strongly non-self-averaging. We further show that the probability distribution of D L has a log-normal left tail and a highly singular, onesided log-stable right tail reminiscent of a Lifshitz singularity.PACS numbers: 05.40.-a, 02.50.-r, 05.10.Ln

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Maximum of a Fractional Brownian Motion: Probabilities of Small Values

Cited by 139 publications

References 8 publications

First passage times for a tracer particle in single file diffusion and fractional Brownian motion

First passage times for a tracer particle in single file diffusion and fractional Brownian motion

Lévy flight movement patterns in marine predators may derive from turbulence cues

Diffusion in periodic, correlated random forcing landscapes

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