Tempered fractional Feynman-Kac equation: Theory and examples

Wu, Xiaochao; Deng, Weihua; Barkai, Eli

doi:10.1103/physreve.93.032151

Cited by 72 publications

(92 citation statements)

References 47 publications

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“…The main challenge now is to develop methods to derive analytical solutions of such equations, that could be applied to physically relevant situations. In fact, even for the simplest scenario of time independent forces and functionals, explicit solutions are sparse and have been restricted to moments or asymptotic expressions for the simplest observables with underlying Lévy-stable [39][40][41][42][43] and tempered Lévy-stable waiting time processes [46,47].…”

Section: Conclusion and Open Questionsmentioning

confidence: 99%

“…, with E α,α being a two-parameter Mittag-Leffler function. This specific case has also been recently discussed in [47] by solving directly for the Laplace-Fourier transform of the joint PDF P (p, k, λ) of a suitable CTRW and then taking its diffusive limit, i.e., (k, λ) → (0, 0). Here, we prove the equivalence of our own result and the approach therein by deriving such a limit solution.…”

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confidence: 99%

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Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Cairoli¹,

Baule²

2017

J. Phys. A: Math. Theor.

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Functionals of a stochastic process Y (t) model many physical time-extensive observables, for instance particle positions, local and occupation times or accumulated mechanical work. When Y (t) is a normal diffusive process, their statistics are obtained as the solution of the celebrated Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y (t) is non-Brownian, e.g., an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate power-law or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a Lévy process whose Laplace exponent is directly related to the memory kernel appearing in the generalized Feynman-Kac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in numerous experiments on biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both spaceand time-dependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the Feynman-Kac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are of particular relevance for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple non-equilibrium model are obtained. An additional aim of this paper is to provide a pedagogical introduction to Lévy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated process.

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Section: Conclusion and Open Questionsmentioning

confidence: 99%

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confidence: 99%

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Cairoli¹,

Baule²

2017

J. Phys. A: Math. Theor.

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“…Research on tempered fractional processes has been expanding at a fast pace. Several models (ARTFIMA, tempered diffusion, tempered stable motions, tempered Lévy flights) have recently been studied and used in a wide range of modern applications such as in the physics and modeling of transient anomalous diffusion (Piryatinska et al (2005), Stanislavsky et al (2008), Sandev et al (2015), Wu et al (2016), Chen et al (2017), Liemert et al (2017), Chen et al (2018)), geophysical flows (Meerschaert et al (2008), ) and finance (Dacorogna et al (1993), Granger and Ding (1996), Cont et al (1997), Ling and Li (2001), Zhang and Xiao (2017)). See also Chevillard (2017) on turbulence modeling based on regularization.…”

Section: Introductionmentioning

confidence: 99%

Tempered fractional Brownian motion: Wavelet estimation, modeling and testing

Boniece

Didier

Sabzikar

2021

Applied and Computational Harmonic Analysis

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The Davenport spectrum is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for non-scaling low frequency behavior. Like the classical fractional Brownian motion vis-à-vis the Kolmogorov spectrum, tempered fractional Brownian motion (tfBm) is a canonical model that displays the Davenport spectrum. The autocorrelation of the increments of tfBm displays semi-long range dependence (hyperbolic and quasi-exponential decays over moderate and large scales, respectively), a phenomenon that has been observed in wide a range of applications from wind speeds to geophysics to finance. In this paper, we use wavelets to construct the first estimation method for tfBm and a simple and computationally efficient test for fBm vs tfBm alternatives. The properties of the wavelet estimator and test are mathematically and computationally established. An application of the methodology to the analysis of geophysical flow data shows that tfBm provides a much closer fit than fBm.where u + = u1 {u>0} and B(du) is an independently scattered Gaussian random measure satisfying EB(du) 2 = σ 2 du. In the boundary case λ = 0 (and 0 < H < 1), tfBm reduces to a fractional Brownian motion (fBm), namely, a Gaussian, stationary-increment, self-similar process (e.g., Embrechts and Maejima (2002), Taqqu (2003)). TfBm is a new canonical model introduced in Meerschaert and Sabzikar that displays the so-named Davenport spectrum (Davenport (1961)). The latter is a modification of the classical Kolmogorov spectrum for the inertial range of turbulence that accounts for low frequency behavior and has * The second author was partially supported by the prime award no. W911NF-14-1-0475 from the Biomathematics subdivision of the Army Research Office, USA. The authors would like to thank Laurent Chevillard for his comments on this work. †

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“…In anomalous diffusion modeling, this is typically reflected in the behavior of the so-named mean squared displacement (MSD) EX 2 (t) ≈ Ct ϑ , C, ϑ ≥ 0, (1.1) of the particle position X(t) over a time interval T t, where the instances ϑ = 1 and ϑ = 1 correspond to classical and anomalous behavior, respectively (e.g., [69,54,94,32,45,108]). In the physics literature, a particle is said to undergo transient anomalous diffusion when the value of the exponent ϑ in (1.1) changes over different time intervals (e.g., [78,95,1,89,103,24,58,25]). Transience may appear in several contexts such as in nanobiophysics [91,70] and particle dispersion [100,104].…”

Section: Introductionmentioning

confidence: 99%

On Fractional Lévy Processes: Tempering, Sample Path Properties and Stochastic Integration

2020

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We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP II, respectively). TFLP and TFLP II make up very broad finite-variance, generally non-Gaussian families of transient anomalous diffusion models that are constructed by exponentially tempering the power law kernel in the moving average representation of a fractional Lévy process. Accordingly, the increment processes of TFLP and TFLP II display semi-long range dependence. We establish the sample path properties of TFLP and TFLP II. We further use a flexible framework of tempered fractional derivatives and integrals to develop the theory of stochastic integration with respect to TFLP and TFLP II, which may not be semimartingales depending on the value of the memory parameter and choice of marginal distribution.

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Tempered fractional Feynman-Kac equation: Theory and examples

Cited by 72 publications

References 47 publications

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Tempered fractional Brownian motion: Wavelet estimation, modeling and testing

On Fractional Lévy Processes: Tempering, Sample Path Properties and Stochastic Integration

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