Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

Bender, Christian; Bormann, Marie; Butko, Yana A.

doi:10.1007/s13540-022-00082-8

Cited by 8 publications

(2 citation statements)

References 27 publications

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“…The superstatistical fBm is indeed a randomly scaled Gaussian process that was studied for fractional anomalous diffusion originally within the framework of the generalized grey Brownian motion (ggBm) [28][29][30][31][32], which recently has been extended to investigate the relation with generalized time-fractional diffusion equations [33,34]. Moreover, randomly scaled Gaussian processes resembling the ggBm have been formulated for modelling and understanding anomalous diffusion also within an under-damped approach [35][36][37].…”

Section: Introductionmentioning

confidence: 99%

The Fokker–Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

2022

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By collecting from literature data experimental evidence of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live Escherichia coli cells, we obtain the probability density function of molecules’ displacement and we derive the corresponding Fokker–Planck equation. Molecules’ distribution emerges to be related to the Krätzel function and its Fokker–Planck equation to be a fractional diffusion equation in the Erdélyi–Kober sense. The irreducibility of the derived Fokker–Planck equation to those of other literature models is also discussed.

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Section: Introductionmentioning

confidence: 99%

The Fokker–Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

2022

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“…In this case, the evolution of the MSD is well defined, and grows exponentially with time in both cases of fractional diffusion [45] and fractional quantum mechanics [13,48]. Note that a general subordination approach for time-fractional evolution equations is studied as well [49].…”

Section: Discussionmentioning

confidence: 98%

Topological Subordination in Quantum Mechanics

2023

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An example of non-Markovian quantum dynamics is considered in the framework of a geometrical (topological) subordination approach. The specific property of the model is that it coincides exactly with the fractional diffusion equation, which describes the geometric Brownian motion on combs. Both classical diffusion and quantum dynamics are described using the dilatation operator xddx. Two examples of geometrical subordinators are considered. The first one is the Gaussian function, which is due to the comb geometry. For the quantum consideration with a specific choice of the initial conditions, it corresponds to the integral representation of the Mittag–Leffler function by means of the subordination integral. The second subordinator is the Dirac delta function, which results from the memory kernels that define the fractional time derivatives in the fractional diffusion equation.

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Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman–Kac Approach

Kay

Giuggioli

2023

J Stat Phys

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We derive, through subordination techniques, a generalized Feynman–Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman–Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman–Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.

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Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

Cited by 8 publications

References 27 publications

The Fokker–Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

The Fokker–Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm

Topological Subordination in Quantum Mechanics

Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman–Kac Approach

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