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1. An Analysis and Global Identification of Smoothless Variable Order of a Fractional Stochastic Differential Equation

   https://doi.org/10.3390/fractalfract7120850  

   Li, Qiao; Zheng, Xiangcheng; Wang, Hong; Yang, Zhiwei; Guo, Xu (December 2023, Fractal and Fractional)

   We establish both the uniqueness and the existence of the solutions to a hidden-memory variable-order fractional stochastic partial differential equation, which models, e.g., the stochastic motion of a Brownian particle within a viscous liquid medium varied with fractal dimensions. We also investigate the inverse problem concerning the observations of the solutions, which eliminates the analytic assumptions on the variable orders in the literature of this topic and theoretically guarantees the reliability of the determination and experimental inference. 

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2. Modelling intermittent anomalous diffusion with switching fractional Brownian motion

   https://doi.org/10.1088/1367-2630/ad00d7  

   Balcerek, Michał; Wyłomańska, Agnieszka; Burnecki, Krzysztof; Metzler, Ralf; Krapf, Diego (October 2023, New Journal of Physics)

   Abstract The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot’s fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking. 

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3. Evaluating Gaussianity of heterogeneous fractional Brownian motion

   https://doi.org/10.1088/1751-8121/adc9e5  

   Balcerek, Michał; Pacheco-Pozo, Adrian; Wylomanska, Agnieszka; Krapf, Diego (April 2025, Journal of Physics A: Mathematical and Theoretical)

   Abstract Heterogeneous diffusion processes are prevalent in various fields, including the motion of proteins in living cells, the migratory movement of birds and mammals, and finance. These processes are often characterized by time-varying dynamics, where interactions with the environment evolve, and the system undergoes fluctuations in diffusivity. Moreover, in many complex systems anomalous diffusion is observed, where the mean square displacement (MSD) exhibits non-linear scaling with time. Among the models used to describe this phenomenon, fractional Brownian motion (FBM) is a widely applied stochastic process, particularly for systems exhibiting long-range temporal correlations. Although FBM is characterized by Gaussian increments, heterogeneous processes with FBM-like characteristics may deviate from Gaussianity. In this article, we study the non-Gaussian behavior of switching fractional Brownian motion (SFBM), a model in which the diffusivity of the FBM process varies while temporal correlations are maintained. To characterize non-Gaussianity, we evaluate the kurtosis, a common tool used to quantify deviations from the normal distribution. We derive exact expressions for the kurtosis of the considered heterogeneous anomalous diffusion process and investigate how it can identify non-Gaussian behavior. We also compare the kurtosis results with those obtained using the Hellinger distance, a classical measure of divergence between probability density functions. Through both analytical and numerical methods, we demonstrate the potential of kurtosis as a metric for detecting non-Gaussianity in heterogeneous anomalous diffusion processes. 

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4. Fractional Brownian motion, the Matérn process, and stochastic modeling of turbulent dispersion

   https://doi.org/10.5194/npg-24-481-2017  

   Lilly, Jonathan M.; Sykulski, Adam M.; Early, Jeffrey J.; Olhede, Sofia C. (January 2017, Nonlinear Processes in Geophysics)

   Stochastic processes exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm), with the spectral slope at high frequencies being associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matérn process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matérn process and its relationship to fBm. An algorithm for the simulation of the Matérn process in O(NlogN) operations is given. Unlike fBm, the Matérn process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence. 

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5. Determining damping terms in fractional wave equations

   Kaltenbacker, Barbara; Rundell, William (January 2022, Inverse problems)

   This paper deals with the inverse problem of recovering an arbitrary number of fractional damping terms in a wave equation. We develop several approaches on uniqueness and reconstruction, some of them relying on Tauberian theorems that provide relations between the asymptotic behaviour of solutions in time and Laplace domains. The possibility of additionally recovering space-dependent coefficients or initial data is discussed. The resulting methods for reconstructing coefficients and fractional orders in these terms are tested numerically. In addition, we provide an analysis of the forward problem consisting of a multiterm fractional wave equation. 

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