We examine two stochastic processes with random parameters, which in their basic versions (i.e., when the parameters are fixed) are Gaussian and display long-range dependence and anomalous diffusion behavior, characterized by the Hurst exponent. Our motivation comes from biological experiments, which show that the basic models are inadequate for accurate description of the data, leading to modifications of these models in the literature through introduction of the random parameters. The first process, fractional Brownian motion with random Hurst exponent (referred to as FBMRE below) has been recently studied, while the second one, Riemann–Liouville fractional Brownian motion with random exponent (RL FBMRE) has not been explored. To advance the theory of such doubly stochastic anomalous diffusion models, we investigate the probabilistic properties of RL FBMRE and compare them to those of FBMRE. Our main focus is on the autocovariance function and the time-averaged mean squared displacement of the processes. Furthermore, we analyze the second moment of the increment processes for both models, as well as their ergodicity properties. As a specific case, we consider the mixture of two-point distributions of the Hurst exponent, emphasizing key differences in the characteristics of RL FBMRE and FBMRE, particularly in their asymptotic behavior. The theoretical findings presented here lay the groundwork for developing new methods to distinguish these processes and estimate their parameters from experimental data.
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February 2025
Research Article|
February 18 2025
Riemann–Liouville fractional Brownian motion with random Hurst exponent
Special Collection:
Anomalous Diffusion and Fluctuations in Complex Systems and Networks
Hubert Woszczek
;
Hubert Woszczek
a)
(Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft, Writing – review & editing)
1
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology
, 50-370 Wrocław, Poland
a)Author to whom correspondence should be addressed: [email protected]
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Agnieszka Wyłomańska
;
Agnieszka Wyłomańska
(Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing)
1
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology
, 50-370 Wrocław, Poland
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Aleksei Chechkin
Aleksei Chechkin
(Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing)
1
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology
, 50-370 Wrocław, Poland
2
Institute for Physics & Astronomy, University of Potsdam
, 14476, Potsdam-Golm, Germany
3
German-Ukrainian Core of Excellence Max Planck Institute of Microstructure Physics
, Weinberg 2, 06120 Halle (Saale), Germany
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Hubert Woszczek
1,a)
Agnieszka Wyłomańska
1
Aleksei Chechkin
1,2,3
1
Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology
, 50-370 Wrocław, Poland
2
Institute for Physics & Astronomy, University of Potsdam
, 14476, Potsdam-Golm, Germany
3
German-Ukrainian Core of Excellence Max Planck Institute of Microstructure Physics
, Weinberg 2, 06120 Halle (Saale), Germany
a)Author to whom correspondence should be addressed: [email protected]
Chaos 35, 023145 (2025)
Article history
Received:
October 15 2024
Accepted:
January 29 2025
Citation
Hubert Woszczek, Agnieszka Wyłomańska, Aleksei Chechkin; Riemann–Liouville fractional Brownian motion with random Hurst exponent. Chaos 1 February 2025; 35 (2): 023145. https://doi.org/10.1063/5.0243975
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