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Theory of Probability and Mathematical Statistics

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Fractional Stokes-Boussinesq-Langevin equation and Mittag-Leffler correlation decay


Authors: V. V. Anh and N. N. Leonenko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 98 (2018).
Journal: Theor. Probability and Math. Statist. 98 (2019), 5-26
MSC (2010): Primary 60G10; Secondary 82C31
DOI: https://doi.org/10.1090/tpms/1060
Published electronically: August 19, 2019
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Abstract: This paper presents some stationary processes which are solutions of the fractional Stokes-Boussinesq-Langevin equation. These processes have reflection positivity and their correlation functions, which may exhibit the Alder-Wainwright effect or long-range dependence, are expressed in terms of the Mittag-Leffler functions. These properties are established rigorously via the theory of the KMO-Langevin equation and a combination of Mittag-Leffler functions and fractional derivatives. A relationship to fractional Riesz-Bessel motion is also investigated. This relationship permits us to study the effects of long-range dependence and second-order intermittency simultaneously.


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Additional Information

V. V. Anh
Affiliation: School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane QLD 4001, Australia
Address at time of publication: School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, People’s Republic of China
Email: v.anh@qut.edu.au

N. N. Leonenko
Affiliation: Cardiff School of Mathematics, Cardiff University, Senghennydd Road, Cardiff CF24 4AG, United Kingdom
Email: LeonenkoN@Cardiff.ac.uk

DOI: https://doi.org/10.1090/tpms/1060
Keywords: Anomalous diffusion, Stokes--Boussinesq--Langevin equation, Langevin equation with delay, long-range dependence, Mittag-Leffler function
Received by editor(s): September 15, 2017
Published electronically: August 19, 2019
Additional Notes: This research was partially supported by the Australian Research Council grant DP160101366. We wish to thank Professor Akihiko Inoue and a referee for many suggestions to rectify some earlier results and improve the paper
Dedicated: This contribution is dedicated to the 85th birthday of Professor Mykhailo Iosipovych Yadrenko
Article copyright: © Copyright 2019 American Mathematical Society