{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:48:27Z","timestamp":1760057307006,"version":"build-2065373602"},"reference-count":150,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,2]],"date-time":"2025-02-02T00:00:00Z","timestamp":1738454400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper investigates the memory of the Ornstein\u2013Uhlenbeck process (OUP) via three ratios of the OUP increments: signal-to-noise, noise-to-noise, and tail-to-tail. Intuition suggests the following points: (1) changing the noise that drives the OUP from Gauss to Levy will not affect the memory, as both noises share the common \u2018independent increments\u2019 property; (2) changing the auto-correlation of the OUP from exponential to slowly decaying will affect the memory, as the change yields a process with long-range correlations; and (3) with regard to Levy driving noise, the greater the noise fluctuations, the noisier the prediction of the OUP increments. This paper shows that intuition is plain wrong. Indeed, a detailed analysis establishes that for each of the three above-mentioned points, the very converse holds. Hence, Levy noise has a significant and counter-intuitive effect on Ornstein\u2013Uhlenbeck memory.<\/jats:p>","DOI":"10.3390\/e27020157","type":"journal-article","created":{"date-parts":[[2025,2,3]],"date-time":"2025-02-03T05:36:32Z","timestamp":1738560992000},"page":"157","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Levy Noise Affects Ornstein\u2013Uhlenbeck Memory"],"prefix":"10.3390","volume":"27","author":[{"given":"Iddo","family":"Eliazar","sequence":"first","affiliation":[{"name":"School of Chemistry, Tel Aviv University, Tel Aviv 6997801, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"823","DOI":"10.1103\/PhysRev.36.823","article-title":"On the theory of the Brownian motion","volume":"36","author":"Uhlenbeck","year":"1930","journal-title":"Phys. Rev."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"8427","DOI":"10.1088\/0305-4470\/30\/24\/009","article-title":"The generalized Ornstein-Uhlenbeck process","volume":"30","author":"Caceres","year":"1997","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"073301","DOI":"10.1063\/1.2206878","article-title":"Generalized ornstein-uhlenbeck processes","volume":"47","author":"Bezuglyy","year":"2006","journal-title":"J. Math. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Maller, R.A., Muller, G., and Szimayer, A. (2009). Ornstein-Uhlenbeck Processes and Extensions, Springer.","DOI":"10.1007\/978-3-540-71297-8_18"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"351","DOI":"10.2307\/1968873","article-title":"The Brownian movement and stochastic equations","volume":"43","author":"Doob","year":"1942","journal-title":"Ann. Math."},{"key":"ref_6","first-page":"133","article-title":"Introduction to Gaussian processes","volume":"168","author":"MacKay","year":"1998","journal-title":"NATO ASI Ser. F Comput. Syst. Sci."},{"key":"ref_7","unstructured":"Ibragimov, I., and Rozanov, Y. (2012). Gaussian Random Processes, Springer."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Lifshits, M. (2012). Lectures on Gaussian Processes, Springer.","DOI":"10.1007\/978-3-642-24939-6"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Lindgren, G. (2012). Stationary Stochastic Processes: Theory and Applications, CRC Press.","DOI":"10.1201\/b12171"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Lindgren, G., Rootzen, H., and Sandsten, M. (2013). Stationary Stochastic Processes for Scientists and Engineers, CRC Press.","DOI":"10.1201\/b15922"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hida, T. (2015). Stationary Stochastic Processes, Princeton University Press. MN-8.","DOI":"10.1515\/9781400868575"},{"key":"ref_12","unstructured":"Gillespie, D.T. (1991). Markov Processes: An Introduction for Physical Scientists, Elsevier."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Liggett, T.M. (2010). Continuous Time Markov Processes: An Introduction, American Mathematical Society.","DOI":"10.1090\/gsm\/113"},{"key":"ref_14","unstructured":"Dynkin, E.B. (2012). Theory of Markov Processes, Dover."},{"key":"ref_15","first-page":"530","article-title":"Sur la theorie du mouvement Brownien","volume":"146","author":"Langevin","year":"1908","journal-title":"Compt. Rendus"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Coffey, W., and Kalmykov, Y.P. (2012). The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, World Scientific.","DOI":"10.1142\/8195"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Pavliotis, G.A. (2014). Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations, Springer.","DOI":"10.1007\/978-1-4939-1323-7"},{"key":"ref_18","unstructured":"Borodin, A.N., and Salminen, P. (2015). Handbook of Brownian Motion: Facts and Formulae, Birkhauser."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1023\/B:JOSS.0000015180.16261.53","article-title":"Relativistic Ornstein-Uhlenbeck process","volume":"88","author":"Debbasch","year":"1997","journal-title":"J. Stat. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3035","DOI":"10.1090\/S0002-9939-00-05345-4","article-title":"Maximal inequalities for the Ornstein-Uhlenbeck process","volume":"128","author":"Graversen","year":"2000","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1007\/s10985-004-4775-9","article-title":"Survival models based on the Ornstein-Uhlenbeck process","volume":"10","author":"Aalen","year":"2004","journal-title":"Lifetime Data Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3759","DOI":"10.1088\/0305-4470\/37\/12\/003","article-title":"A first passage time distribution for a discrete version of the Ornstein-Uhlenbeck process","volume":"37","author":"Larralde","year":"2004","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"122001","DOI":"10.1088\/1751-8113\/41\/12\/122001","article-title":"Markov-breaking and the emergence of long memory in Ornstein-Uhlenbeck systems","volume":"41","author":"Eliazar","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"021115","DOI":"10.1103\/PhysRevE.79.021115","article-title":"From Ornstein-Uhlenbeck dynamics to long-memory processes and fractional Brownian motion","volume":"79","author":"Eliazar","year":"2009","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/s10955-011-0332-6","article-title":"Spherical Ornstein-Uhlenbeck processes","volume":"145","author":"Wilkinson","year":"2011","journal-title":"J. Stat. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"135004","DOI":"10.1088\/1751-8113\/48\/13\/135004","article-title":"Time-changed Ornstein-Uhlenbeck process","volume":"48","author":"Gajda","year":"2015","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"022601","DOI":"10.1103\/PhysRevE.100.022601","article-title":"Active Ornstein-Uhlenbeck particles","volume":"100","author":"Bonilla","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"032123","DOI":"10.1103\/PhysRevE.100.032123","article-title":"Generalized Ornstein-Uhlenbeck model for active motion","volume":"100","author":"Sevilla","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"032607","DOI":"10.1103\/PhysRevE.103.032607","article-title":"Statistical mechanics of active Ornstein-Uhlenbeck particles","volume":"103","author":"Martin","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"035101","DOI":"10.1088\/1361-648X\/ac2c3f","article-title":"Active Ornstein\u2013Uhlenbeck model for self-propelled particles with inertia","volume":"34","author":"Nguyen","year":"2021","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Dabelow, L., and Eichhorn, R. (2021). Irreversibility in active matter: General framework for active Ornstein-Uhlenbeck particles. Front. Phys., 8.","DOI":"10.3389\/fphy.2020.582992"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"054129","DOI":"10.1103\/PhysRevE.107.054129","article-title":"Ornstein-Uhlenbeck process and generalizations: Particle dynamics under comb constraints and stochastic resetting","volume":"107","author":"Trajanovski","year":"2023","journal-title":"Phys. Rev. E"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Trajanovski, P., Jolakoski, P., Kocarev, L., and Sandev, T. (2023). Ornstein-Uhlenbeck Process on Three-Dimensional Comb under Stochastic Resetting. Mathematics, 11.","DOI":"10.3390\/math11163576"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Dubey, A., and Pal, A. (2023). First-passage functionals for Ornstein Uhlenbeck process with stochastic resetting. arXiv.","DOI":"10.1088\/1751-8121\/acf748"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"062142","DOI":"10.1103\/PhysRevE.100.062142","article-title":"Estimation of parameters from time traces originating from an Ornstein-Uhlenbeck process","volume":"100","author":"Strey","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"103125","DOI":"10.1063\/5.0158843","article-title":"Parameter estimation of the fractional Ornstein-Uhlenbeck process based on quadratic variation","volume":"33","author":"Janczura","year":"2023","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"022134","DOI":"10.1103\/PhysRevE.98.022134","article-title":"Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity","volume":"98","author":"Cherstvy","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"062221","DOI":"10.1103\/PhysRevE.99.062221","article-title":"Phase descriptions of a multidimensional Ornstein-Uhlenbeck process","volume":"99","author":"Thomas","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"073012","DOI":"10.1088\/1367-2630\/ab950b","article-title":"Spurious ergodicity breaking in normal and fractional Ornstein-Uhlenbeck process","volume":"22","author":"Mardoukhi","year":"2020","journal-title":"New J. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"2404","DOI":"10.1007\/s10955-020-02669-y","article-title":"Analytical Survival Analysis of the Ornstein-Uhlenbeck Process","volume":"181","author":"Giorgini","year":"2020","journal-title":"J. Stat. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"055002","DOI":"10.1088\/1751-8121\/abd677","article-title":"Statistics of the first passage area functional for an Ornstein-Uhlenbeck process","volume":"54","author":"Kearney","year":"2021","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"032132","DOI":"10.1103\/PhysRevE.103.032132","article-title":"Noise and ergodic properties of Brownian motion in an optical tweezer: Looking at regime crossovers in an Ornstein-Uhlenbeck process","volume":"103","author":"Goerlich","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"014120","DOI":"10.1103\/PhysRevE.105.014120","article-title":"Anomalous scaling and first-order dynamical phase transition in large deviations of the Ornstein-Uhlenbeck process","volume":"105","author":"Smith","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"215003","DOI":"10.1088\/1751-8121\/acc559","article-title":"Mean first exit times of Ornstein-Uhlenbeck processes in high-dimensional spaces","volume":"56","author":"Kersting","year":"2023","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"045001","DOI":"10.1088\/1751-8121\/ada748","article-title":"Generalised Ornstein-Uhlenbeck process: Memory effects and resetting","volume":"58","author":"Trajanovski","year":"2025","journal-title":"J. Phys. Math. Theor."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Adler, R., Feldman, R., and Taqqu, M. (1998). A Practical Guide to Heavy Tails: Statistical Techniques and Applications, Springer.","DOI":"10.21236\/ADA336956"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Nair, J., Wierman, A., and Zwart, B. (2022). The Fundamentals of Heavy-Tails: Properties, Emergence, and Identification, Cambridge University Press.","DOI":"10.1017\/9781009053730"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1029\/WR004i005p00909","article-title":"Noah, Joseph, and operational hydrology","volume":"4","author":"Mandelbrot","year":"1968","journal-title":"Water Resour. Res."},{"key":"ref_49","unstructured":"Bertoin, J. (1998). Levy Processes, Cambridge University Press."},{"key":"ref_50","unstructured":"Ken-Iti, S. (1999). Levy Processes and Infinitely Divisible Distributions, Cambridge University Press."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Barndorff-Nielsen, O.E., Mikosch, T., and Resnick, S.I. (2001). Levy Processes: Theory and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-1-4612-0197-7"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"6843","DOI":"10.1063\/1.1290054","article-title":"Ornstein-Uhlenbeck-Cauchy process","volume":"41","author":"Garbaczewski","year":"2000","journal-title":"J. Math. Phys."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physa.2003.11.007","article-title":"A growth-collapse model: Levy inflow, geometric crashes, and generalized Ornstein-Uhlenbeck dynamics","volume":"334","author":"Eliazar","year":"2004","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"759","DOI":"10.3150\/bj\/1130077593","article-title":"Nonparametric inference for Levy-driven Ornstein-Uhlenbeck processes","volume":"11","author":"Jongbloed","year":"2005","journal-title":"Bernoulli"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10955-004-2710-9","article-title":"Levy, Ornstein-Uhlenbeck, and subordination: Spectral vs. jump description","volume":"119","author":"Eliazar","year":"2004","journal-title":"J. Stat. Phys."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1007\/s10955-004-8782-8","article-title":"Stochastic Ornstein-Uhlenbeck Capacitors","volume":"118","author":"Eliazar","year":"2005","journal-title":"J. Stat. Phys."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"977","DOI":"10.1239\/jap\/1197908818","article-title":"Estimation for non-negative Levy-driven Ornstein-Uhlenbeck processes","volume":"44","author":"Brockwell","year":"2007","journal-title":"J. Appl. Probab."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1080\/15326340701471091","article-title":"Short and long memory fractional Ornstein-Uhlenbeck alpha-stable processes","volume":"23","author":"Magdziarz","year":"2007","journal-title":"Stoch. Model."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.physa.2007.08.016","article-title":"Fractional Ornstein-Uhlenbeck processes. Joseph effect in models with infinite variance","volume":"387","author":"Magdziarz","year":"2008","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_60","first-page":"383","article-title":"Ornstein-Uhlenbeck related models driven by Levy processes","volume":"124","author":"Brockwell","year":"2012","journal-title":"Stat. Methods Stoch. Differ. Equ."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"150602","DOI":"10.1103\/PhysRevLett.110.150602","article-title":"Nonspectral relaxation in one dimensional Ornstein-Uhlenbeck processes","volume":"110","author":"Toenjes","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1007\/s11118-014-9458-x","article-title":"Ornstein-Uhlenbeck processes driven by cylindrical Levy processes","volume":"42","author":"Riedle","year":"2015","journal-title":"Potential Anal."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"052104","DOI":"10.1103\/PhysRevE.93.052104","article-title":"Nonspectral modes and how to find them in the Ornstein-Uhlenbeck process with white \u03bc-stable noise","volume":"93","author":"Thiel","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"1657","DOI":"10.1103\/PhysRevE.50.1657","article-title":"Langevin equations for continuous time Levy flights","volume":"50","author":"Fogedby","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"2736","DOI":"10.1103\/PhysRevE.59.2736","article-title":"Levy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions","volume":"59","author":"Jespersen","year":"1999","journal-title":"Phys. Rev. E"},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/S0301-0104(02)00551-7","article-title":"Stationary states of non-linear oscillators driven by Levy noise","volume":"284","author":"Chechkin","year":"2002","journal-title":"Chem. Phys."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/S0301-0104(02)00671-7","article-title":"Levy flights in external force fields: From models to equations","volume":"284","author":"Brockmann","year":"2002","journal-title":"Chem. Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"739","DOI":"10.1023\/A:1022894030773","article-title":"Levy-driven Langevin systems: Targeted stochasticity","volume":"111","author":"Eliazar","year":"2003","journal-title":"J. Stat. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"1505","DOI":"10.1023\/B:JOSS.0000028067.63365.04","article-title":"Levy flights in a steep potential well","volume":"115","author":"Chechkin","year":"2004","journal-title":"J. Stat. Phys."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"041122","DOI":"10.1103\/PhysRevE.76.041122","article-title":"Stationary states in Langevin dynamics under asymmetric Levy noises","volume":"76","author":"Dybiec","year":"2007","journal-title":"Phys. Rev. E Nonlinear Soft Matter Phys."},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"P07008","DOI":"10.1088\/1742-5468\/2010\/07\/P07008","article-title":"Stationary states in single-well potentials under symmetric Levy noises","volume":"2010","author":"Dybiec","year":"2010","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"162002","DOI":"10.1088\/1751-8113\/45\/16\/162002","article-title":"Langevin unification of fractional motions","volume":"45","author":"Eliazar","year":"2012","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1007\/s10955-012-0465-2","article-title":"Langevin picture of Levy walks and their extensions","volume":"147","author":"Magdziarz","year":"2012","journal-title":"J. Stat. Phys."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"426","DOI":"10.2478\/s13540-012-0031-2","article-title":"Velocity and displacement correlation functions for fractional generalized Langevin equations","volume":"15","author":"Sandev","year":"2012","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1016\/j.physa.2016.09.018","article-title":"Generalized Langevin equation with tempered memory kernel","volume":"466","author":"Liemert","year":"2017","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1016\/j.sigpro.2004.09.016","article-title":"Fractional Ornstein-Uhlenbeck Levy processes and the Telecom process: Upstairs and downstairs","volume":"85","author":"Wolpert","year":"2005","journal-title":"Signal Process."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1080\/0740817X.2016.1172743","article-title":"Levy-driven non-Gaussian Ornstein-Uhlenbeck processes for degradation-based reliability analysis","volume":"48","author":"Shu","year":"2016","journal-title":"IIE Trans."},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s10479-015-1967-5","article-title":"Estimation of Levy-driven Ornstein-Uhlenbeck processes: Application to modeling of CO2 and fuel-switching","volume":"255","author":"Chevallier","year":"2017","journal-title":"Ann. Oper. Res."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s00780-019-00413-3","article-title":"Ruin probabilities for a Levy-driven generalised Ornstein\u2013Uhlenbeck process","volume":"24","author":"Kabanov","year":"2020","journal-title":"Financ. Stochastics"},{"key":"ref_80","unstructured":"Onalan, O. (2009, January 1\u20133). Financial modelling with Ornstein-Uhlenbeck processes driven by Levy process. Proceedings of the World Congress on Engineering, London, UK."},{"key":"ref_81","first-page":"129","article-title":"Fractional Ornstein-Uhlenbeck processes driven by stable Levy motion in finance","volume":"42","author":"Onalan","year":"2010","journal-title":"Int. Res. J. Financ. Econ."},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"3153","DOI":"10.1080\/00036846.2019.1566688","article-title":"Optimal trading strategies for Levy-driven Ornstein\u2013Uhlenbeck processes","volume":"51","author":"Endres","year":"2019","journal-title":"Appl. Econ."},{"key":"ref_83","doi-asserted-by":"crossref","unstructured":"Shlesinger, M.F., and Klafter, J. (1986). Levy walks versus Levy flights. On Growth and Form: Fractal and Non-Fractal Patterns in Physics, Springer.","DOI":"10.1007\/978-94-009-5165-5_29"},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/0378-4371(86)90224-4","article-title":"Levy walks with applications to turbulence and chaos","volume":"140","author":"Shlesinger","year":"1986","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"4760","DOI":"10.1103\/PhysRevE.54.4760","article-title":"Dynamical approach to Levy processes","volume":"54","author":"Allegrini","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.1103\/PhysRevLett.58.1100","article-title":"Levy dynamics of enhanced diffusion: Application to turbulence","volume":"58","author":"Shlesinger","year":"1987","journal-title":"Phys. Rev. Lett."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1070\/PU2003v046n08ABEH001324","article-title":"Self-similar anomalous diffusion and Levy-stable laws","volume":"46","author":"Uchaikin","year":"2003","journal-title":"Phys.-Uspekhi"},{"key":"ref_88","doi-asserted-by":"crossref","unstructured":"Chechkin, A.V., Gonchar, V.Y., Klafter, J., and Metzler, R. (2006). Fundamentals of Levy flight processes. Fractals, Diffusion, and Relaxation in Disordered Complex Systems: Advances in Chemical Physics, Part B, John Wiley & Sons, Inc.","DOI":"10.1002\/0470037148.ch9"},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.chaos.2007.01.055","article-title":"Some fundamental aspects of Levy flights","volume":"34","author":"Metzler","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_90","doi-asserted-by":"crossref","unstructured":"Chechkin, A.V., Metzler, R., Klafter, J., and Gonchar, V.Y. (2008). Introduction to the theory of Levy flights. Anomalous Transport: Foundations and Applications, Wiley-VCH Verlag GmbH & Co. KGaA.","DOI":"10.1002\/9783527622979.ch5"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"2649","DOI":"10.1142\/S0218127408021877","article-title":"Levy flight superdiffusion: An introduction","volume":"18","author":"Dubkov","year":"2008","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1007\/s13538-013-0142-1","article-title":"How physicists made stable Levy processes physically plausible","volume":"43","author":"Schinckus","year":"2013","journal-title":"Braz. J. Phys."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1103\/RevModPhys.87.483","article-title":"Levy walks","volume":"87","author":"Zaburdaev","year":"2015","journal-title":"Rev. Mod. Phys."},{"key":"ref_94","doi-asserted-by":"crossref","unstructured":"Reynolds, A.M. (2018). Current status and future directions of Levy walk research. Biol. Open, 7.","DOI":"10.1242\/bio.030106"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"24336","DOI":"10.1073\/pnas.2001548117","article-title":"Functional advantages of Levy walks emerging near a critical point","volume":"117","author":"Abe","year":"2013","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_96","doi-asserted-by":"crossref","unstructured":"Garg, K., and Kello, C.T. (2021). Efficient Levy walks in virtual human foraging. Sci. Rep., 11.","DOI":"10.1038\/s41598-021-84542-w"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"118001","DOI":"10.1103\/PhysRevLett.127.118001","article-title":"Anomalous diffusion and Levy walks distinguish active from inertial turbulence","volume":"127","author":"Mukherjee","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1016\/j.csbj.2020.11.045","article-title":"Levy walk in swarm models based on Bayesian and inverse Bayesian inference","volume":"19","author":"Gunji","year":"2021","journal-title":"Comput. Struct. Biotechnol. J."},{"key":"ref_99","doi-asserted-by":"crossref","first-page":"484001","DOI":"10.1088\/1751-8121\/ac31a1","article-title":"Bayesian inference of Levy walks via hidden Markov models","volume":"54","author":"Park","year":"2021","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"118835","DOI":"10.1016\/j.jenvman.2023.118835","article-title":"Grazing livestock move by Levy walks: Implications for soil health and environment","volume":"345","author":"Rivero","year":"2023","journal-title":"J. Environ. Manag."},{"key":"ref_101","doi-asserted-by":"crossref","unstructured":"Sakiyama, T., and Okawara, M. (2023). A short memory can induce an optimal Levy walk. World Conference on Information Systems and Technologies, Springer Nature Switzerland.","DOI":"10.1007\/978-3-031-45651-0_42"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"080601","DOI":"10.1103\/PhysRevLett.124.080601","article-title":"Inverse square Levy walks are not optimal search strategies for d \u2265 2","volume":"124","author":"Levernier","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"eabe8211","DOI":"10.1126\/sciadv.abe8211","article-title":"Intermittent inverse-square Levy walks are optimal for finding targets of all sizes","volume":"7","author":"Guinard","year":"2021","journal-title":"Sci. Adv."},{"key":"ref_104","doi-asserted-by":"crossref","unstructured":"Clementi, A., d\u2019Amore, F., Giakkoupis, G., and Natale, E. (2021, January 26\u201330). Search via parallel Levy walks on Z2. Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing, Virtual.","DOI":"10.1145\/3465084.3467921"},{"key":"ref_105","doi-asserted-by":"crossref","unstructured":"Padash, A., Sandev, T., Kantz, H., Metzler, R., and Chechkin, A.V. (2022). Asymmetric Levy flights are more efficient in random search. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6050260"},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"034002","DOI":"10.1088\/1751-8121\/ac3fc1","article-title":"Record statistics for random walks and Levy flights with resetting","volume":"55","author":"Majumdar","year":"2021","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"044147","DOI":"10.1103\/PhysRevE.109.044147","article-title":"Levy flights and Levy walks under stochastic resetting","volume":"109","author":"Zbik","year":"2024","journal-title":"Phys. Rev. E"},{"key":"ref_108","first-page":"L022103","article-title":"Optimizing leapover lengths of Levy flights with resetting","volume":"110","author":"Radice","year":"2024","journal-title":"Phys. Rev."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"062127","DOI":"10.1103\/PhysRevE.101.062127","article-title":"Levy walk dynamics in an external harmonic potential","volume":"101","author":"Xu","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"023002","DOI":"10.1088\/1367-2630\/abd43c","article-title":"Moses, Noah and Joseph effects in Levy walks","volume":"23","author":"Aghion","year":"2021","journal-title":"New J. Phys."},{"key":"ref_111","doi-asserted-by":"crossref","unstructured":"Cleland, J.D., and Williams, M.A.K. (2022). Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Levy Flights. Mathematics, 10.","DOI":"10.3390\/math10183235"},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"409","DOI":"10.3390\/forecast4020023","article-title":"A Monte Carlo Approach to Bitcoin Price Prediction with Fractional Ornstein\u2013Uhlenbeck Levy Process","volume":"4","author":"Mba","year":"2022","journal-title":"Forecasting"},{"key":"ref_113","doi-asserted-by":"crossref","unstructured":"Mariani, M.C., Asante, P.K., Kubin, W., and Tweneboah, O.K. (2022). Data Analysis Using a Coupled System of Ornstein\u2013Uhlenbeck Equations Driven by Levy Processes. Axioms, 11.","DOI":"10.3390\/axioms11040160"},{"key":"ref_114","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10955-021-02815-0","article-title":"Cutoff thermalization for Ornstein\u2013Uhlenbeck systems with small Levy noise in the Wasserstein distance","volume":"184","author":"Barrera","year":"2021","journal-title":"J. Stat. Phys."},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1007\/s10959-022-01170-4","article-title":"Parameter Estimation for Ornstein\u2013Uhlenbeck Driven by Ornstein\u2013Uhlenbeck Processes with Small Levy Noises","volume":"36","author":"Zhang","year":"2023","journal-title":"J. Theor. Probab."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"012135","DOI":"10.1103\/PhysRevE.99.012135","article-title":"Langevin dynamics for a Levy walk with memory","volume":"99","author":"Chen","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_117","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1007\/s10884-022-10138-1","article-title":"The cutoff phenomenon in Wasserstein distance for nonlinear stable Langevin systems with small Levy noise","volume":"36","author":"Barrera","year":"2024","journal-title":"J. Dyn. Differ. Equ."},{"key":"ref_118","doi-asserted-by":"crossref","unstructured":"Liu, Y., Wang, J., and Zhang, M.-g. (2024). Exponential Contractivity and Propagation of Chaos for Langevin Dynamics of McKean-Vlasov Type with Levy Noises. Potential Anal., 1\u201334.","DOI":"10.1007\/s11118-024-10130-y"},{"key":"ref_119","doi-asserted-by":"crossref","first-page":"104341","DOI":"10.1016\/j.spa.2024.104341","article-title":"Exponential ergodicity of Levy driven Langevin dynamics with singular potentials","volume":"172","author":"Bao","year":"2024","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"013024","DOI":"10.1088\/1367-2630\/aaf764","article-title":"Levy-walk-like Langevin dynamics","volume":"21","author":"Wang","year":"2019","journal-title":"New J. Phys."},{"key":"ref_121","doi-asserted-by":"crossref","first-page":"012136","DOI":"10.1103\/PhysRevE.103.012136","article-title":"Levy-walk-like Langevin dynamics affected by a time-dependent force","volume":"103","author":"Chen","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"013109","DOI":"10.1063\/5.0174613","article-title":"Levy-walk-like Langevin dynamics with random parameters","volume":"34","author":"Chen","year":"2024","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_123","unstructured":"Cox, B., Laufer, J.G., Arridge, S.R., Beard, P.C., Laufer, A.J.G., and Arridge, A.S.R. (1984). Long Range Dependence: A Review, In Iowa State University."},{"key":"ref_124","unstructured":"Doukhan, P., Oppenheim, G., and Taqqu, M. (2002). Theory and Applications of Long-Range Dependence, Springer Science & Business Media."},{"key":"ref_125","doi-asserted-by":"crossref","unstructured":"Rangarajan, G., and Ding, M. (2003). Processes with Long-Range Correlations: Theory and Applications, Springer Science & Business Media.","DOI":"10.1007\/3-540-44832-2"},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.aop.2015.09.023","article-title":"How random is a random vector?","volume":"363","author":"Eliazar","year":"2015","journal-title":"Ann. Phys."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"125662","DOI":"10.1016\/j.physa.2020.125662","article-title":"Five degrees of randomness","volume":"568","author":"Eliazar","year":"2021","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_128","doi-asserted-by":"crossref","first-page":"S63","DOI":"10.1121\/1.2016299","article-title":"Perplexity: A measure of the difficulty of speech recognition tasks","volume":"62","author":"Jelinek","year":"1977","journal-title":"J. Acoust. Soc. Am."},{"key":"ref_129","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"2111","DOI":"10.1088\/0022-3719\/3\/10\/013","article-title":"Localization of normal modes in vitreous silica, germania and beryllium fluoride","volume":"3","author":"Bell","year":"1970","journal-title":"J. Phys. C Solid State Phys."},{"key":"ref_131","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1080\/14786437208223861","article-title":"The structure of vitreous silica: Validity of the random network theory","volume":"25","author":"Bell","year":"1972","journal-title":"Philos. Mag."},{"key":"ref_132","doi-asserted-by":"crossref","first-page":"012108","DOI":"10.1103\/PhysRevA.85.012108","article-title":"Collision entropy and optimal uncertainty","volume":"85","author":"Bosyk","year":"2012","journal-title":"Phys. Rev. A"},{"key":"ref_133","doi-asserted-by":"crossref","unstructured":"Skorski, M. (2015, January 15\u201317). Shannon entropy versus renyi entropy from a cryptographic viewpoint. Proceedings of the IMA International Conference on Cryptography and Coding, Oxford, UK.","DOI":"10.1007\/978-3-319-27239-9_16"},{"key":"ref_134","doi-asserted-by":"crossref","first-page":"688","DOI":"10.1038\/163688a0","article-title":"Measurement of diversity","volume":"163","author":"Simpson","year":"1949","journal-title":"Nature"},{"key":"ref_135","doi-asserted-by":"crossref","unstructured":"Hirschman, A.O. (1945). National Power and the Structure of Foreign Trade, University of Califorina Press.","DOI":"10.1525\/9780520378179"},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"427","DOI":"10.2307\/1934352","article-title":"Diversity and evenness: A unifying notation and its consequences","volume":"54","author":"Hill","year":"1973","journal-title":"Ecology"},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1146\/annurev.es.05.110174.001441","article-title":"The measurement of species diversity","volume":"5","author":"Peet","year":"1974","journal-title":"Annu. Rev. Ecol. Syst."},{"key":"ref_138","doi-asserted-by":"crossref","unstructured":"Magurran, A.E. (1988). Ecological Diversity and Its Measurement, Princeton University Press.","DOI":"10.1007\/978-94-015-7358-0"},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1111\/j.2006.0030-1299.14714.x","article-title":"Entropy and diversity","volume":"113","author":"Jost","year":"2006","journal-title":"Oikos"},{"key":"ref_140","unstructured":"Legendre, P., and Legendre, L. (2012). Numerical Ecology, Elsevier."},{"key":"ref_141","unstructured":"Renyi, A. (1960, January 20\u201330). On measures of information and entropy. Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, Berkeley, CA, USA."},{"key":"ref_142","doi-asserted-by":"crossref","unstructured":"Zolotarev, V.M. (1986). One-Dimensional Stable Distributions, American Mathematical Society.","DOI":"10.1090\/mmono\/065"},{"key":"ref_143","doi-asserted-by":"crossref","unstructured":"Borak, S., Hardle, W., and Weron, R. (2005). Stable Distributions, Humboldt-Universitat zu Berlin.","DOI":"10.2139\/ssrn.2894444"},{"key":"ref_144","doi-asserted-by":"crossref","unstructured":"Nolan, J.P. (2020). Univariate Stable Distributions, Springer.","DOI":"10.1007\/978-3-030-52915-4"},{"key":"ref_145","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1145\/358198.358222","article-title":"The weighted median filter","volume":"27","author":"Brownrigg","year":"1984","journal-title":"Commun. ACM"},{"key":"ref_146","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1109\/82.486465","article-title":"Weighted median filters: A tutorial","volume":"43","author":"Yin","year":"1996","journal-title":"IEEE Trans. Circuits Syst. II Analog Digit. Signal Process."},{"key":"ref_147","unstructured":"Justusson, B.I. (2006). Median filtering: Statistical properties. Two-Dimensional Digital Signal Prcessing II: Transforms and Median Filters, Springer."},{"key":"ref_148","doi-asserted-by":"crossref","first-page":"012033","DOI":"10.1088\/1742-6596\/394\/1\/012033","article-title":"Entropy: A new measure of stock market volatility?","volume":"394","author":"Bentes","year":"2012","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_149","doi-asserted-by":"crossref","first-page":"056112","DOI":"10.1103\/PhysRevE.86.056112","article-title":"Alternate entropy measure for assessing volatility in financial markets","volume":"86","author":"Bose","year":"2012","journal-title":"Phys. Rev. E- Nonlinear Soft Matter Phys."},{"key":"ref_150","doi-asserted-by":"crossref","first-page":"74","DOI":"10.3390\/e14010074","article-title":"A new approach to measure volatility in energy markets","volume":"14","author":"Ruiz","year":"2012","journal-title":"Entropy"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/157\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:25:55Z","timestamp":1760027155000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/157"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,2]]},"references-count":150,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["e27020157"],"URL":"https:\/\/doi.org\/10.3390\/e27020157","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,2,2]]}}}