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Designing selfsimilar diffusions

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  • Eliazar, Iddo
  • Arutkin, Maxence

Abstract

Selfsimilar motions emerge universally on the macroscopic level, and the selfsimilarity of their trajectories is characterized by the Hurst exponent. Brownian motion – the paradigmatic model of diffusion – emerges universally from microscopic random walks, and it has the following specific features: its Hurst exponent is half, and the statistics of its positions are Gaussian. Considering Brownian motion as a given selfsimilar ‘input diffusion’, the goal of this paper is to generate a selfsimilar ‘output diffusion’ with the following general features: a desired ‘target’ Hurst exponent, and desired ‘target’ statistics of its positions. To accomplish the goal, the paper acts as follows. (1) Using stochastic differential equations (SDEs), it establishes general results regarding ‘selfsimilar-to-selfsimilar’ SDEs. (2) Applying the Lamperti transform to ‘selfsimilar-to-selfsimilar’ SDEs, it establishes general Lamperti results regarding these SDEs. (3) Following the Ito stochastic calculus, it devises an adaptable Ito design algorithm for selfsimilar diffusions. The results and the algorithm presented here provide researchers with a versatile and practical SDE framework for the design of selfsimilar diffusions – regular and anomalous alike, as well as Gaussian and non-Gaussian alike.

Suggested Citation

  • Eliazar, Iddo & Arutkin, Maxence, 2025. "Designing selfsimilar diffusions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 658(C).
    • Handle: RePEc:eee:phsmap:v:658:y:2025:i:c:s0378437124007799
    DOI: 10.1016/j.physa.2024.130270
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    References listed on IDEAS

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    1. Gorka Muñoz-Gil & Giovanni Volpe & Miguel Angel Garcia-March & Erez Aghion & Aykut Argun & Chang Beom Hong & Tom Bland & Stefano Bo & J. Alberto Conejero & Nicolás Firbas & Òscar Garibo i Orts & Aless, 2021. "Objective comparison of methods to decode anomalous diffusion," Nature Communications, Nature, vol. 12(1), pages 1-16, December.
    2. Lenzi, M.K. & Lenzi, E.K. & Guilherme, L.M.S. & Evangelista, L.R. & Ribeiro, H.V., 2022. "Transient anomalous diffusion in heterogeneous media with stochastic resetting," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 588(C).
    3. Farida Grinberg & Ezequiel Farrher & Luisa Ciobanu & Françoise Geffroy & Denis Le Bihan & N Jon Shah, 2014. "Non-Gaussian Diffusion Imaging for Enhanced Contrast of Brain Tissue Affected by Ischemic Stroke," PLOS ONE, Public Library of Science, vol. 9(2), pages 1-15, February.
    4. Henrik Seckler & Ralf Metzler, 2022. "Bayesian deep learning for error estimation in the analysis of anomalous diffusion," Nature Communications, Nature, vol. 13(1), pages 1-13, December.
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