Designing selfsimilar diffusions

Iddo Eliazar and Maxence Arutkin

Physica A: Statistical Mechanics and its Applications, 2025, vol. 658, issue C

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.

Keywords: Selfsimilarity; Anomalous diffusion; Non-Gaussian diffusion; Stochastic differential equations; Ito stochastic calculus; Lamperti transform (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:658:y:2025:i:c:s0378437124007799

DOI: 10.1016/j.physa.2024.130270

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