Scalings and fractals in information geometry: Ornstein-Uhlenbeck processes

William Oxley, Eun Jin Kim

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We propose a new methodology to understand a stochastic process from the perspective of information geometry by investigating power-law scaling and fractals in the evolution of information. Specifically, we employ the Ornstein-Uhlenbeck process where an initial probability density function (PDF) with a given width ϵ0 and mean value y0 relaxes into a stationary PDF with a width ϵ, set by the strength of a stochastic noise. By utilizing the information length L which quantifies the accumulative information change, we investigate the scaling of L with ϵ. When ϵ = ϵ0, the movement of a PDF leads to a robust power-law scaling with the fractal dimension DF = 2. In general when ϵ ≠ ϵ0, DF = 2 is possible in the limit of a large time when the movement of a PDF is a main process for information change (e.g. y0 ≫ ϵ ≫ ϵ0). We discuss the physical meaning of different scalings due to PDF movement, diffusion and entropy change as well as implications of our finding for understanding a main process responsible for the evolution of information.

Original languageEnglish
Article number113401
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number11
Publication statusPublished - 15 Nov 2018
Externally publishedYes

Bibliographical note

This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Statistical Mechanics: Theory and Experiment. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at 10.1088/1742-5468/aae851

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  • stochastic processes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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