Scalings and fractals in information geometry: Ornstein–Uhlenbeck processes

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 and mean value y 0 relaxes into a stationary PDF with a width epsilon, set by the strength of a stochastic noise. By utilizing the information length which quantifies the accumulative information change, we investigate the scaling of with epsilon. When , the movement of a PDF leads to a robust power-law scaling with the fractal dimension . In general when , is possible in the limit of a large time when the movement of a PDF is a main process for information change (e.g. ). 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.