Exploring Finite-Sized Scale Invariance in Stochastic Variability with Toy Models: The Ornstein–Uhlenbeck Model

Stochastic variability is ubiquitous among astrophysical sources. Quantifying stochastic properties of observed time-series or lightcurves, can provide insights into the underlying physical mechanisms driving variability, especially those of the particles that radiate the observed emission. Toy models mimicking cosmic ray transport are particularly useful in providing a means of linking the statistical analyses of observed lightcurves to the physical properties and parameters. Here, we explore a very commonly observed feature; finite sized self-similarity or scale invariance which is a fundamental property of complex, dynamical systems. This is important to the general theme of physics and symmetry. We investigate it through the probability density function of time-varying fluxes arising from a Ornstein–Uhlenbeck Model, as this model provides an excellent description of several time-domain observations of sources like active galactic nuclei. The probability density function approach stems directly from the mathematical definition of self-similarity and is nonparametric. We show that the OU model provides an intuitive description of scale-limited self-similarity and stationary Gaussian distribution while potentially showing a way to link to the underlying cosmic ray transport. This finite size of the scale invariance depends upon the decay time in the OU model.