Taylor’s Law from Gaussian diffusions

Abstract Taylor’s Law, also known as fluctuation scaling, manifests a power relation between the means and the variances of statistical distributions. The class of Gaussian-selfsimilar stochastic motions offers a plethora of macroscopic diffusion models, regular and anomalous alike. This class includes Brownian motion, scaled Brownian motion, fractional Brownian motion, and more. Within this class, power Brownian motion (PBM) is the sub-class of motions that are also Markovian. Considering conditional distributions of motion positions, this paper establishes that: the Gaussian-selfsimilar class universally generates Taylor’s Law, doing so with both positive and negative Taylor exponents. The paper also unveils a profound interplay between PBM and the universal generation of Taylor’s Law from the Gaussian-selfsimilar class.