Beta Brownian motion

Abstract

Brownian motion (BM) is the paradigmatic model of diffusion. Transcending from diffusion to anomalous diffusion, the principle Gaussian generalizations of BM are Scaled BM (SBM) and Fractional BM (FBM). In the sub/super diffusivity regimes: SBM is characterized by aging/anti-aging, and FBM is characterized by anti-persistence/persistence. BM is neither aging/anti-aging, nor persistent/anti-persistent. Within the realm of diffusion, a recent Gaussian generalization of BM, Weird BM (WBM), was shown to display aging/anti-aging and persistence/anti-persistence. This paper introduces and explores the anomalous-diffusion counterpart of WBM—termed Beta BM (BBM) due to its inherent beta-function kernel structure—and shows that: the weird behaviors of WBM become even weirder when elevating to BBM. Indeed, BBM displays a rich assortment of anomalous behaviors, and an even richer assortment of combinations of anomalous behaviors. In particular, the BBM anomalous behaviors include aging/anti-aging and persistence/anti-persistence—which BBM displays in both the sub and super diffusivity regimes. So, anomalous behaviors that are unattainable by the prominent models of SBM and FBM are well attainable by the BBM model.