Power Brownian Motion: an Ornstein–Uhlenbeck lookout

Abstract

The well-known Ornstein-Uhlenbeck process (OUP) is the central go-to Gaussian model for statistical-equilibrium processes. The recently-introduced power Brownian motion (PBM) is a Gaussian model for diffusive motions, regular and anomalous alike. Using the Lamperti transform, this paper establishes PBM as the ‘diffusion counterpart’ of the OUP. Namely, the paper shows that PBM is for diffusive motions what the OUP is for statistical-equilibrium processes. The intimate parallels between the OUP and PBM are explored and illuminated via four main perspectives. 1) Statistical characterizations. 2) Kernel-integration with respect to Gaussian white noise. 3) Spatio-temporal scaling of the Wiener process. 4) Langevin stochastic dynamics driven by Gaussian white noise. To date, the prominent Gaussian models for anomalous diffusion are fractional Brownian motion (FBM), and scaled Brownian Motion (SBM). Due to its intimate OUP parallels, due to the ‘anomalous features’ it displays, due to the fact that it encompasses SBM, and following a detailed comparison to FBM: this paper argues the case for PBM as a central go-to Gaussian model for regular and anomalous diffusion.