Scaled Brownian motion with random anomalous diffusion exponent

The scaled Brownian motion (SBM) is regarded as one of the paradigmatic random processes, characterized by anomalous diffusion through the diffusion exponent. It is a Gaussian, self-similar process with independent increments and has found applications across various fields, from turbulence and stochastic hydrology to biophysics. In our paper, inspired by recent single particle tracking biological experiments, we introduce a process called scaled Brownian motion with random exponent (SBMRE), which retains the key features of SBM at the level of individual trajectories, but with anomalous diffusion exponents that vary randomly across trajectories.

We discuss the main probabilistic properties of SBMRE, including its probability density function (pdf) and the qth absolute moment. Additionally, we present the expected value of the time-averaged mean squared displacement (TAMSD) and the ergodicity breaking parameter. Furthermore, we analyze the pdf of the first hitting time in a semi-infinite domain, the martingale property of SBMRE, and its stochastic exponential. As special cases, we consider two distributions for the anomalous diffusion exponent, namely the mixture of two point distributions and beta distribution, and explore the asymptotics of the corresponding characteristics. Theoretical results for SBMRE are validated through numerical simulations and compared with the analogous characteristics of SBM.