Riemann–Liouville fractional Brownian motion with random Hurst exponent

We examine two stochastic processes with random parameters, which in their basic versions (i.e., when the parameters are fixed) are Gaussian and display long-range dependence and anomalous diffusion behavior, characterized by the Hurst exponent. Our motivation comes from biological experiments, which show that the basic models are inadequate for accurate description of the data, leading to modifications of these models in the literature through introduction of the random parameters. The first process, fractional Brownian motion with random Hurst exponent (referred to as FBMRE below) has been recently studied, while the second one, Riemann-Liouville fractional Brownian motion with random exponent (RL FBMRE) has not been explored. To advance the theory of such doubly stochastic anomalous diffusion models, we investigate the probabilistic properties of RL FBMRE and compare them to those of FBMRE. Our main focus is on the autocovariance function and the time-averaged mean squared displacement of the processes. Furthermore, we analyze the second moment of the increment processes for both models, as well as their ergodicity properties. As a specific case, we consider the mixture of two-point distributions of the Hurst exponent, emphasizing key differences in the characteristics of RL FBMRE and FBMRE, particularly in their asymptotic behavior. The theoretical findings presented here lay the groundwork for developing new methods to distinguish these processes and estimate their parameters from experimental data.