Fractional Brownian motions ruled by nonlinear equations

In this note we consider generalised diffusion equations in which the diffusivity coefficient is not necessarily constant in time, but instead it solves a nonlinear fractional differential equation involving fractional Riemann–Liouville time-derivative. Our main contribution is to highlight the link between these generalised equations and fractional Brownian motion (fBm). In particular, we investigate the governing equation of fBm and show that its diffusion coefficient must satisfy an additive evolutive fractional equation. We derive in a similar way the governing equation of the iterated fractional Brownian motion.