Fractional Brownian motions ruled by nonlinear equations

TL;DR

This paper explores generalized diffusion equations with time-dependent diffusivity governed by nonlinear fractional differential equations, establishing a link to fractional Brownian motion and deriving their governing equations.

Contribution

It introduces a novel connection between nonlinear fractional differential equations and fractional Brownian motion, including derivation of their governing equations.

Findings

Diffusivity in fBm satisfies an additive evolutive fractional equation

Derived governing equations for fBm and iterated fBm

Highlighted the link between nonlinear fractional equations and fBm

Abstract

In this note we consider generalized 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.