Generating diffusions with fractional Brownian motion

TL;DR

This paper investigates the limiting behavior of systems driven by fractional Brownian motion with different Hurst parameters, revealing convergence to Markov processes and diffusions, bridging homogenization and averaging theories.

Contribution

It demonstrates convergence of fast/slow systems driven by fractional Brownian motion to Markov processes and diffusions without requiring $F$ to average to zero for $H<rac{1}{2}$, extending homogenization and averaging results.

Findings

Fast/slow systems converge to Markov processes.

Solutions converge to diffusions without $F$ averaging for $H<rac{1}{2}$.

$n$-point motions converge to Kunita type SDEs.

Abstract

We study fast / slow systems driven by a fractional Brownian motion $B$ with Hurst parameter $H\in (\frac 13, 1]$. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if $Y^\varepsilon$ denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale $\varepsilon \ll 1$, the solutions of the equation $$ dX^\varepsilon = \varepsilon^{\frac 12-H} F(X^\varepsilon,Y^\varepsilon)\,dB+F_0(X^\varepsilon,Y^\varepsilon)\,dt\; $$ converge to a regular diffusion without having to assume that $F$ averages to $0$, provided that $H< \frac 12$. For $H > \frac 12$, a similar result holds, but this time it does require $F$ to average to $0$. We also prove that the $n$-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuous interpolation between the homogenisation theorem for random ODEs with rapidly oscillating right-hand sides ($H=1$) and the averaging of diffusion processes ($H= \frac 12$).