Generating Diffusions with Fractional Brownian Motion

We study fast/slow systems driven by a fractional Brownian motion B with Hurst parameter H is an element of (1/3, 1]. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if Y-epsilon denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale epsilon << 1, the solutions of the equation dX(epsilon) = epsilon(1/2-H) F (X-epsilon, Y-epsilon) dB + F-0(X-epsilon, Y-epsilon) dt converge to a regular diffusion without having to assume that F averages to 0, provided that H < 1/2. For H > 1/2, 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 time homogenisation theorem for random ODEs with rapidly oscillating right-hand sides (H = 1) and the averaging of diffusion processes (H = 1/2).