BibTex format
@article{Hairer:2022,
author = {Hairer, M and Li, X-M},
journal = {Communications in Mathematical Physics},
title = {Generating diffusions with fractional Brownian motion},
url = {http://arxiv.org/abs/2109.06948v1},
year = {2022}
}
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Citation
RIS format (EndNote, RefMan)
TY - JOUR
AB - We study fast / slow systems driven by a fractional Brownian motion $B$ withHurst parameter $H\in (\frac 13, 1]$. Surprisingly, the slow dynamic convergeson suitable timescales to a limiting Markov process and we describe itsgenerator. More precisely, if $Y^\varepsilon$ denotes a Markov process withsufficiently good mixing properties evolving on a fast timescale $\varepsilon\ll 1$, the solutions of the equation $$ dX^\varepsilon = \varepsilon^{\frac12-H} F(X^\varepsilon,Y^\varepsilon)\,dB+F_0(X^\varepsilon,Y^\varepsilon)\,dt\;$$ converge to a regular diffusion without having to assume that $F$ averagesto $0$, provided that $H< \frac 12$. For $H > \frac 12$, a similar resultholds, but this time it does require $F$ to average to $0$. We also prove thatthe $n$-point motions converge to those of a Kunita type SDE. One nice interpretation of this result is that it provides a continuousinterpolation between the homogenisation theorem for random ODEs with rapidlyoscillating right-hand sides ($H=1$) and the averaging of diffusion processes($H= \frac 12$).
AU - Hairer,M
AU - Li,X-M
PY - 2022///
TI - Generating diffusions with fractional Brownian motion
T2 - Communications in Mathematical Physics
UR - http://arxiv.org/abs/2109.06948v1
ER -