Generating Diffusions with Fractional Brownian Motion
Generating Diffusions with Fractional Brownian Motion
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DOI：
10.1007/s00220-022-04462-2
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发表时间：
2021-09
影响因子：
2.4
通讯作者：
Martin Hairer;Xue-Mei Li
中科院分区：
文献类型：
--
作者：
Martin Hairer;Xue-Mei Li
We study fast/slow systems driven by a fractional Brownian motionBwith Hurst parameter. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, ifdenotes a Markov process with sufficiently good mixing properties evolving on a fast timescale, the solutions of the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} dX^\varepsilon = {\varepsilon }^{\frac{1}{2}-H} F(X^\varepsilon ,Y^\varepsilon )\,dB+F_0(X^\varepsilon ,Y^{\varepsilon })\,dt\; \end{aligned}$$\end{document}converge to a regular diffusion without having to assume thatFaverages to 0, provided that. For, a similar result holds, but this time it does requireFto average to 0. We also prove that then-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 () and the averaging of diffusion processes ().
We study fast/slow systems driven by a fractional Brownian motionBwith Hurst parameter. Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, ifdenotes a Markov process with sufficiently good mixing properties evolving on a fast timescale, the solutions of the equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} dX^\varepsilon = {\varepsilon }^{\frac{1}{2}-H} F(X^\varepsilon ,Y^\varepsilon )\,dB+F_0(X^\varepsilon ,Y^{\varepsilon })\,dt\; \end{aligned}$$\end{document}converge to a regular diffusion without having to assume thatFaverages to 0, provided that. For, a similar result holds, but this time it does requireFto average to 0. We also prove that then-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 () and the averaging of diffusion processes ().