Generating Diffusions with Fractional Brownian Motion

نویسندگان

چکیده

Abstract We study fast/slow systems driven by a fractional Brownian motion B with Hurst parameter $$H\in (\frac{1}{3}, 1]$$ H ∈ ( 1 3 , ] . Surprisingly, the slow dynamic converges on suitable timescales to limiting Markov process and we describe its generator. More precisely, if $$Y^\varepsilon $$ Y ε denotes sufficiently good mixing properties evolving fast timescale $$\varepsilon \ll 1$$ ≪ , solutions of equation $$\begin{aligned} dX^\varepsilon = {\varepsilon }^{\frac{1}{2}-H} F(X^\varepsilon ,Y^\varepsilon )\,dB+F_0(X^\varepsilon ,Y^{\varepsilon })\,dt\; \end{aligned}$$ d X = 2 - F ) B + 0 t converge regular diffusion without having assume that F averages 0, provided $$H< \frac{1}{2}$$ < For $$H > > similar result holds, but this time it does require average 0. also prove n -point motions those Kunita type SDE. One nice interpretation is provides continuous interpolation between homogenisation theorem for random ODEs rapidly oscillating right-hand sides ( $$H=1$$ ) averaging processes $$H= ).

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ژورنال

عنوان ژورنال: Communications in Mathematical Physics

سال: 2022

ISSN: ['0010-3616', '1432-0916']

DOI: https://doi.org/10.1007/s00220-022-04462-2