Slow-Fast Systems with Fractional Environment and Dynamics

Li, Xuemei; Sieber, Julian

doi:10.48550/arxiv.2012.01910

Cited by 4 publications

(5 citation statements)

References 40 publications

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“…It may be surprising that, when H < 1 2 , even though X ε is driven by a fractional Brownian motion and F(x, y) isn't assumed to be centred in the y variable, the limit X is a regular diffusion. This is unlike the case H > 1 2 [31,52] where a noncentred F leads to an averaging result with a process driven by fBm in the limit. This change in behaviour can be understood heuristically as follows.…”

Section: The Markov Semigroup Associated To the Process Y Is Strongly...mentioning

confidence: 74%

“…See also [1,7,12,14] for more recent results with a similar flavour. In the case when the fast dynamics is non-Markovian and solves an equation driven by a fractional Brownian motion, a collection of homogenisation results were obtained in [23][24][25][26], while stochastic averaging results with non-Markovian fast motions are obtained in [51,52] for the case H > 1 2 . The former group of results are proved using rough path techniques, but there is of course an extensive literature on functional limit theorems based on either central or non-central limit theorems, see for example [3,5,6,10,54,62,63].…”

Section: The Markov Semigroup Associated To the Process Y Is Strongly...mentioning

confidence: 99%

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Generating Diffusions with Fractional Brownian Motion

Hairer

2022

Commun. Math. Phys.

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We study fast/slow systems driven by a fractional Brownian motion B with Hurst parameter $$H\in (\frac{1}{3}, 1]$$ H ∈ ( 1 3 , 1 ] . Surprisingly, the slow dynamic converges on suitable timescales to a limiting Markov process and we describe its generator. More precisely, if $$Y^\varepsilon $$ Y ε denotes a Markov process with sufficiently good mixing properties evolving on a fast timescale $$\varepsilon \ll 1$$ ε ≪ 1 , the solutions of the 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 ε = ε 1 2 - H F ( X ε , Y ε ) d B + F 0 ( X ε , Y ε ) d t converge to a regular diffusion without having to assume that F averages to 0, provided that $$H< \frac{1}{2}$$ H < 1 2 . For $$H > \frac{1}{2}$$ 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 ($$H=1$$ H = 1 ) and the averaging of diffusion processes ($$H= \frac{1}{2}$$ H = 1 2 ).

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Section: The Markov Semigroup Associated To the Process Y Is Strongly...mentioning

confidence: 74%

Section: The Markov Semigroup Associated To the Process Y Is Strongly...mentioning

confidence: 99%

Generating Diffusions with Fractional Brownian Motion

Hairer

2022

Commun. Math. Phys.

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“…However, the estimate of P c Res(ψ) is significantly more involved. Such estimates rely on averaging results, which are highly challenging to obtain for fractional noise [22,24]. Here we rely on an explicit pathwise approach possible for additive noise.…”

Section: Proofmentioning

confidence: 99%

“…These results rely on a novel application of the stochastic sewing lemma [23]. Based on this work averaging principles for slow-fast systems driven by independent fractional Brownian motions have been derived in [24]. Here we follow a different more pathwise approach yielding for additive noise stronger error estimates.…”

Section: Introductionmentioning

confidence: 99%

Amplitude equations for SPDEs driven by fractional additive noise with small Hurst parameter

Blömker¹,

Neamţu²

2021

Preprint

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We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter H ∈ (0, 1). Close to a change of stability measured with a small parameter ε, we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. We prove that up to an error term bounded by a power of ε depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so called amplitude equation, and in second order by a fast infinite dimensional Ornstein-Uhlenbeck process. To this aim we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters.

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“…Concerning this kind of slow-fast systems driven by fBM with H ∈ (1/2, 1) and BM, a few related problems have already been studied in [2,3,4]. Though it looks quite difficult to study the case where (w t ) is also fBM, a recent preprint [20] made a first attempt in that direction.…”

Section: Introductionmentioning

confidence: 99%

Averaging principle for slow-fast systems of rough differential equations via controlled paths

Inahama¹

2022

Preprint

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In this paper we prove the strong averaging principle for a slow-fast system of rough differential equations. The slow and the fast component of the system are driven by a rather general random rough path and Brownian rough path, respectively. These two driving noises are assumed to be independent. A prominent example of the driver of the slow component is fractional Brownian rough path with Hurst parameter between 1/3 and 1/2. We work in the framework of controlled path theory, which is one of the most widely-used frameworks in rough path theory. To prove our main theorem, we carry out Khas'minskiȋ's time-discretizing method in this framework.

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Slow-Fast Systems with Fractional Environment and Dynamics

Cited by 4 publications

References 40 publications

Generating Diffusions with Fractional Brownian Motion

Generating Diffusions with Fractional Brownian Motion

Amplitude equations for SPDEs driven by fractional additive noise with small Hurst parameter

Averaging principle for slow-fast systems of rough differential equations via controlled paths

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