Rough Homogenisation with Fractional Dynamics

Gehringer, Johann; Li, Xuemei

doi:10.48550/arxiv.2011.00075

Cited by 2 publications

(3 citation statements)

References 90 publications

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“…Next, since κ(nT ) nT we have that 1 (13). Using the triangle inequality together with the fact that the p-variation norms are monotonically decreasing in p,…”

Section: Examplesmentioning

confidence: 97%

“…In the context of semimartingales and rough paths with jumps [3,8,4], CLT on nilpotent covering graphs and crystal lattices [15,24,16], additive functional of Markov process and random walks in random environment [6]. For homogenization in the continuous settings [5,18,19], and for additive functionals of fractional random fields [11,12,13].…”

Section: Introductionmentioning

confidence: 99%

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Rough invariance principle for delayed regenerative processes

Orenshtein¹

2021

Preprint

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We derive an invariance principle for the lift to the rough path topology of stochastic processes with delayed regenerative increments under an optimal moment condition. An interesting feature of the result is the emergence of area anomaly, a correction term in the second level of the limiting rough path which is identified as the average stochastic area on a regeneration interval. A few applications include random walks in random environment and additive functionals of recurrent Markov chains. The result is formulated in the p-variation settings, where a rough Donsker Theorem is available under the second moment condition. The key renewal theorem is applied to obtain an optimal moment condition.

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“…Next, since κ(nT ) nT we have that 1 (13). Using the triangle inequality together with the fact that the p-variation norms are monotonically decreasing in p,…”

Section: Examplesmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Rough invariance principle for delayed regenerative processes

Orenshtein¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Regarding qualitative properties of slow-fast systems with fractional noise we mention: homogenization results [21], sample path estimates for additive fractional noise with H > 1/2 [18] and averaging principles [22]. Here the slow variable is perturbed by multiplicative fractional noise, with Hurst index H > 1/2 and the fast variable by a Brownian motion.…”

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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Rough Homogenisation with Fractional Dynamics

Cited by 2 publications

References 90 publications

Rough invariance principle for delayed regenerative processes

Rough invariance principle for delayed regenerative processes

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

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