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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.08621 |
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| _version_ | 1866909951846252544 |
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| author | Li, Xue-Mei Piernot, Colin Sobczak, Szymon Ying, Kexing |
| author_facet | Li, Xue-Mei Piernot, Colin Sobczak, Szymon Ying, Kexing |
| contents | We consider a system of multiscale stochastic differential equations whose slow component is drivenby a fractional Brownian motion with Hurst parameter H greater than 1/2. Under ergodic assumptions ensuring the applicability of the fractional averaging and fractional homogenization theorems of Hairer and Li (arXiv:1902.11251, arXiv:2109.06948), we establish a fluctuation result. The deviation of the slow motion, scaled by epsilon^{1/2-H}, from its effective, time-dependent random limit converges, as the time-separation scale epsilon tends to zero, to the solution of a stochastic differential equation driven by a fractional Brownian motion and influenced by an additional space--time Gaussian field. Since the averaging principle and the fractional homogenization hold in different modes of convergence, obtaining the required joint convergence is a delicate matter. Moreover, neither the continuity of the Ito--Lyons solution map nor the martingale method is directly applicable for our purposes, so the proof requires several innovations. To establish the fluctuation theorem, we combine cumulant methods with a residue lemma and formulate the enlarged system as a rough differential equation in a suitable space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_08621 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fluctuations from a random fractional averaging limit Li, Xue-Mei Piernot, Colin Sobczak, Szymon Ying, Kexing Probability We consider a system of multiscale stochastic differential equations whose slow component is drivenby a fractional Brownian motion with Hurst parameter H greater than 1/2. Under ergodic assumptions ensuring the applicability of the fractional averaging and fractional homogenization theorems of Hairer and Li (arXiv:1902.11251, arXiv:2109.06948), we establish a fluctuation result. The deviation of the slow motion, scaled by epsilon^{1/2-H}, from its effective, time-dependent random limit converges, as the time-separation scale epsilon tends to zero, to the solution of a stochastic differential equation driven by a fractional Brownian motion and influenced by an additional space--time Gaussian field. Since the averaging principle and the fractional homogenization hold in different modes of convergence, obtaining the required joint convergence is a delicate matter. Moreover, neither the continuity of the Ito--Lyons solution map nor the martingale method is directly applicable for our purposes, so the proof requires several innovations. To establish the fluctuation theorem, we combine cumulant methods with a residue lemma and formulate the enlarged system as a rough differential equation in a suitable space. |
| title | Fluctuations from a random fractional averaging limit |
| topic | Probability |
| url | https://arxiv.org/abs/2512.08621 |