Bibliographic Information: Lia, M., Lib, Y., Peia, B., Xua, Y. (2024). Averaging principle for semilinear slow-fast rough PDEs. arXiv preprint arXiv:2411.06089v1.
Research Objective: To investigate the averaging principle for a specific class of semilinear slow-fast stochastic partial differential equations (SPDEs) driven by finite-dimensional rough multiplicative noise.
Methodology: The authors utilize controlled rough path theory and Khasminskii's time discretization scheme to analyze the asymptotic behavior of the slow component in the slow-fast SPDE system.
Key Findings: The study demonstrates that the slow component of the system converges strongly to the solution of a corresponding averaged equation under the Hölder topology. This convergence is proven using the framework of controlled rough path theory.
Main Conclusions: The research successfully establishes the strong averaging principle for the considered class of slow-fast rough SPDEs. This implies that the long-term behavior of the slow component can be effectively approximated by an averaged equation, simplifying the analysis and simulation of such complex systems.
Significance: This paper contributes to the field of stochastic analysis, particularly in the context of rough path theory and averaging principles for SPDEs. It extends existing averaging principles for SPDEs driven by semimartingales to a class of SPDEs driven by rough paths, which are not semimartingales.
Limitations and Future Research: The study focuses on a specific class of slow-fast rough SPDEs with certain regularity assumptions on the coefficients. Further research could explore the averaging principle for more general classes of rough SPDEs with weaker assumptions. Additionally, investigating the convergence rates of the averaging principle would be a valuable extension.
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by Miaomiao Li,... at arxiv.org 11-12-2024
https://arxiv.org/pdf/2411.06089.pdfDeeper Inquiries