2020 Fiscal Year Annual Research Report
Stochastic dynamics for singularly perturbed PDEs with fractional Brownian motions
| Project/Area Number |
18F18314
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| Research Institution | Kyushu University |
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Principal Investigator |
稲浜 譲 九州大学, 数理学研究院, 教授 (80431998)
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| Co-Investigator(Kenkyū-buntansha) |
PEI BIN 九州大学, 数理(科)学研究科(研究院), 外国人特別研究員
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| Project Period (FY) |
2018-11-09 – 2021-03-31
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| Keywords | Rough path theory / Averaging principle / Fast-slow system |
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| Outline of Annual Research Achievements |
1, We devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index H (1/3 < H \leq 1/2). Combining the fractional calculus approach to rough path theory and Khasminskii’s classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the L^1 sense. The averaging principle for a fast-slow system in the framework of rough path theory seems new.
2, The main goal of our work is to study an averaging principle for a class of two-time-scale functional stochastic differential equations in which the slow-varying process includes a multiplicative fractional Brownian noise with Hurst parameter 1/2<H<1 and the fast-varying process is a rapidly-changing diffusion. We would like to emphasize that the approach proposed in this paper is based on the fact that a stochastic integral with respect to fractional Brownian motion with Hurst parameter in (1/2 , 1) can be defined by a generalized Stieltjes integral. In particular, to prove a limit theorem for the averaging principle, we will introduce stopping times to control the size of the multiplicative fractional Brownian noise. Then, inspired by the Khasminskii’s approach, an averaging principle is developed in the sense of convergence in the p-th moment uniformly in time.
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| Research Progress Status |
令和2年度が最終年度であるため、記入しない。
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| Strategy for Future Research Activity |
令和2年度が最終年度であるため、記入しない。
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