2018 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 | 非整数ブラウン運動 / neutral terms / two-time-scale / Markov switching |
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| Outline of Annual Research Achievements |
1, We focus on fast-slow stochastic partial differential equations in which the slow variable is driven by a fractional Brownian motion and the fast variable is driven by an additive Brownian motion. We establish an averaging principle in which the fast-varying diffusion process will be averaged out with respect to its stationary measure in the limit process. It is shown that the slow-varying process L^p (p>=2) converges to the solution of the corresponding averaging equation. To reduce the complexity, one can concentrate on the limit process instead of studying the original full fast-slow system. 2, We prove the validity of averaging principles for two-time-scale neutral stochastic delay PDEs driven by fBms under two-time-scale formulation. Firstly, in the sense of mean-square convergence, we obtain not only the averaging principles for stochastic delay PDEs with two-time-scale Markov switching with a single weakly recurrent class but also for the case of two-time-scale Markov switching with multiple weakly irreducible classes. Secondly, averaging principles for neutral stochastic PDEs driven by fBms with random time delays modulated by two-time-scale Markov switching are established. We proved that there is a limit process in which the fast changing noise is averaged out. The limit process is substantially simpler than that of the original full fast-slow system.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
In the past 5 months, for the fast-slow SPDEs in which the slow variable is driven by a fBm and the fast variable is driven by an additive Bm, Dr. Bin Pei has established an averaging principle in which the fast-varying diffusion process will be averaged out with respect to its stationary measure in the limit process. And he also proved the validity of averaging principles for two-time-scale neutral stochastic delay PDEs driven by fBms under two-time-scale formulation. Everything was done as planned. So, I believe that he will do well for the following task.
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| Strategy for Future Research Activity |
Taking this into consideration, the current project undertakes the task of analyzing two-time-scale systems involving fBms. For this fiscal year, we firstly focus on averaging principles for neutral SPDEs with delays driven by fBms under two-time-scale formulation inspired by the Khasminskii’s approach. Then, we will consider the averaging principle for stochastic burgers equation driven by space-time fractional noises. The key is that in the limit, the coefficients are averaged out with respect to the stationary measures of the fast-varying process. We show that the solutions of the averaged SPDEs converge to that of the original SPDEs in the sense of pth moments and also in probability. To proceed, we consider the slow varying diffusion process of multiplicative fBm case. We use fixed point theorem, Young integral, rough path theory and a semigroup approach to overcome the difficulties caused by non-martingale of fBm and no strong solutions for the underlying SPDEs. To proceed, assuming that the switching process is subject to slow and fast variation, either within a weakly irreducible class or within a number of nearly decomposable weakly irreducible classes and consider the averaging principle. Finally, we will returns
to the example to illustrate the utility of our results.
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Research Products
(1 results)
