Stochastic dynamics for singularly perturbed PDEs with fractional Brownian motions
Project/Area Number 
18F18314

Research Category 
GrantinAid for JSPS Fellows

Allocation Type  Singleyear Grants 
Section  外国 
Review Section 
Basic Section 12010:Basic analysisrelated

Research Institution  Kyushu University 
Host Researcher 
稲浜 譲 九州大学, 数理学研究院, 教授 (80431998)

Foreign Research Fellow 
PEI BIN 九州大学, 数理(科)学研究科(研究院), 外国人特別研究員

Project Period (FY) 
20181109 – 20210331

Project Status 
Granted (Fiscal Year 2020)

Budget Amount *help 
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2020: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 2019: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2018: ¥600,000 (Direct Cost: ¥600,000)

Keywords  非整数ブラウン運動 / neutral terms / twotimescale / Markov switching 
Outline of Annual Research Achievements 
1, We focus on fastslow 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 fastvarying diffusion process will be averaged out with respect to its stationary measure in the limit process. It is shown that the slowvarying 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 fastslow system. 2, We prove the validity of averaging principles for twotimescale neutral stochastic delay PDEs driven by fBms under twotimescale formulation. Firstly, in the sense of meansquare convergence, we obtain not only the averaging principles for stochastic delay PDEs with twotimescale Markov switching with a single weakly recurrent class but also for the case of twotimescale Markov switching with multiple weakly irreducible classes. Secondly, averaging principles for neutral stochastic PDEs driven by fBms with random time delays modulated by twotimescale 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 fastslow system.

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 fastslow 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 fastvarying 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 twotimescale neutral stochastic delay PDEs driven by fBms under twotimescale formulation. Everything was done as planned. So, I believe that he will do well for the following task.

Strategy for Future Research Activity 
Taking this into consideration, the current project undertakes the task of analyzing twotimescale systems involving fBms. For this fiscal year, we firstly focus on averaging principles for neutral SPDEs with delays driven by fBms under twotimescale formulation inspired by the Khasminskii’s approach. Then, we will consider the averaging principle for stochastic burgers equation driven by spacetime fractional noises. The key is that in the limit, the coefficients are averaged out with respect to the stationary measures of the fastvarying 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 nonmartingale 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.

Report
(1 results)
Research Products
(1 results)