2019 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 | Averaging principle / Mixed stochastic PDE / Fast slow system |
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| Outline of Annual Research Achievements |
1, We focus on fast-slow systems involving both fractional Brownian motion (fBm) and Brownian motion (Bm). The integral with respect to Bm is the standard Ito integral, and the integral with respect to fBm is the generalised Riemann-Stieltjes integral using the tools of fractional calculus. An averaging principle in which the fast-varying diffusion process of the fast-slow systems acts as a “noise” to be averaged out in the limit is established. It is shown that the slow process has a limit in the mean square sense, which is characterized by the solution of stochastic differential equations driven by fBm whose coefficients are averaged with respect to the stationary measure of the fast-varying diffusion.
2, This project is devoted to a system of SPDEs that have a slow component driven by fBm with the Hurst parameter H > 1/2 and a fast component driven by fast-varying diffusion. It improves previous work in two aspects: Firstly, using a stopping time technique and an approximation of the fBm, we prove an existence and uniqueness theorem for a class of mixed SPDEs driven by both fBm and Brownian motion; Secondly, an averaging principle in the mean square sense for SPDEs driven by fBm subject to an additional fast-varying diffusion process is established. To carry out these improvements, we combine the pathwise approach based on the generalized Stieltjes integration theory with the Ito stochastic calculus. Then, we obtain a desired limit process of the slow component which strongly relies on an invariant measure of the fast-varying diffusion process.
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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 this school year, this project has completed all the contents according to the plan. We have some new research results. This is satisfactory, but further efforts will be needed to achieve more results. This project has devoted to obtaining stochastic averaging principles for two-time-scale stochastic processes that are solutions of a system of SPDEs driven by fBms. Our main objective is to analyze such systems leading to limit systems or reduced systems that are substantially simpler than that of the original systems. To solve the problems in this part, new analytical methods and efficient numerical algorithms have be developed. For examples, 1) This project firstly considered the fast-slow systems involving both fBm and Bm. An averaging principle in which the fast-varying diffusion process of the fast-slow systems acts as a “noise” to be averaged out in the limit was established. 2), This project is devoted to a system of SPDEs that have a slow component driven by fBm with the Hurst parameter H > 1/2 and a fast component driven by fast-varying diffusion. An averaging principle in the mean square sense for SPDEs driven by fBm subject to an additional fast-varying diffusion process is established.
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| Strategy for Future Research Activity |
Recent years have witnessed great interest in models based on fBm. Based on our previous work, the structured analysis framework for our project is now well established but suffers difficulties. My Proposed plan is following:
1, We will study 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 (1/4 < H \leq 1/2). Since rough path theory is now a very hot topic, this direction of research seems the most interesting to me.
2, We will 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.
3. Due to recent developments of Malliavin calculus for rough differential equations, it is now known that, under natural assumptions, the law of a unique solution at a fixed time has a smooth density function. Therefore, it is quite natural to ask whether or when the density is strictly positive.
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