Stochastic dynamics for singularly perturbed PDEs with fractional Brownian motions

具有分数布朗运动的奇扰动偏微分方程的随机动力学

• 批准号: 18F18314
• 负责人: 稲浜 譲
• 金额: $ 1.41万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2018
• 资助国家: 日本
• 起止时间: 2018-11-09 至 2021-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18F18314/
• 关键词: Rough path theory; Averaging principle; Fast-slow system; Mixed stochastic PDE; Fast slow system; 非整数ブラウン運動; neutral terms; two-time-scale; Markov switching

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.

1、我们致力于研究混合分数布朗粗糙路径驱动的快慢微分方程组的平均原理，其中快分量由布朗运动驱动，而慢分量由赫斯特指数H的分数布朗运动驱动（。结合粗糙路径理论的分数阶微积分方法和 Khasminskii 的经典时间离散方法，我们证明慢分量强烈收敛于 1/3 < H \leq 1/2)。相应平均方程在L^1意义上的解在粗糙路径理论框架下的快慢系统的平均原理似乎是新的。2、我们工作的主要目标是研究一类的平均原理。我们想强调的是，两个时间尺度的函数随机微分方程，其中慢变过程包括赫斯特参数 1/2<H<1 的乘法分数布朗噪声，而快变过程是快速变化的扩散。所提出的方法本文基于这样的事实，即可以用广义 Stieltjes 积分来定义关于 (1/2 , 1) 中的 Hurst 参数的分数布朗运动的随机积分，特别是证明平均原理的极限定理。 ，我们将引入停止时间来控制乘法分数布朗噪声的大小，然后，受 Khasminskii 方法的启发，在第 p 次收敛的意义上开发了平均原理。时间上一致的时刻。

项目成果

Pathwise unique solutions and stochastic averaging for mixed SPDEs driven by fractional Brownian motion

分数布朗运动驱动的混合 SPDE 的路径唯一解和随机平均

• 发表时间: 2020
• 作者: ロバート キャンベル;十重田裕一;宗像和重編;Pei Bin
• 通讯作者: Pei Bin