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，我们致力于研究由混合分数布朗式粗糙路径驱动的粗糙微分方程的快速降低系统的平均原理。快速组件是由布朗运动驱动的，而缓慢的组件则由Hurst索引H（1/3 <H \ LEQ 1/2）驱动。将分数计算方法与粗糙路径理论和Khasminskii的经典时间离散方法相结合，我们证明，慢速分量在L^1的感觉中强烈收敛到相应平均方程的解。在粗糙路径理论框架中，快速缓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两次两次尺度功能性随机微分方程的平均原理，在该方程中，缓慢变化的过程包括带有hurst参数1/2 <h <h <h <h <1和快速的差异的多样性棕色噪声和快速差异。我们要强调的是，本文提出的方法是基于以下事实：与（1/2，1）中使用Hurst参数的分数布朗运动相关的随机积分可以通过广义的Stieltjes积分来定义。特别是，为了证明平均原理的限制定理，我们将引入停止时间以控制乘法分数布朗尼噪声的大小。然后，受到Khasminskii方法的启发，在P-thism统一的融合意义上开发了平均原理。

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