G -Brownian Motion as Rough Paths and Differential Equations Driven by G -Brownian Motion
G -Brownian Motion as Rough Paths and Differential Equations Driven by G -Brownian Motion
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DOI：
10.1007/978-3-319-11970-0_6
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发表时间：
2013-06
期刊：
影响因子：
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通讯作者：
X. Geng;Z. Qian;Danyu Yang
中科院分区：
文献类型：
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作者：
X. Geng;Z. Qian;Danyu Yang
本文从粗糙路径理论的角度研究了G-布朗运动的样本路径和由G-布朗运动驱动的随机微分方程(SDE)。作为起点，我们利用粗糙路理论中的技巧，证明了G-布朗运动的拟必然样本路径可以规范地提升到第二水平，从而成为粗糙度为2 <；p<； 3的几何粗糙路径。这一结果使我们能够在粗糙路径的一般框架下引入由路径意义上的由G-布朗运动驱动的粗糙微分方程(RDES)的概念。其次，我们建立了由YG-Brown运动驱动的SDE和RDES之间的基本关系。作为应用，我们在由可微流形驱动的BYG-Brown运动上引入了SDE的概念，并利用路径局部化技术从RDE的角度构造了解。这是在Eells-Elworth-Malliavin思想的基础上发展黎曼流形上的G-Brown运动的起点。本文的最后一部分致力于构造一类广泛而有趣的G-函数，其不变群为正交群。特别地，我们建立了黎曼流形上这类G-Brown运动的生成非线性热方程。我们还发展了由独立兴趣的YG-Brown运动驱动的SDE的Euler-Maruyama逼近。
The present article is devoted to the study of sample paths ofG-Brownian motion and stochastic differential equations (SDEs) driven byG-Brownian motion from the viewpoint of rough path theory. As the starting point, by using techniques in rough path theory, we show that quasi-surely, sample paths ofG-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 <p< 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven byG-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven byG-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven byG-Brownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of developingG-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this article is devoted to such construction for a wide and interesting class ofG-functions whose invariant group is the orthogonal group. In particular, we establish the generating nonlinear heat equation for suchG-Brownian motion on a Riemannian manifold. We also develop the Euler-Maruyama approximation for SDEs driven byG-Brownian motion of independent interest.