The differential structure of spaces of rough paths

粗糙路径空间的微分结构

• 批准号: 2670173
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2670173
• 关键词: differential; structure; spaces; rough; paths

1 Brief description of the context of the research including potential impactRough paths are mathematical objects that have garnered significant attention in recent years thanks to the variety of applications they enjoy both in pure mathematics and applied fields. The space of rough paths is non-linear, which makes it difficult to understand its differential structure. While many theories about how to understand differentiability on non-linear spaces have been developed, a full account of what the differential structure of rough paths spaces has not been developed yet. Smooth functions pervade mathematics, and the definitions of smoothness vary significantly. In real analysis, Ck spaces are spaces of smooth functions, where smoothness is defined as the continuity of the k-th derivative of the function. One important area where smooth functions are of crucial relevance is the study of differential equations. There, we can say that differential equations have solutions which belong to Ck for a certain k. Sobolev spaces are spaces of functions that, in a sense, generalise Ck spaces. While Sobolev spaces are spaces of smooth functions as much as Ck spaces, the requirement for a function to belong to a Sobolev space is that it has a certain degree of smoothness, but smoothness is understood in the weak sense. Sobolev spaces help us in the study of solutions to differential equations by allowing us to find weak solutions to certain partial differential equations in cases where there is no no strong solution, that is a solution that belongs to a Ck space.Understanding the differential structure of rough paths will thus enable us to characterise Sobolev spaces on the space of rough paths. In turn, this will help us better understand the spaces in which to find solutions to certain classes of differential equations where rough paths play a crucial role: stochastic differential equations (SDEs).2 Aims and objectivesThe goal of this research project is to build an account of Sobolev spaces on the spaces of rough paths. Such an account has the potential to allow us to better characterise solutions to SDEs, which in turn have a broad range of pure mathematics and practical applications. 3 Novelty of the research methodologyThe theories being developed are original since only a general account of the differential structure of spaces of rough paths has not been developed yetThis project falls within the EPSRC Mathematics research area, where Mathematical Analysis and Non-Linear Systems are some of the themes or research areas

1 研究背景的简要描述，包括潜在的影响 粗糙路径是近年来引起人们极大关注的数学对象，这要归功于它们在纯数学和应用领域中的各种应用。粗糙路径的空间是非线性的，这使得理解其微分结构变得困难。虽然已经发展了许多关于如何理解非线性空间上的可微性的理论，但尚未发展出对粗糙路径空间的微分结构的全面解释。平滑函数遍及数学，并且平滑的定义差异很大。在实分析中，Ck 空间是平滑函数的空间，其中平滑度定义为函数的 k 阶导数的连续性。光滑函数至关重要的一个重要领域是微分方程的研究。在那里，我们可以说微分方程对于某个 k 具有属于 Ck 的解。 Sobolev 空间是函数空间，在某种意义上概括了 Ck 空间。虽然Sobolev空间与Ck空间一样都是光滑函数的空间，但函数属于Sobolev空间的要求是它具有一定程度的光滑性，但光滑性是在弱意义上理解的。索博列夫空间帮助我们研究微分方程的解，它允许我们在没有强解的情况下找到某些偏微分方程的弱解，即属于 Ck 空间的解。理解粗糙路径的微分结构将使我们能够在粗糙路径空间上表征索博列夫空间。反过来，这将帮助我们更好地理解在其中寻找某些类别的微分方程的解的空间，其中粗糙路径起着至关重要的作用：随机微分方程（SDE）。2 目的和目标本研究项目的目标是在粗糙路径的空间上建立索博列夫空间的解释。这样的解释有可能让我们更好地描述 SDE 的解决方案，而这反过来又具有广泛的纯数学和实际应用。 3 研究方法的新颖性 正在开发的理论是原创的，因为尚未开发出粗糙路径空间微分结构的一般说明 该项目属于 EPSRC 数学研究领域，其中数学分析和非线性系统是其中的一些主题或研究领域